Abstract
We prove that the tagged particles of infinitely many Brownian particles in \( {\mathbb {R}} ^2\) interacting via a logarithmic (two-dimensional Coulomb) potential with inverse temperature \( \beta = 2 \) are sub-diffusive. The associated unlabeled diffusion is reversible with respect to the Ginibre random point field, and the dynamics are thus referred to as the Ginibre interacting Brownian motion. If the interacting Brownian particles have interaction potential \( \Psi \) of Ruelle class and the total system starts in a translation invariant equilibrium state, then the tagged particles are always diffusive if the dimension \( d\) of the space \( {\mathbb {R}}^{d} \) is greater than or equal to two. That is, the tagged particles are always non-degenerate under diffusive scaling. Our result is, therefore, contrary to known results. The Ginibre random point field has various levels of geometric rigidity. Our results reveal that the geometric property of infinite particle systems affects the dynamical property of the associated stochastic dynamics.
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1 Introduction
We consider a system \( {{\textbf {X}}} =( X ^i )_{i\in {\mathbb {N}} }\) of infinitely many Brownian particles moving in \( {\mathbb {R}} ^d\) and interacting through a translation invariant, two-body potential \( \Psi (x) \). \( {{\textbf {X}}} \) is described by the infinite-dimensional stochastic differential equation (ISDE)
where \( B^i \) (\( i\in {\mathbb {N}} \)) denotes independent \( d\)-dimensional Brownian motions and \( \beta \ge 0 \) is the inverse temperature, which is taken as a constant. The solution \( {{\textbf {X}}} \) provides a description of the interacting Brownian motion [5, 15, 16, 36].
The unlabeled process \( {\textsf {X}}=\{ {\textsf {X}}_t \}_{t\in [0,\infty )} \) associated with \( {{\textbf {X}}} \) is given by
where \( \delta _a \) is the delta measure at \( a \in {\mathbb {R}} ^d\) and \( {\textsf {X}}\) is a configuration-valued process by definition.
We suppose that the unlabeled process \( {\textsf {X}}\) is reversible with respect to a translation invariant equilibrium state \( \mu ^{\Psi , \beta }\). In many cases, we expect the existence of such an equilibrium state. For example, if \( \Psi \) is a Ruelle-class potential (i.e., it is super-stable and regular in the sense of Ruelle), then the associated translation invariant canonical Gibbs measures exist. Here, super-stability is a condition that prevents infinitely many particles agglomerating in a bounded domain, and regularity means the integrability of interactions at infinity and therefore provides the Dobrushin–Lanford–Ruelle equation [30].
We investigate the tagged particles \( X ^i = \{ X _t^i \}_{t\in [0,\infty )} \) in the system. Although the total unlabeled system \( {\textsf {X}}\) is a \( \mu ^{\Psi , \beta }\)-reversible Markov process, each tagged particle \( X ^i\) is a non-Markov process because the total system affects it in a complicated fashion. Applying the Kipnis–Varadhan theory, it can nevertheless be seen that the motion of each tagged particle always reverts to Brownian motion under diffusive scaling [4, 8, 19, 26, 35]. That is,
The constant may depend on the initial configuration, and we therefore introduce \( \alpha \), defined as the expectation of \( (1/2) \sigma ^2 \) with respect to the reduced Palm measure of the reversible measure \( \mu ^{\Psi , \beta }\). The constant matrix \( \alpha \) is called the self-diffusion matrix.
Once such convergence of motion is established under this fairly general situation, it is natural and important to inquire about the positivity of the self-diffusion matrix \( \alpha \).
Historically, there was a conjecture that \( \alpha = 0 \) for hard-core potentials and sufficiently large activities in multi-dimensional grand canonical Gibbs measures; cf. [1]. This conjecture seemed to be plausible because the presence of a hard core should suppress the motion of tagged particles. However the contrary was proved in [20]. Indeed, \( \alpha \) is always positive definite if \( d\ge 2 \) and \( \Psi \) is a Ruelle-class potential corresponding to a hard core.
In the setting of discrete spaces, the counterpart is a tagged particle problem of exclusion processes in \( {\mathbb {Z}}^{d} \). As for the simple exclusion processes, Kipnis–Varadhan [14] proved that \( \alpha \) is always positive definite except for the nearest neighborhood jump in one space dimension. Spohn [33] proved that \( \alpha \) is always positive definite for general exclusion processes with Ruelle-class potentials when \( d\ge 2 \). Spohn–Yau [25] proved that bulk diffusion is always positive definite for general exclusion processes with Ruelle-class potentials when \( d\ge 2 \).
We note that the set of the Gibbs measures with Ruelle-class potentials is the standard class of random point fields in both continuous and discrete spaces. We hence consider, on good grounds, that it is reasonable to believe that self-diffusion matrices are always positive definite for \( d\ge 2 \). Nevertheless, we present the antithesis in the present paper.
The Ginibre interacting Brownian motion \( {{\textbf {X}}} = ( X ^i)_{i\in {\mathbb {N}}}\) is a system of infinite-many Brownian particles moving in \( {\mathbb {R}} ^2\) and interacting via the two-dimensional Coulomb potential \( \Psi (x) = - \log |x|\) with inverse temperature \( \beta = 2 \). The stochastic dynamics \( {{\textbf {X}}} = \{ {{\textbf {X}}} _t \} \) are then described by the ISDE
The associated unlabeled process \( {\textsf {X}}\) is reversible with respect to the Ginibre random point field \( \mu _{\textrm{Gin}}\). By definition, \( \mu _{\textrm{Gin}}\) is a random point field on \( {\mathbb {R}} ^2\) for which the n-point correlation function \( \rho _{\textrm{Gin}}^n \) with respect to the Lebesgue measure is given by
where \( {\textsf {K}}_{\textrm{Gin}}\!:\!{\mathbb {R}} ^2 \times {\mathbb {R}} ^2 \!\rightarrow \!{\mathbb {C}}\) is the exponential kernel defined by
Here, we identify \( {\mathbb {R}} ^2 \) as \( {\mathbb {C}}\) by the correspondence \( {\mathbb {R}} ^2 \ni x=(x_1,x_2)\mapsto x_1 + \sqrt{-1} x_2 \in {\mathbb {C}}\), and \( {\bar{y}}=y_1-\sqrt{-1} y_2 \) gives the complex conjugate of y under this identification.
It is known that \( \mu _{\textrm{Gin}}\) is translation and rotation invariant. Furthermore, \( \mu _{\textrm{Gin}}\) is tail trivial [17, 25]. Let \( {\textsf {S}}\) be the configuration space over \( {\mathbb {R}} ^2\) defined by (1.2). Let \( {\textsf {S}}_*\) be a Borel subset of \( {\textsf {S}}\backslash \{ {\textsf {0}} \} \), where \( {\textsf {0}}\) is the zero measure. A measurable map \( {\mathfrak {l}}\!:\!{\textsf {S}}_*\!\rightarrow \!({\mathbb {R}} ^2)^{{\mathbb {N}}}\cup \{ \sum _{m=1} ^{\infty } ({\mathbb {R}} ^2)^m \} \) called a label if \( {\mathfrak {l}}({\textsf {s}})=({\mathfrak {l}}^i({\textsf {s}}))_{i}\) satisfies \( \sum _i \delta _{{\mathfrak {l}}^i({\textsf {s}})}= {\textsf {s}}\). Typically, we take a label \( {\mathfrak {l}}\) such that \( |{\mathfrak {l}}^i({\textsf {s}})|\le |{\mathfrak {l}}^{i+1}({\textsf {s}})|\) for all i.
For \( \mu _{\textrm{Gin}}\circ {\mathfrak {l}}^{-1}\)-a.s. \( {{\textbf {s}}} = (s_i)_{i\in {\mathbb {N}} }\), (1.1) has a solution \( {{\textbf {X}}} = ( X ^i)_{i\in {\mathbb {N}} }\), of which the unlabeled process \( {\textsf {X}}\) is \( \mu _{\textrm{Gin}}\)-reversible [22]. The ISDE has a unique strong solution starting at \( \mu _{\textrm{Gin}}\circ {\mathfrak {l}}^{-1}\)-a.s. \( {{\textbf {s}}} \) under a reasonable constraint. We refer to Section 7 in [28] and references therein for the existence and uniqueness of solutions of (1.1).
Theorem 1.1
Let \( P_{{{\textbf {s}}}}\) be the distribution of the solution \( {{\textbf {X}}} = ( X ^i)_{i\in {\mathbb {N}} }\) of (1.1) staring at \( {{\textbf {s}}} \in ({\mathbb {R}} ^2)^{{\mathbb {N}} }\). Then for each \( i \in {\mathbb {N}}\), under \( P_{{{\textbf {s}}}}\) in \( \mu _{\textrm{Gin}}\circ {\mathfrak {l}}^{-1}\)-probability,
Remark 1.1
The claim in Theorem 1.1 means that for any bounded continuous function F on \( C([0,\infty ); {\mathbb {R}} ^2) \) and \( \kappa > 0 \), it holds that for each \( i \in {\mathbb {N}} \),
Here \( 0 = \{ 0_t \} \) of F(0) denotes the constant path with value 0.
Recently, it has become clear that the Ginibre random point field has various geometric rigidities, specifically a small variance property according to Shirai [31], the number rigidity according to Ghosh and Peres [7], and the dichotomy of reduced Palm measures [27]. These properties are different from those of the Poisson random point field and the Gibbs measure with a Ruelle-class potential, which have been extensively studied as the standard class of random point fields appearing in statistical physics.
These geometric properties affect the dynamical properties. Indeed, from these rigidities, our theorem demonstrates that geometric rigidities yield dynamical rigidity in the sense of the sub-diffusivity of each tagged particle of the natural infinite-particle system given by (1.1).
The self-diffusion matrix \( \alpha [\mu _{\textrm{Gin}}] \) is given by the solution to the Poisson equation of the quotient Dirichlet form on the configuration space (cf. [19, 26]). To prove \( \alpha [\mu _{\textrm{Gin}}] = O \), we use the geometric rigidities of the Ginibre random point field we shall introduce in the following.
The dichotomy of the reduced Palm measures of \( \mu _{\textrm{Gin}}\) was proved in [27], and is the critical geometric rigidity that we use to prove Theorem 1.1. Let \( \mu _{\textrm{Gin}}(\cdot \Vert {\textsf {x}}) \) be the reduced Palm measure of \( \mu _{\textrm{Gin}}\) conditioned at \( {\textsf {x}}\) (see (1.17)).
Lemma 1.1
([27, Theorem 1.1]) Assume that \( {\textsf {x}}({\mathbb {R}} ^2)= m \) and \( {\textsf {y}}({\mathbb {R}} ^2) = n \) for \( m, n \in \{ 0 \} \cup {\mathbb {N}} \), where we take \(\mu _{\textrm{Gin}}(\cdot \Vert {\textsf {x}})= \mu _{\textrm{Gin}}\) if \( m = 0 \). The following then holds.
-
(1)
If \( m \not = n \), then \(\mu _{\textrm{Gin}}(\cdot \Vert {\textsf {x}})\) and \(\mu _{\textrm{Gin}}(\cdot \Vert {\textsf {y}})\) are singular relative to each other.
-
(2)
If \( m = n \), then \(\mu _{\textrm{Gin}}(\cdot \Vert {\textsf {x}})\) and \(\mu _{\textrm{Gin}}(\cdot \Vert {\textsf {y}})\) are mutually absolutely continuous.
We find that \( \mu _{\textrm{Gin}}(\cdot \Vert {\textsf {x}})\) is continuous in \( {{\textbf {x}}}=(x_1,\ldots ,x_m) \in ({\mathbb {R}} ^2)^m\), where \( {\textsf {x}}= \sum _{i=1}^{ m } \delta _{x_i}\). That is, \( ({\mathbb {R}}^2)^{ m } \ni {{\textbf {x}}} \mapsto \int _{{\textsf {S}}} f d\mu _{\textrm{Gin}}(\cdot \Vert {\textsf {x}})\) is continuous for any \( f \in C_b({\textsf {S}})\). This follows from the explicit formula of the Radon–Nikodym density \( {d \mu _{\textrm{Gin}}(\cdot \Vert {\textsf {x}})}/{d \mu _{\textrm{Gin}}(\cdot \Vert {\textsf {y}})} \) in Lemma 1.2.
Lemma 1.2
([27, Theorems 1.2]) Let \( {\textsf {x}}({\mathbb {R}} ^2) ={\textsf {y}}({\mathbb {R}} ^2) = m \in {\mathbb {N}} \). Then
where \( |{{\textbf {x}}} - s_j|= \prod _{i=1}^m |x_i- s_j|\). The normalization constant \( {\mathcal {Z}}_{{\textsf {x}}, {\textsf {y}}}\) is given by
Here, \( {\mathcal {Z}}_{{\textsf {x}}, {\textsf {y}}} \) is the unique smooth function on \( ({\mathbb {R}} ^2)^m \times ({\mathbb {R}} ^2)^m \) defined by continuity when the denominator has vanished. Furthermore, \( \Delta \) denotes the difference product for \( m \ge 2\) and \( \Delta ({{\textbf {x}}}) = 1 \) for \( m = 1 \).
Intuitively, the dichotomy in Lemma 1.1 indicates the following phenomena. Suppose that we remove a finite unknown number m of particles \( \{ s_{i_1},\ldots ,s_{i_m} \}\) from a sample point \( {\textsf {s}}= \sum _i \delta _{s_i}\) of \( \mu _{\textrm{Gin}}\). We then deduce the number m from information of \( {\textsf {s}}_{\diamond }:= \sum _{i \in {\mathbb {N}} \backslash \{i_1,\ldots ,i_m\}} \delta _{s_i}\). Such a structure is the same as periodic random point fields. Although a sample point \( {\textsf {s}}\) of \( \mu _{\textrm{Gin}}\) has enough randomness, we can infer the number of the removed particles exactly for \(\mu _{\textrm{Gin}}\)-a.s. \({\textsf {s}}\).
For \( d= 1\), we proved that the non-collision of particles always implies sub-diffusivity [24]. (See also [9, 32].) Using the variational formula of the self-diffusion constant, the proof in [24] relies on the construction of a sequence of functions that reduces this constant to zero. This crucially uses the total order structure of non-collision particle systems in \( {\mathbb {R}} \), which is specific in one-dimension.
A key point of the proof of Theorem 1.1 is to construct such a sequence of functions without using the total order structure. Indeed, we shall use the above-mentioned geometric rigidity of the Ginibre random point field to accomplish this procedure.
We shall present general theorems concerning the sub-diffusivity of the interacting Brownian motions for \( d\ge 2 \) and prove Theorem 1.1 as a specific example of the general theorems (Theorems 1.2 and 1.3).
Let \( S_{R}=\{ x \in {\mathbb {R}} ^d; |x|< R\} \). Let \({\textsf {S}}\) be the configuration space over \({\mathbb {R}} ^d\).
We endow \({\textsf {S}}\) with the vague topology, under which \({\textsf {S}}\) is a Polish space. Let \( {\mathcal {B}}( {\textsf {S}}) \) be the Borel \( \sigma \)-field of \( {\textsf {S}}\). A probability measure \( \mu \) on \( ({\textsf {S}}, {\mathcal {B}}( {\textsf {S}}) )\) is called a random point field and also a point process.
Let \( \{ \vartheta _x \}_{x\in {\mathbb {R}} ^d}\) be the translation operator on \( {\textsf {S}}\) such that for \({\textsf {s}}=\sum _i\delta _{s_i}\),
Then, \( \vartheta _x \!:\!{\textsf {S}}\!\rightarrow \!{\textsf {S}}\) is a homeomorphism for each \( x \in {\mathbb {R}} ^d\), and \( {\textsf {s}}\mapsto \vartheta _x ({\textsf {s}})\) is a continuous function of \( x \in {\mathbb {R}} ^d\) for each \( {\textsf {s}}\in {\textsf {S}}\). Furthermore, \( (x, {\textsf {s}}) \mapsto \vartheta _x ({\textsf {s}})\) is continuous. A random point field \( \mu \) on \( {\mathbb {R}} ^d\) is called translation invariant if
We assume the following.
\( (\) A1 \()\) \( \mu \) is translation invariant and \( \mu (\{ {\textsf {s}}({\mathbb {R}} ^d) = \infty \} ) = 1 \).
The translation invariance implies \( \mu (\{ {\textsf {s}}({\mathbb {R}} ^d) = \infty \} ) = 1 \) if \( \mu \) is not a zero measure. Thus, the second assumption in \( (\) A1 \()\) yields no restriction in practice.
A symmetric and locally integrable function \( \rho ^n \!:\!( {\mathbb {R}} ^d) ^n\!\rightarrow \![0,\infty ) \) is called the n-point correlation function of \( \mu \) with respect to the Lebesgue measure if
for any sequence of disjoint bounded measurable sets \( A_1,\ldots ,A_m \in {\mathcal {B}}(S) \) and a sequence of natural numbers \( k_1,\ldots ,k_m \) satisfying \( k_1+\cdots + k_m = n \). When \( {\textsf {s}}(A_i) - k_i < 0\), according to our interpretation, \({{\textsf {s}}(A_i) ! }/{({\textsf {s}}(A_i) - k_i )!} = 0\) by convention. We make an assumption.
\( (\) A2 \()\) \( \mu \) has a locally bounded k-point correlation function for each \( k \in {\mathbb {N}} \).
We set the projections \( \pi _{R}, \pi _{R}^{c}\!:\!{\textsf {S}} \!\rightarrow \!{\textsf {S}} \) such that
For two measures \( \nu _1\) and \( \nu _2 \) on a measurable space \( (\Omega , {\mathcal {B}})\), we write \( \nu _1 \le \nu _2 \) if \( \nu _1(A)\le \nu _2(A)\) for all \( A\in {\mathcal {B}}\). Let \( \Psi \!:\!{\mathbb {R}} ^d\!\rightarrow \!{\mathbb {R}} \cup \{ \infty \} \) be a measurable function satisfying \( \Psi (x) = \Psi (- x )\). We take \( \Psi \) as an interaction potential of \( \mu \). Let \( {\textsf {S}}_{R}^m = \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}(S_{R})= m \} \). We set \( \Lambda _{R} ^m = \Lambda (\cdot \cap {\textsf {S}}_{R}^m )\), where \( \Lambda \) is the Poisson random point field whose intensity is the Lebesgue measure.
For \( {\textsf {x}} = \sum _i \delta _{x_i} \in {\textsf {S}}\), let \( {\mathcal {H}}_{R}\) be the Hamiltonian on \( S_{R}\) such that
Definition 1.1
[23] We say a random point field \( \mu \) on \( {\mathbb {R}} ^d\) is a \( \Psi \)-quasi-Gibbs measure with inverse temperature \( \beta \ge 0 \) if there exists a sequence of measures \( \{\mu _{R, k }^{m}\} \) on \( {\textsf {S}}\) such that, for each \( R, m \in {\mathbb {N}}\),
and, for all \( R, k, m \in {\mathbb {N}}\) and \( \mu \)-a.s. \( {\textsf {s}}\), the regular conditional measures
satisfy
Here, \( C_{1.1} \) is a positive constant depending on \( \beta \), \(R\), k, m and \( \pi _{R}^{c}({\textsf {s}}) \).
Remark 1.2
The definition of quasi-Gibbs measure in Definition 1.1 is a particular case of that of [23]. Because we consider the tagged particle problem in the present paper, we adopt a more restrictive definition of quasi-Gibbs measures as above.
\( (\) A3 \()\) \( \mu \) is a \( \Psi \)-quasi-Gibbs measure with inverse temperature \( \beta \ge 0 \) such that \(\Psi ( x ) < \infty \) for \( x \ne 0 \) and that there exist an upper semi-continuous function \( {\hat{\Psi }} \) locally bounded from below and a constant \(C_{1.2} > 0 \) satisfying \( c_{{1.2}}^{-1} {\hat{\Psi }}(x) \le \Psi (x) \le c_{{1.2}}{\hat{\Psi }}(x) \).
Let \( {\textsf {S}}_{\textrm{s,i}}\) be the set consisting of infinite, single configurations such that
We endow \( {\textsf {S}}_{\textrm{s,i}}\) with the vague topology. Let \( W ({\textsf {S}}_{\textrm{s,i}})\) be the set consisting of \( {\textsf {S}}_{\textrm{s,i}}\)-valued continuous paths on \( [0,\infty )\). We write \( {\textsf {w}}= \{ {\textsf {w}}_t \} \in W ({\textsf {S}}_{\textrm{s,i}})\) as
Here, \( I_i \) is an interval of the form \( [0,b_i)\) or \( (a_i,b_i)\), where \( 0 \le a_i < b_i \le \infty \). For \( {\textsf {w}}\in W ({\textsf {S}}_{\textrm{s,i}})\), the set \( \{ ( w ^i, I_i ) \}_{i \in {\mathbb {N}}} \) is uniquely determined (except labeling).
We write \( {\textsf {w}}= \{ ( w ^i, I_i ) \}_{i \in {\mathbb {N}}} \) if the representation of \( {\textsf {w}}\) is \( \{ ( w ^i, I_i ) \}_{i \in {\mathbb {N}}} \). Let
For \( {\textsf {w}}\in W_{\textrm{NE}} ({\textsf {S}}_{\textrm{s,i}})\), \( w^i \in C([0,\infty );{\mathbb {R}}^d)\) holds for all \( i \in {\mathbb {N}}\). Thus, \( W_{\textrm{NE}} ({\textsf {S}}_{\textrm{s,i}})\) is the set consisting of non-exploding and non-entering paths. For a label \( {\mathfrak {l}}\) on \( {\textsf {S}}_{\textrm{s,i}}\), we have a unique map \( {\mathfrak {l}}_{\textrm{path}}\!:\! W_{\textrm{NE}} ({\textsf {S}}_{\textrm{s,i}})\!\rightarrow \! C([0,\infty );({\mathbb {R}}^d)^{{\mathbb {N}}}) \) such that
In Sect. 2.1, we introduce the Dirichlet forms \( ({\mathscr {E}}^{\mu }, \underline{{\mathscr {D}}}^{\mu }) \) and \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\) on \( L^{2}(\mu )\). We call \( ({\mathscr {E}}^{\mu }, \underline{{\mathscr {D}}}^{\mu }) \) and \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\) the lower and upper Dirichlet forms, respectively. Such Dirichlet forms exist under \( (\) A2 \()\) and \( (\) A3 \()\). In Sect. 2.3, we introduce the perpendicular Dirichlet form \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) \), which is a new Dirichlet form provided in the present paper and plays vital role in the proof of the main theorems.
From \( (\) A2 \()\) and \( (\) A3 \()\), we have a \( \mu \)-reversible diffusion \( (P_{{\textsf {s}}}, {\textsf {X}}_t)\) associated with \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\) on \( L^{2}(\mu )\) [23]. We set \( P _{\mu } (\cdot )= \int P_{{\textsf {s}}} (\cdot ) \mu (d{\textsf {s}}) \). From \( (\) A1 \()\)–\( (\) A3 \()\) and \( d\ge 2 \), the \( \mu \)-reversible diffusion \( (P_{{\textsf {s}}}, {\textsf {X}}_t)\) has the non-explosion and non-collision properties (see Lemma 10.2 in [28]):
From (1.12) and (1.13), we have a continuous labeled process \( {{\textbf {X}}} = {\mathfrak {l}}_{\textrm{path}}({\textsf {X}})\). By construction, \( {{\textbf {X}}} _0 = {\mathfrak {l}}({\textsf {X}}_0)\), \( {{\textbf {X}}} = (X^i)_{i\in {\mathbb {N}} } \), and \( {\textsf {X}} = \{ {\textsf {X}}_t \}_{t\in [0,\infty )} \) is such that \( {\textsf {X}}_t = \sum _{i\in {\mathbb {N}} } \delta _{X_t^i}\).
To prove sub-diffusivity, we introduce the new Dirichlet form \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } )\) in Lemma 2.7. From Lemmas 2.7(3) and 2.2, we have
Here, for non-negative, symmetric bilinear forms \( ({\mathscr {E}}^i, {\mathscr {D}}^i ) \), \( i=1,2\), we write \( ({\mathscr {E}}^1, {\mathscr {D}}^1 ) \le ({\mathscr {E}}^2, {\mathscr {D}}^2 )\) if \( {\mathscr {D}}^1 \supset {\mathscr {D}}^2 \) and \( {\mathscr {E}}^1 (f,f) \le {\mathscr {E}}^2 (f,f) \) for all \( f \in {\mathscr {D}}^2 \).
Taking (1.14) into account, we make an assumption.
\( (\) A4 \()\) \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) = ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\).
For \( {\textsf {x}}, {\textsf {s}}\in {\textsf {S}}\), we write \( {\textsf {x}}\prec {\textsf {s}}\) if \( {\textsf {x}}(\{ x \} ) \le {\textsf {s}}(\{ x \} )\) for all \( x \in {\mathbb {R}} ^d\). For \( {\textsf {s}}, {\textsf {x}}\in {\textsf {S}}\) such that \( {\textsf {x}}\prec {\textsf {s}}\), the difference \( {\textsf {s}}- {\textsf {x}}\) belongs to \( {\textsf {S}}\). We set
By definition \( {\textsf {A}} - {\textsf {y}}= \emptyset \) if no \( {\textsf {s}}\in {\textsf {A}}\) satisfies \( {\textsf {y}}\prec {\textsf {s}}\).
Let \( {\textsf {S}}_{ m }= \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}({\mathbb {R}} ^d) = m \} \) for \( m \in \{ 0 \} \cup {\mathbb {N}} \). By definition, \( {\textsf {S}}_{ 0 }\) consists of the zero measure. Let \( {\mathfrak {u}} ({{\textbf {x}}}) = \sum _i\delta _{x_i}\) for \( {{\textbf {x}}} = (x_i)_i\). Then \( {\textsf {S}}_{ m }= {\mathfrak {u}} ( ({\mathbb {R}}^d)^m) \). Let \( {\check{\mu }}^m \) be the mth factorial moment measure of \( \mu \) such that
for \(\{ A_i \} \) such that \( A_i\cap A_j = \emptyset \) for \(i\ne j\) [11]. Let \( {\widetilde{\mu }}_m= {\check{\mu }}^m \circ {\mathfrak {u}} ^{-1 }\) be the measure supported on \( {\textsf {S}}_{ m }\). For \( {\widetilde{\mu }}_m\)-a.e. \( {\textsf {x}}\in {\textsf {S}}_{ m }\), the reduced Palm measure \( \mu (\cdot \Vert {\textsf {x}})\) exists and satisfies, by definition, for any \({\textsf {A}} \in {\mathcal {B}}({\textsf {S}}_{ m })\), and \( {\textsf {B}} \in {\mathcal {B}}({\textsf {S}}) \),
For each \( {\textsf {B}} \in {\mathcal {B}}({\textsf {S}}) \), \( \mu ({\textsf {B}} \Vert {\textsf {x}})\) becomes a \( {\mathcal {B}}({\textsf {S}}_{ m })\)-measurable function in \( {\textsf {x}}\).
We write \( \nu _1 \ll \nu _2\) if \( \nu _1 \) is absolutely continuous with respect to \( \nu _2\), where \( \nu _1\) and \( \nu _2 \) are measures. We write \( \nu _1 \approx \nu _2 \) if \( \nu _1 \ll \nu _2 \) and \( \nu _2 \ll \nu _1 \).
Taking the dichotomy of the Ginibre random point field in Lemma 1.1 into account, we introduce the following concept.
Definition 1.2
We call \( \mu \) k-decomposable with \( \{ {\textsf {S}}_{ m } ^{\diamond }\}_{m=0}^{ k }\) if for \( 0 \le m \le k \)
We call \( \mu \) irreducibly k-decomposable with \( \{ {\textsf {S}}_{ m } ^{\diamond }\}_{m=0}^{ k }\) if, in addition, for \( 1 \le m \le k \)
Remark 1.3
(1) Let \( {\textsf {0}}\) be the zero measure. Then \( {\textsf {S}}_{ 0 }= \{ {\textsf {0}} \} \) and \( \mu (\cdot \Vert {\textsf {0}}) = \mu \). Hence, we have \( {\widetilde{\mu }}_0 = \mu (\cdot \Vert {\textsf {0}}) = \mu \). Thus, (1.20) implies \( \mu ( {\textsf {S}}_{ 0 } ^{\diamond }) = 1 \).
(2) From (1.18) and (1.20), \( \mu \) and \( \mu (\cdot \Vert {\textsf {x}})\) are singular relative to each other for \( {\textsf {x}}\in {\textsf {S}}_{ m }\) and \( \mu ({\textsf {S}}_{ m } ^{\diamond }) = 0 \) for \( m \ge 1 \).
Example 1.1
(Ginibre random point field) From Lemmas 1.1 and 1.2, we easily see that the Ginibre random point field is irreducibly k-decomposable for all \( k \in {\mathbb {N}}\).
In Lemma 5.5, we construct a reduced Palm measure \( \mu (\cdot \Vert {\textsf {y}})\) for \( {\textsf {y}}\) such that \( {\textsf {y}}({\mathbb {R}} ^d) = \infty \) if \( \mu \) is irreducibly decomposable in the sense of Definition 1.2. Such Palm measures are called dual reduced Palm measures. The concept of dual reduced Palm measures plays an important role of the proof of main theorems.
\( (\) A5 \()\)
\( \mu \) is irreducibly one-decomposable with \( \{ {\textsf {S}}_{ 0 } ^{\diamond }, {\textsf {S}}_{ 1 } ^{\diamond }\}\).
Theorem 1.2
Assume \( d\ge 2 \). Assume \( (\) A1 \()\)–\( (\) A5 \()\). Let \( {{\textbf {X}}} = {\mathfrak {l}}_{\textrm{path}}( {\textsf {X}}) = ( X ^i)_{i\in {\mathbb {N}} }\) be the labeled process defined after (1.13) with \( {{\textbf {X}}} _0 = {\mathfrak {l}}({\textsf {s}})\). Then, for each \( i \in {\mathbb {N}}\),
Remark 1.4
(1) In Theorems 1.2–1.4, we assume \( d \ge 2\). This assumption is used only for the non-collision property of tagged particles.
(2) In Lemma 5.7, we deduce \( (\) A5 \()\) from \( (\) A6 \()\) introduced below. Thus, we obtain \( (\) A5 \()\) for the Ginibre random point field from Lemma 1.1 and Lemma 5.7.
We write \( \mu _{x}= \mu ( \cdot \Vert \delta _x ) \). From \( (\) A1 \()\) and \( (\) A2 \()\), \( \mu _{x}\) exists for all \( x \in {\mathbb {R}} ^d\). We can and do take \( \mu _{x}\circ \vartheta _x ^{-1} = \mu _{0}\) for all \( x \in {\mathbb {R}}^d \). We make an assumption.
\( (\) A6 \()\) (1) \( \mu \) and \( \mu _{0}\) are singular relative to each other.
(2) \( \mu _{0}\approx \mu _{x}\) for all \( x \in {\mathbb {R}} ^d\).
Theorem 1.3
Assume \( d\ge 2 \). Assume \( (\) A1 \()\)–\( (\) A4 \()\) and \( (\) A6 \()\). We then obtain the same result as in Theorem 1.2.
We prepare a set of notations for functions on \( {\textsf {S}}\) following [18].
Let \( {\textsf {S}}_{R}^m = \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}(S_{R}) = m \} \) for \( R\in {\mathbb {N}} \) and \( m \in \{ 0 \}\cup {\mathbb {N}} \). Then
Let \( S_{R}^m = S_{R} \times \cdots \times S_{R}\) be the m-product of \( S_{R}\). We call \({{\textbf {x}}}_{R}^m ({\textsf {s}}) \in S_{R}^m \) an \(S_{R}^m \)-coordinate of \({\textsf {s}}\in {\textsf {S}}_{R}^m \) if \( \pi _{R}({\textsf {s}})=\sum _{i=1}^m \delta _{x_{R}^{i}({\textsf {s}})}\), where \( {{\textbf {x}}}_{R}^m ({\textsf {s}})=(x_{R}^{i}({\textsf {s}}))_{i=1}^m \).
For \(f: {\textsf {S}} \rightarrow {\mathbb {R}} \) and \(R, m \in {\mathbb {N}} \), let \( f_{ R,{\textsf {s}}}^m ({{\textbf {x}}}) \) be the function satisfying
Note that \( f_{ R,{\textsf {s}}}^m\) is unique and \( f({\textsf {s}})=\sum _{m=0}^\infty f_{ R,{\textsf {s}}}^m({{\textbf {x}}}_{R}^m ({\textsf {s}}))\) for each \( R\in {\mathbb {N}} \) and \( {\textsf {s}}\in {\textsf {S}}\). Here by convention, \( {{\textbf {x}}}_{R}^0 ({\textsf {s}}) = \emptyset \) for \( {\textsf {s}}\in {\textsf {S}}_{R}^0 \). We see \( f_{ R,{\textsf {s}}}^0 (\emptyset )= f ( {\textsf {0}}) \) because \( {\textsf {S}}_{R}^0 \) consists of the zero configuration \( {\textsf {0}}\). The function \( f_{ R,{\textsf {s}}}^0 \) is thus constant on \( {\textsf {S}}_{R}^0 \). Although the \(S_{R}^m \)-coordinate \( {{\textbf {x}}}_{R}^m ({\textsf {s}})\) of \({\textsf {s}}\) is not unique, \( f_{ R,{\textsf {s}}}^m \) is well defined by (1.24). For a bounded set A, we set \( {{\textbf {x}}}_{A }^m ({\textsf {s}}) \) and \( f_{A,{\textsf {s}}}^m ({{\textbf {x}}} )\) similarly as above by replacing \( S_{R}\) by A.
A function \( f\!:\!{\textsf {S}}\!\rightarrow \!{\mathbb {R}} \) is called smooth if \(f_{ R,{\textsf {s}}}^m ({{\textbf {x}}})\) is smooth in \( {{\textbf {x}}} \) on \(S_{R}^m \) for all \(R, m \in {\mathbb {N}} \), \({\textsf {s}}\in {\textsf {S}} \), and local if f is \( \sigma [\pi _{R}]\)-measurable for some \( R\in {\mathbb {N}} \). Let
Let \( \mu ^{[1]}\) be the one-Campbell measure of \( \mu \) such that
where \( \rho ^1 \) is the one-point correlation function of \( \mu \) with respect to the Lebesgue measure and \( \mu _{x}= \mu ( \cdot \Vert \delta _x ) \). \( \rho ^1\) exists and is constant by \( (\) A1 \()\) and \( (\) A2 \()\).
We now recall the concept of the logarithmic derivative of \( \mu \) from [22].
Definition 1.3
[22] The logarithmic derivative \( {\textsf {d}}^{\mu }\) of \(\mu \) is an \( {\mathbb {R}} ^d\)-valued function such that \( {\textsf {d}}^{\mu }\in L_{\textrm{loc}}^{1}(\mu ^{[1]}) ^{d} \) and that, for all \(\varphi \in C_{0}^{\infty } ({\mathbb {R}} ^d) \otimes {\mathscr {D}} _{\circ \textrm{b}}\),
Here \( L_{\textrm{loc}}^{1}(\mu ^{[1]}) = \bigcap _{R=1}^{\infty } L^1 (\mu ^{[1]}_{R} )\) and \( \mu ^{[1]}_{R} = \mu ^{[1]}(\,\cdot \, \cap \{ S_{R}\times {\textsf {S}}\} )\).
Once \( {\textsf {d}}^{\mu }\) is calculated, we obtain the ISDE describing the labeled process \( {{\textbf {X}}} = ( X ^i)_{i\in {\mathbb {N}} }\). Let \( {\textsf {X}}_t^{i \diamond } = \sum _{j \not = i } \delta _{X_t^j}\). We consider the ISDE
Then, under \( (\) A2 \()\) and \( (\) A3 \()\), (1.31) has a weak solution for \( \mu \)-a.s. \( {\textsf {s}}\) such that the associated unlabeled process \( {\textsf {X}}\) is a \( \mu \)-reversible diffusion associated with the Dirichlet form \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\) on \( L^{2}(\mu )\) [22]. Under mild constraints, a weak solution of (1.31) is unique in law for \( \mu \)-a.s. \( {\textsf {s}}\) (see [13, 28]).
The logarithmic derivative \( {\textsf {d}}^{\mu _{\textrm{Gin}}} \) of \( \mu _{\textrm{Gin}}\) is given by
Hence, taking \( \mu = \mu _{\textrm{Gin}}\) in (1.31), we obtain the ISDE (1.1) (see [22, 28]).
We explain the idea of the proof of the main theorems (Theorems 1.1–1.4). Let \( \mu _{0}\) be the reduced Palm measure conditioned at the origin. It is known that the self-diffusion matrix \( \alpha = (\alpha _{ p, q })_{ p, q =1}^d \) satisfies the variational formula
Here, \( D^{\textrm{trn}} = (D^{\textrm{trn}}_{ p } )_{ p = 1}^d \) is the generator of the translation \( \vartheta _x \) on \( {\textsf {S}}\) defined by
where \( {{\textbf {e}}}_{ p }\) is the unit vector in the p-direction, \( \delta _{ p , q }\) is the Kronecker delta, \( {\mathbb {D}}\) is the carré du champ defined by (2.4), and \( {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\) is the subset of \( {\mathscr {D}}_{\bullet }\) given by (3.29). In [19, 26], \( \alpha _{ p, p }\) was given by replacing \( {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\) by \( {\mathscr {D}}_{\circ }^{{\textsf {Y}}}\), where \( {\mathscr {D}}_{\circ }^{{\textsf {Y}}}\) is given after (3.28). Because we shall prove that the closures of \( {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\) and \( {\mathscr {D}}_{\circ }^{{\textsf {Y}}}\) coincide in Lemma 3.5, \( \alpha _{ p, p }\) in (1.33) equals that given in [19, 26]. We set
In Lemma 4.3, we shall derive \( \alpha _{ p, p } = 0 \) from the following assumption.
\( (\) A7 \()\) For \( 1 \le p \le d \), we find an \( {\mathscr {E}}^{{\textsf {Y}}}\)-Cauchy sequence \( \{ \chi _{ L , p }\} \) in \( {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\) such that
Theorem 1.4
(1) Let \( d\ge 2 \). Assume (A1)–(A5). Then, (A7) holds.
(2) Let \( d\ge 2 \). Assume (A1)–(A4) and (A7). We then obtain the same result as in Theorem 1.2.
To verify \( (\) A7 \()\), we shall construct a sequence of functions \( \chi _{ L , p }\) such that
At first glance, it is difficult to construct such a sequence of functions satisfying these two conditions. This is because the second condition suggests that the limit function is a constant, while the first condition states that it is not. To resolve this issue, we focus on the tail \( \sigma \)-field \( \textrm{Tail}({\textsf {S}}) = \bigcap _{R\in {\mathbb {N}} } \sigma [\pi _{R}^{c}]\).
We note that, from (2.2)–(2.4), all tail measurable functions f satisfy
We also remark that a tail measurable function \( f \in {\mathscr {D}}_{\bullet }\) is not necessarily continuous under the vague topology. Indeed, it happens that, in general,
even if \(f_{ R,{\textsf {s}}}^m ({{\textbf {x}}}) \) in (1.23)–(1.27) for f is constant for each \( R, m \in {\mathbb {N}} \) and \( {\textsf {s}}\in {\textsf {S}}\).
Let \( {\textsf {S}}_{ \infty }= \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}({\mathbb {R}} ^d) = \infty \} \). If f is tail measurable, then f is constant on \( {\textsf {S}}\backslash {\textsf {S}}_{ \infty }\). In contrast, f is not necessarily constant on \( {\textsf {S}}_{ \infty }\), as we see in (1.39), even if f is tail measurable. We note \( \mu ({\textsf {S}}_{ \infty }) = 1\) by \( (\) A1 \()\). Thus, it may possible to construct a sequence of tail measurable functions f satisfying both (1.37) and (1.38).
The Ginibre random point field is tail trivial [17, 25]. Hence, a tail measurable function becomes a constant for \( \mu _{\textrm{Gin}}\)-a.s. and thus does not satisfy (1.37). Hence, we shall introduce the \( \sigma \)-field \( {\mathcal {G}}_{\infty } \) in (5.41). This \( \sigma \)-field is larger than \( \textrm{Tail}({\textsf {S}})\) and we can construct a sequence of \( {\mathcal {G}}_{\infty } \)-measurable functions satisfying both (1.37) and (1.38).
From \( (\) A5 \()\), we shall construct the function \( \chi _{ L , p }( {\textsf {s}}) \) in (6.4). Using \( (\) A1 \()\), we deduce that \( \{\chi _{ L , p }\} \) satisfies (1.37). Furthermore, the function \( \chi _{ L , p }( {\textsf {s}}) \) is \( {\mathcal {G}}_{\infty } \)-measurable, and thus satisfies (1.38).
We explain the difficulty of vanishing the self-diffusion in multi-dimensions in the domain \( {\mathscr {D}}_{\circ }^{{\textsf {Y}}}\) of local functions.
To provide a comparison, we begin by outlining the construction in one dimension. We can construct \( \chi _L \) under the non-collision condition in one dimension, as follows [24].
Here we label the particle \( {\textsf {s}}= \sum _i \delta _{s_i}\) such that \( 0 \le s_1< \cdots < s_L \) and that \( \{ s_1,\ldots , s_L \} \) is the first L-particles of \( {\textsf {s}}\) in \( (0,\infty )\). Because of the non-collision condition, we have \( \chi _L \in {\mathscr {D}}_{\circ }^{{\textsf {Y}}}\). Then we see that \( \chi _L \) satisfies (1.37) and (1.38).
In multi-dimensions, we have no such a well-behaved label. Instead, we can define \( \chi _L \) such that
Here \( {\textsf {s}}(S_L)\) is the number of the particles in \( S_L \). Then \( \chi _L \) satisfies (1.37) and (1.38) for \( \mu _{0}\)-a.s. \( {\textsf {s}}\). Unfortunately, \( \chi _L \) has a discontinuity on \( \partial {\textsf {S}}(L) \), where \( \partial {\textsf {S}}(L) = \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}(\partial S_L) \ge 1 \} \), and not an element of \( {\mathscr {D}}_{\circ }^{{\textsf {Y}}}\). Hence, we choose a smooth cut-off function \( \varphi _L\in C_0^{\infty }({\mathbb {R}} ^d) \) such that \( \varphi _L = 1\) on \( S_L \) and \( 0 \le \varphi _L \le 1 \) on \( {\mathbb {R}} ^d\). We set \( {\widetilde{\chi }}_L ({\textsf {s}}) = ({\widetilde{\chi }}_{L,p} ({\textsf {s}})) _{p=1}^d\) by
Then we have \( {\widetilde{\chi }}_{L,p} \in {\mathscr {D}}_{\circ }^{{\textsf {Y}}}\) and, for \( x = (x_1,\ldots , x_d) \in {\mathbb {R}} ^d\),
Because of the presence of the term \( x_p \frac{\partial \varphi _L}{\partial x_q} (x)\) in (1.41) and (1.42), it is challenging to find \( \varphi _L \in C_0^{\infty }({\mathbb {R}} ^d) \) such that \( {\widetilde{\chi }}_L \) satisfies (1.37) and (1.38). The author has been trying to construct \( {\widetilde{\chi }}_L \) of type (1.40) for a long time but has been unsuccessful.
The remainder of the paper is organized as follows. In Sect. 2, we introduce the three types of Dirichlet forms for the unlabeled dynamics. In Sect. 3, we present various diffusion processes related to the tagged particle problem and the associated Dirichlet forms: namely, one-labeled processes (Sect. 3.1), tagged particle processes (Sect. 3.2), and environment processes (Sect. 3.3). In Sect. 4, we present a sufficient condition such that the limit self-diffusion matrix vanishes. In Sect. 5.1, we introduce the concept of the dual reduced Palm measures conditioned at infinitely many particles for irreducibly decomposable random point fields. This concept is one of the main tools of our analysis. In Sect. 5.2, we introduce the mean-rigid \( \sigma \)-field \( {\mathcal {G}}_{\infty } \), which yields the mean-rigid conditioning of random point fields. This \( \sigma \)-field is also a key point of the proof of the main theorems. In Sect. 6, we complete the proof of Theorems 1.1–1.4.
2 Dirichlet forms of unlabeled dynamics
In Sect. 2, we introduce three types of the Dirichlet forms describing the unlabeled dynamics: the perpendicular, lower, and upper Dirichlet forms. From \( (\) A4 \()\), we deduce these three Dirichlet forms are the same in Lemma 2.8. This result yields the identity of the corresponding three Dirichlet forms of the environment process in Lemma 3.5.
We say that a non-negative symmetric bilinear form \( ({\mathscr {E}}, {\mathscr {D}}_0 ) \) is closable on \( L^{2}(\mu )\) if \( \lim _{n\rightarrow \infty }{\mathscr {E}}(f_n, f_n ) = 0 \) for any \( {\mathscr {E}}\)-Cauchy sequence \( f_n\in {\mathscr {D}}_0\) such that \( \lim _{n\rightarrow \infty } \Vert f_n \Vert _{L^{2}(\mu )} = 0 \). If \( ({\mathscr {E}}, {\mathscr {D}}_0) \) is closable on \( L^{2}(\mu )\), then there exists a closed extension of \( ({\mathscr {E}}, {\mathscr {D}}_0) \). The smallest closed extension \( ({\mathscr {E}}, {\mathscr {D}} ) \) of \( ({\mathscr {E}}, {\mathscr {D}}_0) \) is called the closure of \( ({\mathscr {E}}, {\mathscr {D}}_0) \). We refer to [6] for detail.
For non-negative symmetric bilinear forms \( ({\mathscr {E}}_1, {\mathscr {D}}_1 ) \) and \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \) on \( L^{2}(\mu )\), we say \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \) is an extension of \( ({\mathscr {E}}_1, {\mathscr {D}}_1 ) \) if
Suppose that \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \) is an extension of \( ({\mathscr {E}}_1, {\mathscr {D}}_1 ) \). Then, \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \le ({\mathscr {E}}_1, {\mathscr {D}}_1 ) \). The following simple fact will be used repeatedly in the present paper.
Lemma 2.1
Let \( ({\mathscr {E}}_1, {\mathscr {D}}_1 ) \) and \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \) be non-negative symmetric bilinear forms on \( L^{2}(\mu )\). Let \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \) be an extension of \( ({\mathscr {E}}_1, {\mathscr {D}}_1 ) \). Let \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \) be closable on \( L^{2}(\mu )\). Then, \( ({\mathscr {E}}_1, {\mathscr {D}}_1 ) \) is closable on \( L^{2}(\mu )\).
Proof
If \( \{ f_n \} \) is a Cauchy sequence of \( ({\mathscr {E}}_1, {\mathscr {D}}_1 ) \), then \( \{ f_n \} \) is a Cauchy sequence of \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \) by (2.1). Then \( \lim _{n\rightarrow \infty } {\mathscr {E}}_2 (f_n, f_n ) = 0 \) because of the closability of \( ({\mathscr {E}}_2, {\mathscr {D}}_2 ) \) on \( L^{2}(\mu )\). Hence, \( \lim _{n\rightarrow \infty } {\mathscr {E}}_1 (f_n, f_n ) = 0 \) from (2.1). This completes the proof. \(\square \)
We say a closed non-negative symmetric bilinear form \( ({\mathscr {E}}, {\mathscr {D}} ) \) on \( L^{2}(\mu )\) is a symmetric Dirichlet form [6] if any \( u \in {\mathscr {D}}\) satisfies
For a symmetric Dirichlet form, there exists an associated symmetric Markovian \( L^2\)-semi-group. In addition, if the Dirichlet form is strongly local and quasi-regular and the state space is homeomorphic to a complete separable metric space, then the associated symmetric diffusion process exists [3]. In general, a Dirichlet form is not necessarily symmetric. In the present paper, a Dirichlet form means a symmetric Dirichlet form.
2.1 Dirichlet forms associated with the unlabeled diffusions
In Sect. 2.1, we prepare results for the Dirichlet forms associated with the unlabeled diffusions from [12, 18, 23]. Let \( {\textsf {S}}_{R}^m = \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}(S_{R})= m \} \) as before. Let \( {\mathscr {D}}_{\bullet }\) be as in (1.28). For \(f,g \in {\mathscr {D}}_{\bullet }\) and \( {\textsf {s}}=\sum _i \delta _{s_i}\), we set
Here \( \frac{\partial }{\partial s_{i}} = (\frac{\partial }{\partial s_{i,1}},\ldots ,\frac{\partial }{\partial s_{i,d}})\), \( f_{ R,{\textsf {s}}}^m \) is as in (1.23)–(1.27) for \( f \in {\mathscr {D}}_{\bullet }\), and \({{\textbf {x}}}_{R}^m ({\textsf {s}})\) is an \(S_{R}^m \)-coordinate of \({\textsf {s}}\) introduced after (1.22). Note that \({\mathbb {D}}_{R}^{m}[f,g] ({\textsf {s}})\) is independent of the choice of the \(S_{R}^m \)-coordinate \({{\textbf {x}}}_{R}^m ({\textsf {s}})\) and is well defined. Let
Then \( {\mathbb {D}}_{R}[ f, f ] ({\textsf {s}}) \) is non-decreasing in \( R\) for all \( f \in {\mathscr {D}}_{\bullet }\) and \( {\textsf {s}}\in {\textsf {S}}\). Hence, we set
We set the carré du champs \( {\mathbb {D}}[ f, g ]\) by polarization. We set
Using the method in [12, 18, 23], we deduce from \( (\) A3 \()\) that \( ({\mathscr {E}}^{\mu }_{R},{\mathscr {D}}_{ R, \bullet }^{\mu })\) is closable on \( L^{2}(\mu )\). Hence, we denote by \( ({\mathscr {E}}^{\mu }_{R},\underline{{\mathscr {D}}}_{R}^{\mu }) \) its closure. Clearly, the sequence of the closed forms \( ({\mathscr {E}}^{\mu }_{R},\underline{{\mathscr {D}}}_{R}^{\mu }) \) is increasing in the sense that
Let \( ({\mathscr {E}}^{\mu }, \underline{{\mathscr {D}}}^{\mu }) \) be the increasing limit of \( ({\mathscr {E}}^{\mu }_{R},\underline{{\mathscr {D}}}_{R}^{\mu }) \), \( R\in {\mathbb {N}}\), such that
Then \( ({\mathscr {E}}^{\mu }, \underline{{\mathscr {D}}}^{\mu }) \) is a closed form on \( L^{2}(\mu )\). We easily see that \( ({\mathscr {E}}^{\mu }, {\mathscr {D}}_{\bullet }^{\mu })\) is closable on \( L^{2}(\mu )\) and its closure coincides with \( ({\mathscr {E}}^{\mu }, \underline{{\mathscr {D}}}^{\mu }) \).
Let \( {\mathscr {D}}_{\circ }\) be as in (1.28). Clearly, \( {\mathscr {D}}_{\circ }\subset {\mathscr {D}}_{\bullet }\). We set
Note that \( ({\mathscr {E}}^{\mu }_{R},{\mathscr {D}}_{ R, \bullet }^{\mu })\) is an extension of \( ({\mathscr {E}}^{\mu }_{R},{\mathscr {D}}_{R,\circ } ^{\mu })\) and that \( ({\mathscr {E}}^{\mu }_{R},{\mathscr {D}}_{ R, \bullet }^{\mu })\) is closable on \( L^{2}(\mu )\). Hence, \( ({\mathscr {E}}^{\mu }_{R},{\mathscr {D}}_{R,\circ } ^{\mu })\) is closable on \( L^{2}(\mu )\) by Lemma 2.1. Then we denote the closure of \( ({\mathscr {E}}^{\mu }_{R},{\mathscr {D}}_{R,\circ } ^{\mu })\) on \( L^{2}(\mu )\) as \( ( {\mathscr {E}}^{\mu }_{R}, {\overline{{\mathscr {D}}}}_{R}^{\mu })\). By construction, \( ({\mathscr {E}}^{\mu }_{R}, \underline{{\mathscr {D}}}^{\mu }_{R} ) \) is an extension of \( ( {\mathscr {E}}^{\mu }_{R}, {\overline{{\mathscr {D}}}}_{R}^{\mu }) \). In particular, we have
If \( f \in {\mathscr {D}}_{R,\circ } ^{\mu }\), then f is \( \sigma [\pi _{R}]\)-measurable. Then \( {\mathbb {D}}_{R}[f,f] = {\mathbb {D}}_{R+1}[f,f]\). Hence
From (2.10), \( ( {\mathscr {E}}^{\mu }_{R}, {\overline{{\mathscr {D}}}}_{R}^{\mu })\) is decreasing in \( R\). Let \( ( {\mathscr {E}}^{\mu }, \cup _{R\in {\mathbb {N}} }{\overline{{\mathscr {D}}}}_{R}^{\mu }) \) be the decreasing limit. Note that \( ({\mathscr {E}}^{\mu }, \underline{{\mathscr {D}}}^{\mu }) \) is an extension of \( ( {\mathscr {E}}^{\mu }, \cup _{R\in {\mathbb {N}} }{\overline{{\mathscr {D}}}}_{R}^{\mu }) \) and \( ({\mathscr {E}}^{\mu }, \underline{{\mathscr {D}}}^{\mu }) \) is a closed form on \( L^{2}(\mu )\). Hence, the decreasing limit \( ( {\mathscr {E}}^{\mu }, \cup _{R\in {\mathbb {N}} }{\overline{{\mathscr {D}}}}_{R}^{\mu }) \) is closable on \( L^{2}(\mu )\) by Lemma 2.1.
We denote the closure of \( ( {\mathscr {E}}^{\mu }, \cup _{R\in {\mathbb {N}} }{\overline{{\mathscr {D}}}}_{R}^{\mu })\) on \( L^{2}(\mu )\) by \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\):
Lemma 2.2
Assume \( (\) A1 \()\)–\( (\) A3 \()\). Let \( {\mathscr {D}}_{\circ }^{\mu } = \{ f \in {\mathscr {D}}_{\circ }; {\mathscr {E}}^{\mu }( f, f ) < \infty , f \in L^{2}(\mu )\} \). Then \( ({\mathscr {E}}^{\mu }, {\mathscr {D}}_{\circ }^{\mu } )\) is closable on \( L^{2}(\mu )\) and its closure \( ( {\mathscr {E}}^{\mu }, \overline{{\mathscr {D}}_{\circ }^{\mu }} ) \) satisfies \(\overline{{\mathscr {D}}_{\circ }^{\mu }} = {\overline{{\mathscr {D}}}}^{\mu } \) and
Proof
From (2.7)–(2.11), it is clear that (2.12) holds. If f is \( \sigma [\pi _{R}]\)-measurable, then \( {\mathbb {D}}[f,f] = {\mathbb {D}}_{R}[f,f]\). Hence from (2.8) and (2.11),
Then \( ({\mathscr {E}}^{\mu }, {\mathscr {D}}_{\circ }^{\mu } )\) is closable on \( L^{2}(\mu )\) by Lemma 2.1 because \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu } )\) is an extension of \( ({\mathscr {E}}^{\mu }, {\mathscr {D}}_{\circ }^{\mu } )\) from (2.13) and \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu } )\) is closed on \( L^{2}(\mu )\). Thus, from this and (2.13),
Because \( ( {\mathscr {E}}^{\mu }_{R}, {\overline{{\mathscr {D}}}}_{R}^{\mu } )\) is the closure of \( ({\mathscr {E}}^{\mu }_{R},{\mathscr {D}}_{R,\circ } ^{\mu })\), \( {\mathscr {D}}_{\circ }^{\mu }\supset {\mathscr {D}}_{R,\circ } ^{\mu }\), and (2.10) holds, we have \( \overline{{\mathscr {D}}_{\circ }^{\mu }} \supset {\overline{{\mathscr {D}}}}_{R}^{\mu } \) for all \( R\). Hence, \( \overline{{\mathscr {D}}_{\circ }^{\mu }} \supset \cup _{R\in {\mathbb {N}}}{\overline{{\mathscr {D}}}}_{R}^{\mu } \). From this and (2.11), we deduce
We thus obtain \( \overline{{\mathscr {D}}_{\circ }^{\mu }} = {\overline{{\mathscr {D}}}}^{\mu } \) from (2.14) and (2.15). \(\square \)
It is known that \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\) on \( L^{2}(\mu )\) is a quasi-regular Dirichlet form and the associated \( \mu \)-reversible diffusion \( (P_{{\textsf {s}}}, {\textsf {X}}_t)\) satisfies (1.13) (see [23], Lemma 10.2 in [28], Lemma 2.5 in [12]). Hence, we have the labeled process \( {{\textbf {X}}} = {\mathfrak {l}}_{\textrm{path}}({\textsf {X}})\).
2.2 Perpendicular carré du champs \( {\mathbb {D}}^{\perp }\)
In Sect. 2.2, we introduce the concept of the carré du champ perpendicular to the generator of the translation operator \( D^{\textrm{trn}} = ( D^{\textrm{trn}}_{ p } )_{ p =1 }^{d}\) defined by (1.34).
For a set \( A \subset {\mathbb {R}} ^d\) and \( {\textsf {s}}=\sum _i \delta _{s_i} \in {\textsf {S}}\), we set
Note that the orthogonal projection of \(\partial / \partial s_i \) onto the subspace perpendicular to \(\partial / \partial \Gamma ( A ) \) is then given by
We note that \( {\textsf {s}}( A )\) becomes the number of particles in A for \( {\textsf {s}}= \sum _i \delta _{s_i}\).
The orthogonality above is with respect to the inner product such that
where \( s_i = (s_{i,p})_{p=1}^d,\, s_j = (s_{j,q})_{q=1}^d \in {\mathbb {R}} ^d\). For each \( s_i \in A \), we have
Let \( A \subset B \). Then
Let \( T _1 = S_{1} \) and \( T _{R}= S_{R}\backslash S_{{ R-1}} \) for \( R\ge 2 \). Note that \( \{ T _{R}\}_{ R\in {\mathbb {N}} } \) is a partition of \( {\mathbb {R}} ^d\). For each \( R\in {\mathbb {N}} \), let \( \{ {\textsf {T}}_{ R}^{ m } \}_{ m \in \{ 0 \} \cup {\mathbb {N}} } \) be the partition of \( {\textsf {S}}\) such that \( {\textsf {T}}_{ R}^{ m } = \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}( T _{R})= m \} \). We set the carré du champs such that
Here for f, the functions \( f_{ T _{R},{\textsf {s}}}^m\) are given by (1.23)–(1.27).
Lemma 2.3
For each \( f \in {\mathscr {D}}_{\bullet }\) and \( R\in {\mathbb {N}} \),
Proof
For \( {\textsf {s}}= \sum _i \delta _{s_i}\in {\textsf {T}}_{ Q }^m \), we see from (2.16)
Hence from (2.18), (2.19), and (2.21), we deduce
Summing both sides over \( Q = 1,\ldots , R\), we obtain (2.20). \(\square \)
Let \( {\textsf {S}}_{ R}^m= \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}(S_{ R})= m \}\). Note that \( S_{ R-1}\cup T _{R}= S_{ R}\) and \( S_{ R-1}\cap T _{R}= \emptyset \) for \( R\in {\mathbb {N}} \). We set, for \( R\in {\mathbb {N}} \) such that \( 2 \le R\),
Lemma 2.4
Let \( {\mathbb {T}}^{\gamma }_{R}\) be as in (2.19). For \( R\ge 2\), we have
Proof
From (2.22) and (2.23), we have
Using \( {\textsf {s}}( S_{ R-1})+ {\textsf {s}}( T _{R}) = {\textsf {s}}(S_{R})\) and \( \frac{\partial }{\partial \Gamma ( S_{ R-1}) } + \frac{\partial }{\partial \Gamma ( T _{R}) }= \frac{\partial }{\partial \Gamma ( S_{ R}) } \), we see
Putting (2.26) into (2.25) and using (2.19) and (2.23), we have
We have thus completed the proof of (2.24). \(\square \)
Let \( {\mathbb {D}}^{\perp }_1 = {\mathbb {T}}^{\perp }_{ 1 }\) and
Lemma 2.5
For each \( f \in {\mathscr {D}}_{\bullet }\) and \( R\in {\mathbb {N}} \),
Proof
Using (2.24), we have
Note that \( S_1 = T _1 \) by definition. Hence from (2.19) and (2.23), we have
Combining (2.29) and (2.30), we deduce
Using (2.20), (2.27), and (2.31), we obtain (2.28). \(\square \)
We now introduce the perpendicular carré du champs \( {\mathbb {D}}^{\perp }\) such that
Then from (2.27), \( {\mathbb {D}}^{\perp }_{R} [f,f] ({\textsf {s}}) \) is increasing in \( R\) and satisfies
Lemma 2.6
For each \( f \in {\mathscr {D}}_{\bullet }\) and \( {\textsf {s}}\in {\textsf {S}}\),
In particular, \( {\mathbb {D}}^{\perp }[f,f] ({\textsf {s}}) < \infty \) holds for each \( f \in {\mathscr {D}}_{\circ }\) and \( {\textsf {s}}\in {\textsf {S}}\).
Proof
Note that both \( {\mathbb {D}}^{\perp }_{R} [f,f] ({\textsf {s}}) \) and \( {\mathbb {D}}_{R} [ f, f ] ({\textsf {s}}) \) are increasing in \( R\) and thus have the limits \( {\mathbb {D}}^{\perp }[f,f] ({\textsf {s}}) \) and \( {\mathbb {D}}[ f, f ] ({\textsf {s}}) \). Then, we obtain (2.34) from (2.28).
For each \( f \in {\mathscr {D}}_{\circ }\), we there exists an \( R\in {\mathbb {N}} \) such that f is \( \sigma [\pi _{R}]\)-measurable. Because \( {\mathbb {D}}[f,f] ({\textsf {s}}) = {\mathbb {D}}_{R}[f,f] ({\textsf {s}}) \) and \( {\textsf {s}}(S_{R}) < \infty \), we have \( {\mathbb {D}}[f,f] ({\textsf {s}}) = {\mathbb {D}}_{R}[f,f] ({\textsf {s}}) < \infty \). Hence from (2.34), we obtain \( {\mathbb {D}}^{\perp }[f,f] ({\textsf {s}}) < \infty \). \(\square \)
2.3 Identity of the perpendicular, lower, and upper Dirichlet forms
In Sect. 2.3, we introduce the concept of the perpendicular Dirichlet form. We shall prove that the perpendicular, lower, and upper Dirichlet forms coincide under \( (\) A4 \()\). We set for \( {\textsf {s}}= \sum _i \delta _{s_i}\)
If \( \upsilon \in \Upsilon _{R,1}^{m}\), then \( \upsilon \) is a partial derivative on \( T _{R}\). To be precise, \( \upsilon \in \Upsilon _{R,1}^{m}\) is a partial derivative on \( T _{R}^{ m }\). We disregard m and simply call a partial derivative on \( T _{R}\) for convenience. Because \( \upsilon \) is a local operator, we can regard \( \upsilon \) as a partial derivative on any domain A including \( T _{R}\).
For \(f\!:\!{\textsf {S}}\!\rightarrow \!{\mathbb {R}} \) and \( \upsilon \in \Upsilon _{R,1}^{m}\), we denote \( f \in \textrm{Dom}(\upsilon ) \) if \( f_{ T _{R}, {\textsf {s}}}^m ({{\textbf {x}}})\) is in the domain of \( \upsilon \), where \( f_{ T _{R}, {\textsf {s}}}^m \) is the representation of f defined by (1.23)–(1.27).
Let \( {\mathfrak {u}} ({{\textbf {x}}}) = \sum _i\delta _{x_i}\) for \( {{\textbf {x}}} = (x_i)_i\) as before. We set \( \upsilon f ({\textsf {s}}) = 0 \) for \( {\textsf {s}}\notin {\textsf {T}}_{R}^m \) and
For \(f\!:\!{\textsf {S}}\!\rightarrow \!{\mathbb {R}} \) and \( \upsilon \in \Upsilon _{R, k}^{m,n}\), \( k=2,3 \), we define \( f \in \textrm{Dom}(\upsilon )\) similarly. For \( \upsilon \in \Upsilon _{R,2 } ^{ m , n }\), we set \( \upsilon f ({\textsf {s}}) = 0 \) for \( {\textsf {s}}\notin {\textsf {S}}_{ R- 1 }^m \cap {\textsf {T}}_{ R}^n \) and
For \( \upsilon \in \Upsilon _{R,3 } ^{ m , n }\), we set \( \upsilon f ({\textsf {s}}) = 0 \) for \( {\textsf {s}}\notin {\textsf {T}}_{ R}^m \cap {\textsf {S}}_{R}^n \) and
Let \( {\mathbb {D}}_{\upsilon } [f,g] = \frac{1}{2}( \upsilon f, \upsilon g )_{{\mathbb {R}} ^d}\) be the carré du champ generated by \( \upsilon \). Then,
Here \( {\mathbb {D}}^{\perp }\) is the perpendicular carré du champ defined by (2.32) and \( \Upsilon \) is
where the right-hand side is a disjoint union of \( \Upsilon _{R,1}^{m}, \Upsilon _{R,2 } ^{ m , n }\), and \(\Upsilon _{R,3 } ^{ m , n }\). Thus, \( \Upsilon \) is a collection of partial derivatives constituting \( {\mathbb {D}}^{\perp }\). We set
Here \( {\mathbb {D}}^{\perp }_{R}\) is as in (2.27). In (2.37), we suppress \( \mu \) from the notation.
Lemma 2.7
Assume \( (\) A2 \()\) and \( (\) A3 \()\). Then the following hold.
(1) \( ( {\mathscr {E}}^{\perp } _{R}, {\mathscr {D}}_{\bullet }^{\perp }) \) and \( ( {\mathscr {E}}^{\perp } , {\mathscr {D}}_{\bullet }^{\perp }) \) are closable on \( L^{2}(\mu )\).
(2) Let \( ( {\mathscr {E}}^{\perp } _{R}, {\mathscr {D}}_{ R}^{\perp }) \) and \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) \) be the closures of \( ( {\mathscr {E}}^{\perp } _{R}, {\mathscr {D}}_{\bullet }^{\perp }) \) and \( ( {\mathscr {E}}^{\perp } , {\mathscr {D}}_{\bullet }^{\perp }) \) on \( L^{2}(\mu )\), respectively. Then, \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) \) is the increasing limit of \( ( {\mathscr {E}}^{\perp } _{R}, {\mathscr {D}}_{ R}^{\perp }) \).
(3) \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) \le ( {\mathscr {E}}^{\mu }, {\underline{{\mathscr {D}}}}^{\mu }) \).
Proof
For \( \upsilon \in \Upsilon \), let \( {\mathscr {E}}^{\upsilon } (f,g)= \int _{{\textsf {S}}} {\mathbb {D}}_{\upsilon }[f, g] d\mu \). Then, \( ( {\mathscr {E}}^{\upsilon }, {\mathscr {D}}_{\bullet }^{\perp })\) is closable on \( L^{2}(\mu )\) by \( (\) A3 \()\). Thus from (2.18), (2.22), (2.27), and (2.32), \( ( {\mathscr {E}}^{\perp } _{R}, {\mathscr {D}}_{\bullet }^{\perp }) \) and \( ( {\mathscr {E}}^{\perp } , {\mathscr {D}}_{\bullet }^{\perp }) \) are countable sums of closable forms \( ( {\mathscr {E}}^{\upsilon }, {\mathscr {D}}_{\bullet }^{\perp })\) on \( L^{2}(\mu )\). Hence, we obtain (1).
From (1) and (2.33), \( ( {\mathscr {E}}^{\perp } , {\mathscr {D}}_{\bullet }^{\perp }) \) is the increasing limit of closable forms \( ( {\mathscr {E}}^{\perp } _{R}, {\mathscr {D}}_{\bullet }^{\perp }) \) on \( L^{2}(\mu )\). From this we obtain (2).
From (2.6), (2.34), and (2.37), \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}_{\bullet }^{\perp }) \le ({\mathscr {E}}^{\mu }, {\mathscr {D}}_{\bullet }^{\mu })\). This yields (3). \(\square \)
Lemma 2.8
Assume (A2)–(A4). Then \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) = ( {\mathscr {E}}^{\mu }, {\underline{{\mathscr {D}}}}^{\mu }) = ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\).
Proof
From Lemmas 2.7(3) and 2.2, we obtain
By \( (\) A4 \()\), we have \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) = ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\). This and (2.38) complete the proof. \(\square \)
Let \( T_t^{\mu }\) be the Markovian semi-group on \( L^{2}(\mu )\) associated with \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\) on \( L^{2}(\mu )\). Then there exists an unlabeled diffusion \( (P_{{\textsf {s}}}, {\textsf {X}}_t)\) associated with the Dirichlet form \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\) on \( L^{2}(\mu )\) [18, 22, 23, 28]. By construction, \( (P_{{\textsf {s}}}, {\textsf {X}}_t)\) is \( \mu \)-reversible and \( T_t^{\mu } f ({\textsf {s}}) = E_{{\textsf {s}}}[ f ({\textsf {X}}_t )] \) for each \( f \in L^{2}(\mu )\). From Lemma 2.8, \( (P_{{\textsf {s}}}, {\textsf {X}}_t)\) is also associated with \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) \) and \( ({\mathscr {E}}^{\mu }, \underline{{\mathscr {D}}}^{\mu }) \).
3 One-labeled, tagged particle, and environment processes
In Sect. 3, we introduce the three stochastic processes related to the tagged particle problem: one-labeled, tagged particle, and environment processes.
We can obtain these three stochastic processes from the labeled process \( {{\textbf {X}}} =(X^i)_{i\in {\mathbb {N}}}\) by change of coordinate. These stochastic processes are diffusion processes and we identify the associated Dirichlet forms. We remark that the original labeled process \( {{\textbf {X}}} \) does not have any associated Dirichlet form.
3.1 One-labeled processes
Let \( (P_{{\textsf {s}}}, {\textsf {X}}_t)\) be the \( \mu \)-reversible diffusion associated with \( ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\) on \( L^{2}(\mu )\) given in Sect. 2.1. From \( (\) A1 \()\)–\( (\) A3 \()\), \( (P_{{\textsf {s}}}, {\textsf {X}}_t)\) satisfies (1.13). Then using \( {\mathfrak {l}}_{\textrm{path}}\) defined by (1.12), we have the corresponding labeled process \( {{\textbf {X}}} = {\mathfrak {l}}_{\textrm{path}}({\textsf {X}})\). From the labeled process \( {{\textbf {X}}} = ( X ^i)_{i\in {\mathbb {N}} }\), we construct the one-labeled processes \( ( X ^i, {\textsf {X}}^{i \diamond })\), \( i \in {\mathbb {N}}\), where \( {\textsf {X}}_t^{i \diamond } = \sum _{j\ne i }^{\infty } \delta _{ X _t^j } \).
The same construction is also possible for the unlabeled processes given by \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) \) and \( ( {\mathscr {E}}^{\mu }, {\underline{{\mathscr {D}}}}^{\mu }) \) on \( L^{2}(\mu )\). If we assume \( (\) A4 \()\) in addition, then these three processes are the same by Lemma 2.8. We shall present the Dirichlet form associated with the one-labeled processes \( ( X ^i, {\textsf {X}}^{i \diamond })\), \( i \in {\mathbb {N}}\). The Dirichlet form is independent of \( i \in {\mathbb {N}}\) because of the symmetry of the particle system.
Let \( \nabla \) be the nabla in \({\mathbb {R}} ^d\). We set for \( f, g \in C_0^{\infty }({\mathbb {R}} ^d) \)
We naturally regard \( \nabla \) and \( {\mathbb {D}}\) as the carré du champs on \( C_0^{\infty }({\mathbb {R}} ^d) \otimes {\mathscr {D}}_{\bullet }\) in such a way that, for \( f = f_1 \otimes f_2 \) and \( g = g_1 \otimes g_2 \),
We regard \( {\mathbb {D}}_{ R}^{m} \), \( {\mathbb {D}}^{\perp } \), and \( {\mathbb {D}}_{R}^{\perp } \) as the carré du champs on \( C_0^{\infty }({\mathbb {R}} ^d) \otimes {\mathscr {D}}_{\bullet }\) in the same fashion. Let \( \mu ^{[1]}\) be the one-Campbell measure of \( \mu \) given by (1.29). Let
Lemma 3.1
Assume \( (\) A1 \()\)–\( (\) A3 \()\). Then \( ( {\mathscr {E}}_R^{[1],\perp }, {\mathscr {D}}_{\bullet }^{[1],\perp } )\), \( ({\mathscr {E}}^{[1]}_{R},{\mathscr {D}}_{\bullet }^{[1]}) \), and \( ({\mathscr {E}}^{[1]}_{R},{\mathscr {D}}_{\circ }^{[1]}) \) are closable on \( L^2 (\mu ^{[1]}) \).
Proof
Let \( \mu _{R, {\textsf {s}}}\) be the regular conditional probability measure such that
Let \( \mu _{R, {\textsf {s}}}^{[1]} \) be the one-Campbell measure of \( \mu _{R, {\textsf {s}}}\). We set for \( m \in {\mathbb {N}} \)
Let \( {\mathcal {H}}_{R}({\textsf {x}}) \) be as in (1.7). Recall that \( \mu \) is a \( \Psi \)-quasi-Gibbs measure. Then using (1.9), (3.2), and (3.3), we have for \( ( x, {\textsf {x}}) \in S_{R} \times {\textsf {S}}_{R}^{m-1} \)
Here, \( \Lambda _{R} ^{[1], 1 } (d x d{\textsf {x}})= 1_{S_{R}}(x)dx \delta _{{\textsf {0}}}({\textsf {x}})\) and \( \Lambda _{R} ^{[1], m } (d x d{\textsf {x}}) = 1_{S_{R}}(x)dx \Lambda _{R} ^{m-1} (d{\textsf {x}}) \) for \( m \ge 2 \). Furthermore, \( C_{3.1} \) is a positive constant depending on \( \beta \), \( R\), m, and \( \pi _{R}^{c}({\textsf {s}}) \). We set
Using \( (\) A3 \()\), (3.4), and (3.5) and applying the method in [18, Lemma 3.2], we see that \( ({\mathscr {E}}_{ R, {\textsf {s}}}^{[1], m}, {\mathscr {D}}_{\bullet }^{[1]}) \) is closable on \( L^2 (\mu _{ R, {\textsf {s}}}^{[1], m})\). Let
We write \( \pi _{R}({\textsf {s}}) = {\textsf {x}}\) and \( \pi _{R}^{c}({\textsf {s}}) = {\textsf {y}}\). Thus \( {\textsf {s}}= ({\textsf {x}}, {\textsf {y}})\). From (3.3), we have
From (3.6) and (3.7), we deduce that \( ( {\mathscr {E}}_{R}^{[1],m},{\mathscr {D}}_{\bullet }^{[1]}) \) on \( L^2 (\mu ^{[1]}) \) is the integral of closable bilinear forms \( ({\mathscr {E}}_{ R, {\textsf {s}}}^{[1], m}, {\mathscr {D}}_{\bullet }^{[1]}) \) on \( L^2 (\mu _{ R, {\textsf {s}}}^{[1], m})\). Hence using the method in “Proof of Theorem 4” in [18, p.130], we see that \( ( {\mathscr {E}}_{R}^{[1],m},{\mathscr {D}}_{\bullet }^{[1]}) \) is closable on \( L^2 (\mu ^{[1]}) \).
From (3.6) and (3.7), \( {\mathscr {E}}_{R}^{[1]} = \sum _{m=1}^{\infty } {\mathscr {E}}_{R}^{[1],m} \). Because \( ( {\mathscr {E}}_{R}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \) is a countable sum of closable forms \( ( {\mathscr {E}}_{R}^{[1],m},{\mathscr {D}}_{\bullet }^{[1]}) \) on \( L^2 (\mu ^{[1]}) \), \( ( {\mathscr {E}}_{R}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \) is closable on \( L^2 (\mu ^{[1]}) \).
The proof of the closability of \( ( {\mathscr {E}}_R^{[1],\perp }, {\mathscr {D}}_{\bullet }^{[1],\perp } )\) on \( L^{2}(\mu )\) is similar to that of \( ({\mathscr {E}}^{[1]}_{R},{\mathscr {D}}_{\bullet }^{[1]}) \) on \( L^{2}(\mu )\). Hence we omit it. \(\square \)
We set
We define \( {\mathscr {D}}_{\circ }^{[1]}\) by (3.9) through replacing \( {\mathscr {D}}_{\bullet }\) with \( {\mathscr {D}}_{\circ }\). We define \( {\mathscr {E}}^{[1],\perp } \) by (3.8) through replacing \( {\mathbb {D}}\) with \( {\mathbb {D}}^{\perp } \). We define \( {\mathscr {D}}_{\bullet }^{[1],\perp } \) by (3.9) through replacing \( {\mathscr {D}}_{\bullet }\) and \( {\mathscr {E}}^{[1]}\) with \( {\mathscr {D}}_{\bullet }^{\perp }\) and \( {\mathscr {E}}^{[1],\perp } \), respectively. Here \( {\mathscr {D}}_{\bullet }^{\perp }\) was given by (2.37). We deduce from (2.34) and \( {\mathscr {D}}_{\bullet }\supset {\mathscr {D}}_{\circ }\)
Lemma 3.2
Assume (A1)–(A3). Then \( ( {\mathscr {E}}^{[1],\perp }, {\mathscr {D}}_{\bullet }^{[1],\perp } )\), \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \), and \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\circ }^{[1]}) \) are closable on \( L^2 (\mu ^{[1]}) \).
Proof
Because \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \) is the increasing limit of \( ( {\mathscr {E}}_{R}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \) and \( ( {\mathscr {E}}_{R}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \) is closable on \( L^{2}(\mu ^{[1]})\) by Lemma 3.1, \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \) is closable on \( L^{2}(\mu ^{[1]})\).
Note that \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \) is an extension of \( ({\mathscr {E}}^{[1]}_{R},{\mathscr {D}}_{\circ }^{[1]}) \) and that \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \) is closable on \( L^{2}(\mu ^{[1]})\). Hence, \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\circ }^{[1]}) \) is closable on \( L^{2}(\mu ^{[1]})\) by Lemma 2.1.
We see that \( ( {\mathscr {E}}^{[1],\perp }, {\mathscr {D}}_{\bullet }^{[1],\perp } ) \) is the increasing limit of \( ( {\mathscr {E}}_R^{[1],\perp }, {\mathscr {D}}_{\bullet }^{[1],\perp } )\) as \( R\rightarrow \infty \). From Lemma 3.1, \( ( {\mathscr {E}}_R^{[1],\perp }, {\mathscr {D}}_{\bullet }^{[1],\perp } )\) is closable on \( L^{2}(\mu ^{[1]})\). Hence, \( ( {\mathscr {E}}^{[1],\perp }, {\mathscr {D}}_{\bullet }^{[1],\perp } ) \) is closable on \( L^{2}(\mu ^{[1]})\). \(\square \)
Let \( ({\mathscr {E}}^{[1], \perp }, {\mathscr {D}}^{[1], \perp } ) \), \( ({{\mathscr {E}}}^{[1]}, \underline{{\mathscr {D}}}^{[1]}) \), and \( ({\mathscr {E}}^{[1]}, \overline{{\mathscr {D}}}^{[1]}) \) be the closures of \( ( {\mathscr {E}}^{[1],\perp }, {\mathscr {D}}_{\bullet }^{[1],\perp } )\), \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\bullet }^{[1]}) \), and \( ({\mathscr {E}}^{[1]},{\mathscr {D}}_{\circ }^{[1]}) \) on \( L^2 (\mu ^{[1]})\), respectively.
Lemma 3.3
Assume (A1)–(A4). Then
The diffusion \( ( X ^i, {\textsf {X}}^{i \diamond })\) is associated with the closed forms in (3.11) on \( L^{2}(\mu ^{[1]})\).
Proof
We see \( {\overline{{\mathscr {D}}_{\bullet }^{\perp }}}= {\mathscr {D}}^{\perp } \) by definition and \( {\overline{{\mathscr {D}}}}^{\mu } = \overline{{\mathscr {D}}_{\circ }^{\mu } }\) by Lemma 2.2. From Lemma 2.8, \( ({\mathscr {E}}^{\perp } , {\mathscr {D}}^{\perp } ) = ( {\mathscr {E}}^{\mu }, {\underline{{\mathscr {D}}}}^{\mu }) = ( {\mathscr {E}}^{\mu }, {\overline{{\mathscr {D}}}}^{\mu })\). Combining these, we have \( {\overline{{\mathscr {D}}_{\bullet }^{\perp }}} = \overline{{\mathscr {D}}_{\circ }^{\mu } } \). Hence,
It is easy to see that
From (3.12) and (3.13), we obtain
From (3.10), we see
Hence combining (3.14) and (3.15), we obtain (3.11).
Using [21, Theorem 2.4], we see that \( ( X ^i, {\textsf {X}}^{i \diamond })\) is the diffusion associated with the Dirichlet form \( ({\mathscr {E}}^{[1]}, \overline{{\mathscr {D}}}^{[1]}) \) on \( L^{2}(\mu ^{[1]})\). Hence, (3.11) yields the second claim. \(\square \)
3.2 Tagged particle processes
The tagged particle problem of interacting Brownian motions is to prove the diffusive scaling limit of each particle in the system [4, 8, 19, 20]. The standard device for this problem is to introduce the environment process seen from the tagged particle [8]. We use this device and define it by changing the coordinates as follows. For the labeled process \( {{\textbf {X}}} = (X^i)_{i\in {\mathbb {N}}}\) given after (1.13), we set
Here, X denotes the tagged particle and \( {{\textbf {Y}}}=(Y^{i})_{i\in {\mathbb {N}}}\) is the labeled environment process seen from the tagged particle. The unlabeled environment process \( {\textsf {Y}}= \{ {\textsf {Y}}_t \}_t \) is given by
For \( R,T \in {\mathbb {N}} \) and \( {{\textbf {w}}}=(w^i)_{ i \in {\mathbb {N}} } \in C([0,\infty );{\mathbb {R}} ^d)^{{\mathbb {N}}}\), we set
For \( {{\textbf {w}}}=(w^i)_{ i \in {\mathbb {N}} } \in C([0,\infty );{\mathbb {R}} ^d)^{{\mathbb {N}}}\), let \( {\textsf {w}} \) be such that \( {\textsf {w}}_t = \sum _{i=1}^{\infty } \delta _{w^i(t)}\). If \( I_{R, T }({{\textbf {w}}}) < \infty \) for all \( R,T \in {\mathbb {N}}\), then \( {\textsf {w}} \) is an \( {\textsf {S}}\)-valued continuous path. Conversely, if \( I_{R, T }({{\textbf {w}}}) = \infty \) for some R or \(T \in {\mathbb {N}}\), then \( {\textsf {w}} \) is not necessarily an \( {\textsf {S}}\)-valued continuous path. See [28, Remark 3.10].
Example 3.1
We present the ISDE of \( ( X, {{\textbf {Y}}} )\) for the Ginibre interacting Brownian motion. Using (1.1) and (3.16), we see that X and \( {{\textbf {Y}}}=(Y^i)_{i\in {\mathbb {N}}}\) satisfy
Here, \( {\tilde{B}}_{t}^i = ({1}/{\sqrt{2}}) (B^{i+1}_{t} - B^{1}_{t}) \). \( \{ {\tilde{B}}^i \}_{i\in {\mathbb {N}}} \) are not independent and each \( {\tilde{B}}^i \) is equivalent in law to the standard Brownian motion in \( {\mathbb {R}} ^2\). The second equation above is self-contained as an equation of \( {{\textbf {Y}}}\). We see that the unlabeled process \( {\textsf {Y}}\) of \( {{\textbf {Y}}} \) is a diffusion process with invariant probability measure \( \mu _{\textrm{Gin}, 0 }\) in Sect. 3.3. This property is critical in the Kipnis–Varadhan theory to the tagged particle problem [4, 14, 19].
Although \( ( X, {{\textbf {Y}}} )\) is a diffusion with state space \({\mathbb {R}} ^d \times ({\mathbb {R}} ^d) ^{{\mathbb {N}} } \), there exists no associated Dirichlet space. Indeed, suitable invariant measures are lacking for \( ( X, {{\textbf {Y}}} )\). In contrast, \( ( X, {\textsf {Y}})\) is a diffusion with an invariant measure. As a result, it has the associated Dirichlet space. This fact is important in the analysis of the Dirichlet form version of the Kipnis–Varadhan theory in [19]. We shall specify the Dirichlet form associated with \( ( X, {\textsf {Y}})\).
Let \( D^{\textrm{trn}} \) be the generator of the translation operator given by (1.34). Let
We introduce the bilinear form \( ({\mathscr {E}}^{X{\textsf {Y}}}, {\mathscr {D}}_{\bullet }^{X{\textsf {Y}}})\) such that
Here, \( {\mathscr {D}}^{\textrm{trn}}\) is the domain of \( D^{\textrm{trn}} \): i.e., \( {\mathscr {D}}^{\textrm{trn}}\) is the set of functions for which the limit in (1.34) exists for all \( {\textsf {s}}\in {\textsf {S}}\) and \( 1 \le p \le d \). We define \( {\mathscr {E}}^{X{\textsf {Y}},\perp } \) through (3.20) by replacing \( {\mathbb {D}}\) with \( {\mathbb {D}}^{\perp }\). We define \( {\mathscr {D}}_{\circ }^{X{\textsf {Y}}} \) by replacing \( {\mathscr {D}}_{\bullet }\) with \( {\mathscr {D}}_{\circ }\) in \( {\mathscr {D}}_{\bullet }^{X{\textsf {Y}}}\). We also set \( {\mathscr {D}}_{\bullet }^{X{\textsf {Y}}, \perp } \) by replacing \( ({\mathscr {D}}_{\bullet }, {\mathscr {E}}^{X{\textsf {Y}}}) \) with \( ({\mathscr {D}}_{\bullet }^{\perp }, {\mathscr {E}}^{X{\textsf {Y}},\perp }) \) in \( {\mathscr {D}}_{\bullet }^{X{\textsf {Y}}}\).
Lemma 3.4
Make the same assumptions as Lemma 3.3. Then the following hold.
(1) \( ( {\mathscr {E}}^{X{\textsf {Y}},\perp }, {\mathscr {D}}_{\bullet }^{X{\textsf {Y}}, \perp } )\), \( ({\mathscr {E}}^{X{\textsf {Y}}}, {\mathscr {D}}_{\bullet }^{X{\textsf {Y}}})\), and \( ({\mathscr {E}}^{X{\textsf {Y}}}, {\mathscr {D}}_{\circ }^{X{\textsf {Y}}} )\) are closable on \( L ^2 (dx \times \mu _{0}) \).
(2) Let \( ({\mathscr {E}}^{X{\textsf {Y}},\perp }, {\mathscr {D}}^{X{\textsf {Y}},\perp } ) \), \( ({\mathscr {E}}^{X{\textsf {Y}}}, \underline{{\mathscr {D}}}^{X{\textsf {Y}}})\) and \( ({\mathscr {E}}^{X{\textsf {Y}}}, \overline{{\mathscr {D}}}^{X{\textsf {Y}}})\) be the closures of the closable forms in (1) on \( L ^2 (dx \times \mu _{0}) \), respectively. Then,
(3) The diffusion \( ( X, {\textsf {Y}})\) is associated with \( ({\mathscr {E}}^{X{\textsf {Y}},\perp }, {\mathscr {D}}^{X{\textsf {Y}},\perp }) \) on \( L^2 ( dx \times \mu _{0}) \), where X and \( {\textsf {Y}}\) are as in (3.16) and (3.17), respectively.
Proof
Let \( \{ \vartheta _x \}_{x\in {\mathbb {R}} ^d}\) be as in (1.3). Let \( \iota \) be the transformation on \( {\mathbb {R}} ^d \times {\textsf {S}}\) defined by \( \iota ( x, {\textsf {s}}) = ( x, \vartheta _x ({\textsf {s}}))\). Using this and \( (\) A1 \()\), we deduce that
Hence, \( \iota \) is the unitary transformation between \( L^{2}(\mu ^{[1]})\) and \( L ^2 (dx \times \mu _{0}) \) such that
Recall that \( \vartheta _x ({\textsf {s}}) = \sum _i \delta _{s_i - x} \) by (1.3). Hence
Hence from (3.19), (3.24), and \(\iota ( x, {\textsf {s}}) = ( x, \vartheta _x ({\textsf {s}})) \), we deduce that
From (3.20), (3.22), and (3.25), we see the isometry of the bilinear forms such that
Using (3.23) and (3.26) together with Lemma 3.2, we obtain (1).
Combining (3.11) and (3.26), we obtain (2) immediately.
We regard \( \iota \) as the transformation of \( C([0,\infty ); {\mathbb {R}} ^d \times {\textsf {S}})\), denoted by the same symbol \( \iota \), such that \( \iota ( X^1, {\textsf {X}}^{1 \diamond } ) = \{ \iota ( X _t^1, {\textsf {X}}_t^{1 \diamond } ) \}_{t\in [0,\infty )}\). Then
Using Lemma 3.3, (3.23), (3.26), and (3.27), we obtain (3). \(\square \)
3.3 Environment processes
Let \( {\textsf {Y}}\) be the environment process given by (3.17). Note that \( {\textsf {Y}}\) itself is a diffusion. Hence, we specify the Dirichlet form associated with \( {\textsf {Y}}\).
Let \( D^{\textrm{trn}} \), \( {\mathbb {D}}\), and \( {\mathbb {D}}^{\perp }\) be as in (1.35), (2.4) and (2.32), respectively.
Let \( ( {\mathscr {E}}^{{\textsf {Y}},\perp }, \ {\mathscr {D}}_{\bullet }^{{\textsf {Y}}, \perp } ) \) be the bilinear form defined by
Let \( ({\mathscr {E}}^{{\textsf {Y}}}, {\mathscr {D}}_{\bullet }^{{\textsf {Y}}})\) and \( ({\mathscr {E}}^{{\textsf {Y}}}, {\mathscr {D}}_{\circ }^{{\textsf {Y}}})\) be the bilinear forms defined by
For a random variable Z and a probability measure \( \nu \), we write \( Z {\mathop {=}\limits ^{\textrm{law}}}\nu \) if the law of Z coincides with \( \nu \).
Lemma 3.5
Make the same assumptions as Lemma 3.3. Then the following hold.
(1) \( ({\mathscr {E}}^{{\textsf {Y}}, \perp },{\mathscr {D}}_{\bullet }^{{\textsf {Y}},\perp } ) \), \( ({\mathscr {E}}^{{\textsf {Y}}},{\mathscr {D}}_{\bullet }^{{\textsf {Y}}})\), and \( ({\mathscr {E}}^{{\textsf {Y}}},{\mathscr {D}}_{\circ }^{{\textsf {Y}}})\) are closable on \( L^{2}(\mu _{0})\).
(2) Let \( ({\mathscr {E}}^{{\textsf {Y}}, \perp }, {\mathscr {D}}^{{\textsf {Y}}, \perp } ) \), \( ({\mathscr {E}}^{{\textsf {Y}}}, \underline{{\mathscr {D}}}^{{\textsf {Y}}}) \), and \( ({\mathscr {E}}^{{\textsf {Y}}}, \overline{{\mathscr {D}}}^{{\textsf {Y}}}) \) be the closures of the closable forms in (1) on \( L^{2}(\mu _{0})\), respectively. Then,
(3) Let \( ( X, {\textsf {Y}})\) be as in (3.16) and (3.17). Suppose that \( ( X _0, {\textsf {Y}}_0) {\mathop {=}\limits ^{\textrm{law}}}\zeta \times f d\mu _{0}\), where \( \zeta \) is a probability measure on \( {\mathbb {R}} ^d\) and \( 0 \le f \in {\mathscr {D}}_{\bullet }\) such that \( \int _{{\textsf {S}}} f d\mu _{0}=1\). The distribution of \( {\textsf {Y}}\) in \( ( X, {\textsf {Y}})\) is then the same as that of the diffusion \( {\textsf {Y}}\) associated with \( ({\mathscr {E}}^{{\textsf {Y}}, \perp }, {\mathscr {D}}^{{\textsf {Y}}, \perp } ) \) such that \( {\textsf {Y}}_0 {\mathop {=}\limits ^{\textrm{law}}}f d\mu _{0}\).
Proof
For \( \varphi \in C_0^{\infty }({\mathbb {R}}^d)\) and \( f \in {\mathscr {D}}_{\bullet }^{{\textsf {Y}},\perp } \), we see
Indeed, (3.31) is a straightforward calculation and (3.32) follows from the following.
Integrating (3.33) over \( {\mathbb {R}}^d \times {\textsf {S}}\) with respect to \( dx \times \mu _{0}\) and using
we obtain (3.32).
Let \( \Vert \varphi \Vert _{L^2(\mu _{0})} \ne 0 \). Let \( \{ f_n \} \) be an \( {\mathscr {E}}^{{\textsf {Y}},\perp }\)-Cauchy sequence in \( {\mathscr {D}}_{\bullet }^{{\textsf {Y}},\perp }\) such that \( \lim _{n\rightarrow \infty } \Vert f_n \Vert _{L^2 (\mu _{0})} = 0 \). Then from (3.31) and (3.32), we see that \( \{\varphi \otimes f_n \}\) is an \( {\mathscr {E}}^{X{\textsf {Y}},\perp }\)-Cauchy sequence such that
By Lemma 3.4(1), \( ( {\mathscr {E}}^{X{\textsf {Y}},\perp }, {\mathscr {D}}_{\bullet }^{{\textsf {Y}},\perp } )\) is closable on \( L ^2 (dx \times \mu _{0}) \). Hence we deduce
From (3.32), (3.34), \( \Vert \varphi \Vert _{L^2(dx )} \ne 0 \), and \( \lim _{n\rightarrow \infty } \Vert f_n \Vert _{L^2 (\mu _{0})} = 0 \), we deduce
as \( n \rightarrow \infty \). This implies the first claim in (1). The proof of the second and third claim in (1) is same. Hence, we omit it. Claim (2) follows from Lemmas 3.4(2) and (3.32). Claim (3) follows from Lemmas 3.4(3), (3.31), and (3.32). \(\square \)
4 Vanishing the self-diffusion matrix
In Sect. 4, we present a sufficient condition of vanishing the self-diffusion matrix. The goal of Sect. 4 is to prove Lemma 4.4, which is an “in \( \mu _0\)-measure” version of Theorem 1.4 (2).
In Sect. 4, \( \mu \) is a random point field on \( {\mathbb {R}} ^d\) satisfying \( (\) A1 \()\)–\( (\) A4 \()\).
Let \( ({\mathscr {E}}^{{\textsf {Y}}},{\mathscr {D}}_{\bullet }^{{\textsf {Y}}})\) be the bilinear form defined by (3.29). We set
From this, we have a decomposition of the bilinear form such that
Using (4.1) and the obvious inequalities \( {\mathscr {E}}^{{\textsf {Y}}, i }(f,f) \le {\mathscr {E}}^{{\textsf {Y}}}(f,f) \), we extend the domain of \( {\mathscr {E}}^{{\textsf {Y}}, i }\) from \( {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\) to \( \underline{{\mathscr {D}}}^{{\textsf {Y}}}\), where \( i=1,2\). Hence, (4.2) yields
Because of Lemma 3.5, \( ( {\mathscr {E}}^{{\textsf {Y}}}, {\mathscr {D}}_{\bullet }^{{\textsf {Y}}})\) is closable on \( L^{2}(\mu _{0})\). Meanwhile, each of \( ({\mathscr {E}}^{{\textsf {Y}},1},{\mathscr {D}}_{\bullet }^{{\textsf {Y}}})\) and \( ({\mathscr {E}}^{{\textsf {Y}},2},{\mathscr {D}}_{\bullet }^{{\textsf {Y}}})\) is not necessarily closable on \( L^{2}(\mu _{0})\). Still, (4.3) makes sense for \( f, g \in \underline{{\mathscr {D}}}^{{\textsf {Y}}}\).
\( \underline{{\mathscr {D}}}^{{\textsf {Y}}}\) is a Hilbert space with inner product \( {\mathscr {E}}^{{\textsf {Y}}}+ (\cdot , * )_{L^{2}(\mu _{0})}\) and \( \underline{{\mathscr {D}}}^{{\textsf {Y}}}\) is a pre-Hilbert space with non-negative bilinear form \( {\mathscr {E}}^{{\textsf {Y}}}\). Let \( \sim \) be the equivalence relation on \( \underline{{\mathscr {D}}}^{{\textsf {Y}}}\) such that \( f \sim g \) if and only if \( {\mathscr {E}}^{{\textsf {Y}}}(f-g,f-g) = 0 \). The quotient space \( \underline{{\mathscr {D}}}^{{\textsf {Y}}}/ \sim \) is a pre-Hilbert space with inner product \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}}}\) such that
where \( {\tilde{f}}= f/\sim \) and \( {\tilde{g}} = g /\sim \). The completion \( {\tilde{{\mathscr {D}}}}^{{\textsf {Y}}}\) of \( \underline{{\mathscr {D}}}^{{\textsf {Y}}}/ \sim \) is then a Hilbert space with inner product \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}}}\).
Let \( D^{\textrm{trn}} _{ p } \) be as defined in (1.34). For \( 1 \le p \le d \) and \( g \in {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\), we set
By the Schwartz inequality, (4.2), and (4.4), we obtain for any \( g \in {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\)
From (4.6), we regard \( F_{ p }\) as a bounded linear functional on \( \underline{{\mathscr {D}}}^{{\textsf {Y}}}\) and \( {\tilde{{\mathscr {D}}}}^{{\textsf {Y}}}\), and we denote it by the same symbol \( F_{ p }\).
Lemma 4.1
For \( 1 \le p \le d \), there exists a unique solution \( \psi _{ p }\in {\tilde{{\mathscr {D}}}}^{{\textsf {Y}}}\) of the equation
Proof
Because we regard \( F_{ p } \) as a bounded linear functional of the Hilbert space \( {\tilde{{\mathscr {D}}}}^{{\textsf {Y}}}\) with inner products \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}}}\), Lemma 4.1 is obvious from the Riesz theorem. \(\square \)
We consider a resolvent equation. For each \( \lambda > 0 \) and \( 1 \le p \le d \), let \( \psi _{\lambda , p }\in \underline{{\mathscr {D}}}^{{\textsf {Y}}}\) be the unique solution of the equation such that for any \( g \in \underline{{\mathscr {D}}}^{{\textsf {Y}}}\)
Lemma 4.2
For each \( 1 \le p \le d \), the following hold.
(1) \( \{ \psi _{\lambda , p }\}_{\lambda > 0 }\) is an \( {\mathscr {E}}^{{\textsf {Y}}}\)-Cauchy sequence in \( \underline{{\mathscr {D}}}^{{\textsf {Y}}}\) satisfying
(2) \( \{ {\tilde{\psi }} _{\lambda , p }\}_{\lambda > 0 }\) is an \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}}}\)-Cauchy sequence in \( {\tilde{{\mathscr {D}}}}^{{\textsf {Y}}}\) satisfying
Here, \( \psi _{ p }\in {\tilde{{\mathscr {D}}}}^{{\textsf {Y}}}\) is the limit of \( \{ {\tilde{\psi }} _{\lambda , p }\}_{\lambda > 0 }\) in \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}}}\).
Proof
Lemma 4.2 follows from the standard argument; see [14, 26]. \(\square \)
Lemma 4.3
Assume \( (\) A7 \()\). Then, for \( 1 \le p \le d \),
In particular, \( \alpha _{p,p} = 0 \) for \( 1 \le p \le d \).
Proof
Let \( g \in {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\). From (4.1) and (4.3), we have
Here we used (1.36) and the Cauchy–Schwartz inequality to the last line. Hence
Because \( \{ \chi _{ L , p }\} \) is an \( {\mathscr {E}}^{{\textsf {Y}}}\)-Cauchy sequence, \( \{ {\tilde{\chi }}_{ L , p }\} \) is an \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}}}\)-Cauchy sequence in \( {\tilde{{\mathscr {D}}}}^{{\textsf {Y}}}\) by (4.4). By (4.4) and (4.12), \( \{ {\tilde{\chi }}_{ L , p }\} \) is a weak convergent sequence in the Hilbert space \( {\tilde{{\mathscr {D}}}}^{{\textsf {Y}}}\). By (4.5) and Lemma 4.1, \( \psi _{ p }\) is the limit of \( \{ {\tilde{\chi }}_{ L , p }\} \). Hence,
We write \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}},1}( f ) = {\tilde{{\mathscr {E}}}}^{{\textsf {Y}},1}(f,f)\). Then,
Taking \( L \rightarrow \infty \) and then \( \lambda \rightarrow 0 \) in the last line, each term vanishes by (4.10), (4.13), and (1.36). Hence, we find that the first term in (4.11) converges to zero.
We calculate the second term in (4.11). We write \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}},2}( f ) = {\tilde{{\mathscr {E}}}}^{{\textsf {Y}},2}(f,f)\). Then,
From (4.10), (4.13), (1.36), and \( {\tilde{{\mathscr {E}}}}^{{\textsf {Y}},2}( {\tilde{\chi }}_{ L , p }) = {\mathscr {E}}^{{\textsf {Y}},2}(\chi _{ L , p })\), the last line converges to zero as \( L \rightarrow \infty \) and then \( \lambda \rightarrow 0 \). Hence, the second term in (4.11) converges to zero. Collecting these, we have (4.11). From \( \psi _{\lambda , p }\in {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\), we obtain the second claim. \(\square \)
Using Lemma 4.3 and [19], we have Lemma 4.4 below. The result follows from the Dirichlet form version of the Kipnis–Varadhan theory. We address the reader to Subsection 4.2 of the preprint “Ginibre interacting Brownian motion in infinite dimensions is sub-diffusive" on arXiv:2109.14833v3 [math.PR] for a simplified proof of Lemma 4.4.
Lemma 4.4
Assume (A1)–(A4) and (A7). Let \( ( X, {\textsf {Y}})\) be as in Lemma 3.4. Then,
5 Dual reduced Palm measures and mean-rigid conditioning
5.1 Dual reduced Palm measures
In Sect. 5.1, we introduce the concept of the dual reduced Palm measures. We construct the translation invariant dual reduced Palm measures in Lemmas 5.6 and 5.7.
Let \( {\textsf {S}}_{ m }= \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}({\mathbb {R}} ^d) = m \} \) for \( m \in \{ 0 \} \cup {\mathbb {N}} \) as before. Let \( {\widetilde{\mu }}_m= {\check{\mu }}^m \circ {\mathfrak {u}} ^{-1}\) for \( m \in {\mathbb {N}}\), where \( {\check{\mu }}^m \) are defined by (1.16) and \( {\mathfrak {u}} ({{\textbf {x}}}) = \sum _i\delta _{x_i}\) for \( {{\textbf {x}}} = (x_i)_i\). Note that \( {\widetilde{\mu }}_m\) are measures on \( {\textsf {S}}_{ m }\) but usually not probability measures. Let \( {\widetilde{\mu }}= \sum _{m=0}^{\infty } {\widetilde{\mu }}_m\) be the measure on \( \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}({\mathbb {R}} ^d) < \infty \} \), where \( {\widetilde{\mu }}_0 = \delta _{{\textsf {0}}}\) is degenerated to the zero measure \( {\textsf {0}}\). Let \( \mu (\cdot \Vert {\textsf {x}})\), \( {\textsf {x}}\in {\textsf {S}}_{ m }\), be the reduced Palm measure defined by (1.17).
Recall that \( {\textsf {S}}_{R}^m = \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {s}}(S_{R})= m \} \). Note that \( {\textsf {S}}_{ m }\cap {\textsf {S}}_{R}^m \subset \pi _{R}({\textsf {S}})\) and that \( {\textsf {s}}(S_{R}^c) = 0 \) for \( {\textsf {s}}\in {\textsf {S}}_{ m }\cap {\textsf {S}}_{R}^m \). We repeatedly use this fact in Sect. 5.1.
Lemma 5.1
Assume (A2), (A3), and (1.21). Then the following hold.
for each \( {\textsf {x}}\in {\textsf {S}}_{ m }\cap {\textsf {S}}_{R}^m \). Here, \( \cdot \approx * \) means \( \cdot \) and \( * \) are mutually absolutely continuous.
Proof
From \( (\) A2 \()\), we have the m-point correlation function \( \rho ^m \) and the density function \( \sigma _R^m \) on \( S_{R}\) of \( \mu \) such that
where \( {{\textbf {y}}}_{n-m}= (y_1,\ldots ,y_{n-m}) \in ({\mathbb {R}} ^d)^{n-m} \). From \( (\) A3 \()\), (1.9) and (1.7) hold. Let \( \mu _{ R, k , {\textsf {s}}}^n \) be the regular conditional measures given by (1.8). Hence, for \( \mu \)-a.s. \( {\textsf {s}}\), the Radon-Nikodym density \( \sigma _{ R, k , {\textsf {s}}}^n = d\mu _{ R, k , {\textsf {s}}}^n /d\Lambda _{R}^n \) and the n-point correlation function \( \rho _{ R, k , {\textsf {s}}}^n \) of \( \mu _{ R, k , {\textsf {s}}}^n \) exists. Furthermore, there exists a constant \( c_{5.1} (R, k , {\textsf {s}}, n ) > 0 \) such that for any \( {{\textbf {x}}}_n=(x_1,\ldots ,x_n) \in S_{R}^n \)
Hence, (5.3) also holds for \( \sigma _{ R, k , {\textsf {s}}}^n \) and \( \rho _{ R, k , {\textsf {s}}}^n \).
From \( (\) A3 \()\), \( \Psi \) is locally bounded from below, and \(\Psi ( x ) < \infty \) for \( x \ne 0 \). Using these and that \( \sigma _{R, k , {\textsf {s}}}^n ({{\textbf {x}}}_{n}) \) is symmetric in \( {{\textbf {x}}}_n\), we see for \( \mu \)-a.s. \( {\textsf {s}}\)
Hence using this and (5.3) for \( \sigma _{ R, k , {\textsf {s}}}^n \) and \( \rho _{ R, k , {\textsf {s}}}^n \), we deduce for \( \mu \)-a.s. \( {\textsf {s}}\)
Integrating (5.4) for \( {\textsf {s}}\) with respect to \( \mu \) and taking \( k \rightarrow \infty \), we obtain from (1.8)
We deduce from (5.3) and (5.5) that \( \mu \circ \pi _{R}^{-1}\approx {\widetilde{\mu }}\circ \pi _{R}^{-1}\), which yields (5.1).
For \( {\textsf {A}} \in {\mathcal {B}}({\textsf {S}}) \), we set \( {\mathbb {A}}_{R}^c= (\pi _{R}^{c})^{-1}( \pi _{R}^{c}({\textsf {A}})) \). Then for \( {\textsf {x}}\in {\textsf {S}}_{ m }\cap {\textsf {S}}_{R}^m \),
This yields (5.2). \(\square \)
We note that \( {\textsf {S}}\) is homeomorphic to a complete separable metric space. Hence, for \( \mu \)-a.s. \( {\textsf {y}}\), we define a random point field \( \mu _{R, {\textsf {y}}}\) on \( S_{R}\) by the regular conditional probability such that
Note that \( {\textsf {A}} = \pi _{R}({\textsf {A}}) + \pi _{R}^{c}({\textsf {A}})\) and that \( \mu _{R, {\textsf {y}}}= \mu _{R, \pi _{R}^{c}({\textsf {y}}) }\). Then we have
In Lemma 5.2, \( \mu \) is irreducibly k-decomposable with \( \{ {\textsf {S}}_{ m } ^{\diamond }\}_{m=0}^{ k }\) in the sense of Definition 1.2. From Remark 1.3(2), \( \mu ({\textsf {S}}_{ m } ^{\diamond }) = 0 \) for \( m \ge 1 \). Thus, it is not obvious that \( \mu _{R, {\textsf {y}}}\) exists for \( {\textsf {y}}\in {\textsf {S}}_{ m } ^{\diamond }\), \( m \ge 1 \). We resolve this in Lemma 5.2.
Let \( \mu _m ^{\diamond }\) be a random point field such that
Note that \( {\textsf {S}}_{R}^m + \pi _{R}({\textsf {y}}) = \{ {\textsf {s}}+ \pi _{R}({\textsf {y}}); {\textsf {s}}\in {\textsf {S}}_{R}^m \} \subsetneqq {\textsf {S}}_{R}^m \) if \( {\textsf {y}}(S_{R}) \ge 1 \). We set
If \( \mu \) is k-decomposable, then \( {\textsf {S}}_{ 0 } ^{\diamond } ({\textsf {y}})- {\textsf {y}}\subset {\textsf {S}}_{ m }\) for \( {\textsf {y}}\in {\textsf {S}}_{ m } ^{\diamond }\) from (1.19).
Lemma 5.2
Assume (A2), (A3), and (1.21). Let \( \mu \) be irreducibly k-decomposable with \( \{ {\textsf {S}}_{ m } ^{\diamond }\}_{m=0}^{ k }\) in the sense of Definition 1.2. Then for \( 1 \le m \le k \),
Proof
Using (5.8) and (5.2), we have for all \( {\textsf {x}}\in {\textsf {S}}_{ m }\) such that \( {\textsf {x}}(S_{R}) = m \)
From \( \mu _{R, {\textsf {y}}}= \mu _{R, \pi _{R}^{c}({\textsf {y}})}\), \( \mu _{R, {\textsf {y}}}\) exists for \( \mu \circ (\pi _{R}^{c})^{-1}\)-a.s. \( {\textsf {y}}\). Combining this with (5.12), we obtain (5.10). From (1.19), \( {\textsf {S}}_{ 0 } ^{\diamond } ({\textsf {y}})- {\textsf {y}}\subset {\textsf {S}}_{ m }\) for \( {\textsf {y}}\in {\textsf {S}}_{ m } ^{\diamond }\). Note that we condition \( \mu \) on \( S_{R}^c\) as \( \mu _{R, {\textsf {y}}}= \mu (\cdot \vert \pi _{R}^{c}({\textsf {s}}) = \pi _{R}^{c}({\textsf {y}})) \). Hence, \( {\textsf {S}}_{ 0 } ^{\diamond } ({\textsf {y}})- {\textsf {y}}\subset {\textsf {S}}_{R}^m \) for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\in {\textsf {S}}_{ m } ^{\diamond }\). We have thus obtained (5.11). \(\square \)
By (5.10), \( \mu _{R, {\textsf {y}}}\) exists for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\) for \( 1 \le m \le k \). Let \( \rho _{R, {\textsf {y}}}^m \) be the m-point correlation function of \( \mu _{R, {\textsf {y}}}\). Then \( \rho _{R, {\textsf {y}}}^m \) exists on \( S_{R}\) and satisfies
From (5.13), the mth factorial moment measure \( {\widetilde{\mu }}_{R, {\textsf {y}}}^m \) of \( \mu _{R, {\textsf {y}}}\) exists on \( S_{R}\) for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\). Hence, the reduced Palm measure \( \mu _{R, {\textsf {y}}}(\cdot \Vert {\textsf {x}}) \) of \( \mu _{R, {\textsf {y}}}\) exists for \( {\widetilde{\mu }}_{R, {\textsf {y}}}^m \)-a.e. \( {\textsf {x}}\) for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\). Let \( {\textsf {S}}_{R}\) be the configuration space over \( S_{R}\). We regard \( {\textsf {S}}_{R}^m \) as a subset of \( {\textsf {S}}_{R}\). Then by definition, for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\),
for all \({\textsf {A}} \in {\mathcal {B}}({\textsf {S}}_{R}^m )\), and \( {\textsf {B}} \in {\mathcal {B}}({\textsf {S}}_{R}) \), and, equivalently, for \( {\widetilde{\mu }}_{R, {\textsf {y}}}^m \)-a.e. \( {\textsf {x}}\),
Originally, \( \mu _{R, {\textsf {y}}}(\cdot \Vert {\textsf {x}}) \) is the random point field on \( S_{R}\). We can regard \( \mu _{R, {\textsf {y}}}( \cdot \Vert {\textsf {x}})\) as a random point field on \( {\mathbb {R}} ^d\) by taking
We denote this extension by the same symbol \( \mu _{R, {\textsf {y}}}(\cdot \Vert {\textsf {x}}) \). By definition, \( \mu _{R, {\textsf {y}}}(\cdot \Vert {\textsf {x}}) \) degenerates into \( \pi _{R}^{c}({\textsf {y}})\) outside \( S_{R}\). In the following, \( \mu _{R, {\textsf {y}}}(\cdot \Vert {\textsf {x}}) \) is thus a random point field on \( {\mathbb {R}} ^d\). Let \( {\textsf {S}}({\textsf {y}})= \{ {\textsf {s}}\in {\textsf {S}}; {\textsf {y}}\prec {\textsf {s}}\} \) and \( {\textsf {S}}_{ 0 } ^{\diamond } ({\textsf {y}})= {\textsf {S}}({\textsf {y}})\cap {\textsf {S}}_{ 0 } ^{\diamond }\).
Lemma 5.3
Make the same assumptions as Lemma 5.2. Then for \( 1 \le m \le k \)
Proof
The reduced Palm measure \( \mu _{R, {\textsf {y}}}(\cdot \Vert {\textsf {x}}) \) satisfies for \( 1 \le m \le k \)
Similarly as (5.5), we deduce \( \sigma _{R, {\textsf {y}}}^m ({{\textbf {x}}}_{m}) = 0 \Longleftrightarrow \rho _{R, {\textsf {y}}}^m ({{\textbf {x}}}_m) = 0 \). Hence,
Here \( {\widetilde{\mu }}_{R, {\textsf {y}}} = \sum _{ n =1}^{\infty } {\widetilde{\mu }}_{R, {\textsf {y}}}^{ n }\). Taking \( {\textsf {x}}= \pi _{R}({\textsf {y}})\) in (5.17) and using (5.17) and (5.18), we see that the reduced Palm measure \( \mu _{R, {\textsf {y}}}(\cdot \Vert \pi _{R}({\textsf {y}})) \) satisfies
Using (5.14), we can replace \( {\textsf {S}}({\textsf {y}})\) in (5.19) by \( {\textsf {S}}_{ 0 } ^{\diamond } ({\textsf {y}})={\textsf {S}}({\textsf {y}})\cap {\textsf {S}}_{ 0 } ^{\diamond }\). Thus we obtain
Note that \( \mu _{R, {\textsf {y}}}= \mu _{R, \pi _{R}^{c}({\textsf {y}})}\). By (5.15), all removed particles of \( \mu _{R, {\textsf {y}}}( \cdot \Vert {\textsf {x}}) \circ (\pi _{R}^{c})^{-1} \) are in \( S_{R}\), that is, \( {\textsf {x}}(S_{R}) = m \). Hence, from (1.19) and (5.15), we have for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\)
Hence from (5.20) and (5.21), we obtain
This implies (5.16). \(\square \)
For \( R\in {\mathbb {N}}\) and \( {\textsf {y}}\in {\textsf {S}}\), let \( {\mathscr {I}}_{R,{\textsf {y}}} \) and \( {\mathscr {I}}_{\infty ,{\textsf {y}}} \) be the \( \sigma \)-fields such that
Lemma 5.4
Make the same assumptions as Lemma 5.2. Let \( \mu (\cdot \vert {\mathscr {I}}_{R,{\textsf {y}}} ) \) be the regular conditional probability measure with respect to \( {\mathscr {I}}_{R,{\textsf {y}}} \) for \( R\in {\mathbb {N}}\cup \{ \infty \} \) and \( {\textsf {y}}\in {\textsf {S}}_{ m } ^{\diamond }\). Then the following hold for \( 1 \le m \le k \) and \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\).
(1) For each \( {\textsf {B}} \in {\mathcal {B}}({\textsf {S}}) \),
(2) For \( \mu \)-a.s. \( {\textsf {s}}\),
(3) Let \( {\textsf {B}} \in {\mathcal {B}}({\textsf {S}}) \) be such that \( {\textsf {B}} \subset {\textsf {S}}_{ m }\cap {\textsf {S}}_{R}^m \) for some \( R\in {\mathbb {N}}\). Then \( \mu ({\textsf {B}} \vert {\mathscr {I}}_{\infty ,{\textsf {y}}} ) ({\textsf {s}}) \) is constant and \( \mu ( {\textsf {S}}_{ m }+ {\textsf {y}} \vert {\mathscr {I}}_{\infty ,{\textsf {y}}} ) ({\textsf {s}}) = 1 \) for \( \mu \)-a.s. \( {\textsf {s}}\).
Proof
We easily see that \( {\mathscr {I}}_{R,{\textsf {y}}} \supset {\mathscr {I}}_{R',{\textsf {y}}} \) for \( R\le R' \le \infty \) and that \( \{ \mu ({\textsf {B}} \vert {\mathscr {I}}_{R,{\textsf {y}}} ) \} \) is bounded in \( L^{2}(\mu )\). Hence using the martingale convergence theorem (cf. [29, I (2.4) Corollary]), we have (5.23), which implies (1).
From (1), it is easy to see that \(\{ \mu (\cdot \vert {\mathscr {I}}_{R,{\textsf {y}}} ) ({\textsf {s}}) \} _{ R\in {\mathbb {N}}} \) is tight for \( \mu \)-a.s. \( {\textsf {s}}\). Hence, we denote an arbitrary convergent subsequence by the same symbol \( \mu (\cdot \vert {\mathscr {I}}_{R,{\textsf {y}}} ) ({\textsf {s}}) \) and its limit by \( \mu ' (\cdot ) ({\textsf {s}}) \). Note that the measurable space \( ({\textsf {S}}, {\mathcal {B}}({\textsf {S}}) ) \) is countably determined, that is, any probability measures on \( ({\textsf {S}}, {\mathcal {B}}({\textsf {S}}) )\) are determined by a countable system of elements of \( {\mathcal {B}}({\textsf {S}}) \) (cf. [10, p.14]). Let \( \{ {\textsf {B}}_n \}_{n\in {\mathbb {N}}} \) be a countable system of subsets determines the probabilities on \( ({\textsf {S}}, {\mathcal {B}}({\textsf {S}}) )\). Then from (5.23), for \( \mu \)-a.s. \( {\textsf {s}}\),
Hence from (5.25), we deduce \( \mu ' (\cdot ) ({\textsf {s}}) = \mu (\cdot \vert {\mathscr {I}}_{\infty ,{\textsf {y}}} ) ({\textsf {s}}) \) for \( \mu \)-a.s. \( {\textsf {s}}\). This implies (2).
Let \( T _1 = S_{1} \) and \( T _{ Q}= S_{Q}\backslash S_{{Q-1}} \) for \( Q\ge 2 \). For m and \( R\), let
Then \( {\textsf {T}}_{R,{\textsf {y}}}^{m , {{\textbf {n}}}}\cap {\textsf {T}}_{R,{\textsf {y}}}^{m , {{\textbf {n}}}'}= \emptyset \) for \( {{\textbf {n}}}\ne {{\textbf {n}}}'\). Thus, \( \{ {\textsf {T}}_{R,{\textsf {y}}}^{m , {{\textbf {n}}}}\}_{{{\textbf {n}}}\in {{\textbf {N}}}_{R}^m } \) is a partition of \( {\textsf {U}}_{R,{\textsf {y}}}^m \). This together with \( {\textsf {B}} \subset {\textsf {S}}_{ m }\cap {\textsf {S}}_{R}^m \) yields \( {\textsf {B}} + {\textsf {y}}\subset {\textsf {U}}_{R,{\textsf {y}}}^m \) and
Using (5.22) and \( {\textsf {B}} \subset {\textsf {S}}_{ m }\cap {\textsf {S}}_{R}^m \), we have
From (5.26), (5.27), \( {\textsf {B}} + {\textsf {y}}\subset {\textsf {U}}_{R,{\textsf {y}}}^m \), and that \( \{ {\textsf {T}}_{R,{\textsf {y}}}^{m , {{\textbf {n}}}}\}_{{{\textbf {n}}}\in {{\textbf {N}}}_{R}^m } \) is a partition of \( {\textsf {U}}_{R,{\textsf {y}}}^m \), we see
Using this, (5.24), and that \( \{{\textsf {U}}_{R,{\textsf {y}}}^m \}_{R=1}^{\infty } \) is increasing, we see
From (5.28) and \( \mu (\cup _{R=1}^{\infty }{\textsf {U}}_{R,{\textsf {y}}}^m ) = 1 \), we obtain the first claim in (3). Using (5.14) and Lemma 5.3, we see for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\) and any \( R\in {\mathbb {N}}\),
From this, (5.28), and \( \mu (\cup _{R=1}^{\infty }{\textsf {U}}_{R,{\textsf {y}}}^m ) = 1 \), we get the second claim in (3). \(\square \)
We now construct the dual reduced Palm measure conditioned on infinite-many points \( {\textsf {y}}\in {\textsf {S}}_{ m } ^{\diamond }\). For \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\) and \( {\textsf {B}} \in {\mathcal {B}}({\textsf {S}}_{ m }) \), let
From Lemma 5.4(3), \( \mu ({\textsf {B}} + {\textsf {y}} \vert {\mathscr {I}}_{\infty ,{\textsf {y}}} ) ({\textsf {s}})\) is constant and \( \mu ({\textsf {S}}_{ m }+ {\textsf {y}} \vert {\mathscr {I}}_{\infty ,{\textsf {y}}} ) = 1 \) for \( \mu \)-a.s. \( {\textsf {s}}\). Thus \( \mu ( {\textsf {B}} \Vert {\textsf {y}}) \) is independent of \( {\textsf {s}}\) and satisfies \( \mu ( {\textsf {S}}_{ m }\Vert {\textsf {y}}) = 1\). Hence, we extend the domain of \( \mu ( \cdot \Vert {\textsf {y}}) \) to \( {\mathcal {B}}({\textsf {S}}) \) by \( \mu ( \cdot \Vert {\textsf {y}}) = \mu ( \cdot \cap {\textsf {S}}_{ m }\Vert {\textsf {y}})\).
We regard (5.30) and (5.31) in Lemma 5.5 as the dual relation to \( \mu ( {\textsf {S}}_{ m } ^{\diamond }\Vert {\textsf {y}}) = 1 \) for \( {\textsf {y}}\in {\textsf {S}}_{ m }\) in (1.20) for the original reduced Palm measure. This relation is a result of decomposability of \( \mu \). We call \( \mu ( \cdot \Vert {\textsf {y}}) \) the dual reduced Palm measure because \( \mu ( \cdot \Vert {\textsf {y}}) \) is conditioned at an element \( {\textsf {y}}\) of the dual set \( {\textsf {S}}_{ m } ^{\diamond }\) to \( {\textsf {S}}_{ m }\) and \( \mu ( \cdot \Vert {\textsf {y}}) \) is supported on \( {\textsf {S}}_{ m }\) such that \( \mu ( {\textsf {S}}_{ m }\Vert {\textsf {y}}) =1 \) for \( {\textsf {y}}\in {\textsf {S}}_{ m } ^{\diamond }\).
We convert the convergence of the Palm measures in Lemma 5.4 to that of the reduced Palm measures.
Lemma 5.5
Make the same assumptions as Lemma 5.2. Let \( 1 \le m \le k \).
(1) For \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\),
(2) \( \mu ( {\textsf {B}} \Vert \cdot ) \) is a \( {\mathcal {B}}({\textsf {S}})\)-measurable function for each \( {\textsf {B}} \in {\mathcal {B}}({\textsf {S}}) \).
(3) For \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\), \( \mu ( \cdot \Vert {\textsf {y}}) \) is a probability measure on \( ({\textsf {S}},{\mathcal {B}}({\textsf {S}}) ) \) such that
Proof
Because \( \mu ( \cdot \Vert {\textsf {y}}) = \mu ( \cdot \cap {\textsf {S}}_{ m }\Vert {\textsf {y}})\) by definition, we assume \( {\textsf {B}} \subset {\textsf {S}}_{ m }\). We see
Here the convergence takes place for \( \mu \)-a.s. \( {\textsf {s}}\) and in \( L^1(\mu )\). From (5.22)
From (5.32), (5.33), and \( {\textsf {B}} \subset {\textsf {S}}_{ m }\), we see (5.30). From (5.14) and (5.30), we obtain (2). From Lemma 5.4(3) and (5.29), we have (3). \(\square \)
Lemma 5.6
Make the same assumptions as Lemma 5.2. In addition, assume that \( \mu \) and \( {\textsf {S}}_{ 0 } ^{\diamond }\) are translation invariant. Then, for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\) and \( 1 \le m \le k \),
Proof
Because \( {\textsf {S}}_{ 0 } ^{\diamond }\) is translation invariant, we have
Then using (5.30), (5.34), and \( (\) A1 \()\), we have, for \( \mu _m ^{\diamond }\)-a.s. \( {\textsf {y}}\) and all \( \textrm{a}\in {\mathbb {R}} ^d\),
We have thus completed the proof of Lemma 5.6. \(\square \)
Lemma 5.7
Assume (A1)–(A3) and \( (\) A6 \()\). Then \( (\) A5 \()\) holds with \( \{ {\textsf {S}}_{ 0 } ^{\diamond }, {\textsf {S}}_{ 1 } ^{\diamond }\} \) such that both \( {\textsf {S}}_{ 0 } ^{\diamond }\) and \({\textsf {S}}_{ 1 } ^{\diamond }\) are translation invariant.
Proof
Let \( {\textsf {K}}_{\epsilon }\subset {\textsf {S}}\) be an increasing sequence of compact sets such that \( \mu ({\textsf {K}}_{\epsilon }) > 1-\epsilon \) as \( \epsilon \downarrow 0 \). We set
Note that the translation operator \( \vartheta _x \) on \( {\textsf {S}}\) is a homeomorphism for each \( x \in {\mathbb {R}} ^d\) and that \( {\textsf {s}}\mapsto \vartheta _x ({\textsf {s}})\) is continuous in \( x \in {\mathbb {R}} ^d\) for each \( {\textsf {s}}\). Hence, \( (x,{\textsf {s}}) \mapsto \vartheta _x ({\textsf {s}}) \) is continuous.
Let \( R< \infty \). Because \( \{ |x|\le R\} \times {\textsf {K}}_{\epsilon }\) is compact and the map \( (x,{\textsf {s}}) \mapsto \vartheta _x ({\textsf {s}}) \) is continuous, we deduce that \( {\textsf {K}}^{*}_{R, \epsilon }\) is a compact set in \( {\textsf {S}}\). Hence, \( {\textsf {K}}^{*}_{\infty }\) is a Borel set because \( {\textsf {K}}^{*}_{\infty }\) is the increasing limit of \( {\textsf {K}}^{*}_{\infty , \epsilon }\) as \( \epsilon \downarrow 0 \) and \( {\textsf {K}}^{*}_{\infty , \epsilon }\) is the increasing limit of compact sets \( {\textsf {K}}^{*}_{R, \epsilon }\) as \( R\uparrow \infty \).
From \( \mu (\cup _{ 0< \epsilon < 1 } {\textsf {K}}_{\epsilon }) = 1 \) and \( \cup _{ 0< \epsilon < 1 } {\textsf {K}}_{\epsilon }\subset {\textsf {K}}^{*}_{\infty }\), we see \( \mu ({\textsf {K}}^{*}_{\infty }) = 1 \). By construction, \( {\textsf {K}}^{*}_{\infty , \epsilon }\) is translation invariant for each \( \epsilon \). Hence, \( {\textsf {K}}^{*}_{\infty }\) is translation invariant. Let \( {\textsf {x}}= \delta _x\) and \({\textsf {K}}^{*}_{\infty }({\textsf {x}})= \{ {\textsf {s}}\in {\textsf {K}}^{*}_{\infty }; {\textsf {x}}\prec {\textsf {s}}\} \). Then we have for \( {\widetilde{\mu }}_1\)-a.e. \( {\textsf {x}}\)
Recall that \( {\textsf {S}}_{1}= \{ {\textsf {x}}= \delta _x; x \in {\mathbb {R}}^d \} \). Because \( \mu \), \( {\textsf {K}}^{*}_{\infty }\), and \( {\textsf {S}}_{1}\) are translation invariant, we can refine the property above so that (5.35) holds for all \( {\textsf {x}}\in {\textsf {S}}_{1}\). Let
Then \( {\textsf {K}} _{\infty }^{*,1}\) is translation invariant. Because \( {\textsf {K}}^{*}_{\infty }({\textsf {x}})- {\textsf {x}}\subset {\textsf {K}} _{\infty }^{*,1}\) and \( \mu ({\textsf {K}}^{*}_{\infty }({\textsf {x}})- {\textsf {x}}\Vert {\textsf {x}}) = 1\) for all \( {\textsf {x}}\in {\textsf {S}}_{1}\), we obtain
Let \( {\textsf {S}}_{ 1 } ^{\diamond }= {\textsf {K}} _{\infty }^{*,1}\) and \( {\textsf {S}}_{ 0 } ^{\diamond }= ({\textsf {K}} _{\infty }^{*,1}+ {\textsf {S}}_{1}) \backslash {\textsf {K}} _{\infty }^{*,1}\). Then \( {\textsf {S}}_{ 0 } ^{\diamond }\cap {\textsf {S}}_{ 1 } ^{\diamond }= \emptyset \), and thus we obtain (1.18). (1.19) follows from \( {\textsf {S}}_{ 0 } ^{\diamond }= ({\textsf {K}} _{\infty }^{*,1}+ {\textsf {S}}_{1}) \backslash {\textsf {K}} _{\infty }^{*,1}\subset {\textsf {K}} _{\infty }^{*,1}+ {\textsf {S}}_{1}= {\textsf {S}}_{ 1 } ^{\diamond }+ {\textsf {S}}_{1}\).
From \( (\) A6 \()\)(1) and (5.37), \( \mu ( {\textsf {K}} _{\infty }^{*,1}) = 0 \). From this and \( \mu ({\textsf {K}}^{*}_{\infty }) = 1 \), we deduce
Hence, (1.20) follows from (5.37) and (5.38). By construction, \( {\textsf {S}}_{ 0 } ^{\diamond }\) and \( {\textsf {S}}_{ 1 } ^{\diamond }\) are translation invariant. Irreducibility follows from \( (\) A6 \()\)(2), which completes the proof. \(\square \)
5.2 Mean-rigid conditioning
In Sect. 5.2, we introduce the concept of the mean-rigid \( \sigma \)-field \( {\mathcal {G}}_{\infty } \). We define the functions \( \textrm{N}_{R}\) and \( \textrm{M} _{R}\) on \( {\textsf {S}}\) such that
Let \( T _1 = S_{1} \) and \( T _{R}= S_{R}\backslash S_{{ R-1}} \) for \( R\ge 2 \). Replacing \( S_{R}\) by \( T _{R}\) in (5.39), we define \( \textrm{N}_{ T _{R}}\) and \( \textrm{M}_{ T _{R}}\). For a function f, a \( \sigma \)-field \( {\mathcal {F}} \), and a random point field \( \mu \), we say f is \( {\mathcal {F}} \)-measurable for \( \mu \)-a.s. if a \( \mu \)-version of f is \( {\mathcal {F}} \)-measurable.
Definition 5.1
(1) A random point field \( \mu \) on \( {\mathbb {R}} ^d\) is said to be number-rigid if, for each \( R\in {\mathbb {N}} \), the function \( \textrm{N}_{R}({\textsf {s}}) \) (resp. \( \textrm{N}_{ T _{R}}\)) is \( \sigma [\pi _{R}^{c}] \)-measurable (resp. \( \sigma [\pi _{ T _{R}}^c]\)-measurable) for \( \mu \)-a.s.
(2) A random point field \( \mu \) on \( {\mathbb {R}} ^d\) is said to be mean-rigid if \( \mu \) is number-rigid and, for each \( R\in {\mathbb {N}} \), the function \( \textrm{M} _{R}({\textsf {s}}) \) (resp. \( \textrm{M}_{ T _{R}} ({\textsf {s}}) \)) is \( \sigma [\pi _{R}^{c}] \)-measurable (resp. \( \sigma [\pi _{ T _{R}}^c]\)-measurable) for \( \mu \)-a.s.
If \( \mu \) is mean-rigid, then, by definition, for each \( R\in {\mathbb {N}} \) and for \( \mu \)-a.s. \( {\textsf {s}}\), there exist \( a = a (\pi _{R}^{c}({\textsf {s}})) \), \( b = b (\pi _{R}^{c}({\textsf {s}}))\), \( a' = a' (\pi _{ T _{R}}^c ({\textsf {s}})) \), and \( b' = b' (\pi _{ T _{R}}^c ({\textsf {s}}))\) such that
In [2, 8], rigidity is posed for all Borel sets with Lebesgue-negligible boundary. Our concept of number and mean rigidity is slightly weaker than that in [2, 8]. We pose rigidity only for \( S_{R}\) and \( T _{R}\). This reduction is enough for our purpose.
Mean rigidity is a critical property for sub-diffusivity. The Ginibre random point field is number-rigid but not mean-rigid. Hence, we introduce the algorithm to transform a function on \( {\textsf {S}}\) with a number-rigid random point field to a function that is measurable concerning a mean-rigid \( \sigma \)-field.
For \( R\in {\mathbb {N}} \), let \( {\mathcal {G}}_{R} \) and \( {\mathcal {H}}_{R} \) be the sub \( \sigma \)-fields of \( {\mathcal {B}}({\textsf {S}}_{ 1 } ^{\diamond }) \) given by
We set the mean-rigid \( \sigma \)-field \( {\mathcal {G}}_{\infty } \) as follows.
Lemma 5.8
(1) \( {\mathcal {G}}_{R} \supset {\mathcal {G}}_{R+1} \) and \( {\mathcal {H}}_{R} \supset {\mathcal {G}}_{R} \) for each \( R\in {\mathbb {N}} \).
(2) \( \vartheta _{\textrm{a}} ({\mathcal {G}}_{\infty } ) = {\mathcal {G}}_{\infty } \) for all \( \textrm{a}\in {\mathbb {R}} ^d\), where \( \vartheta _{\textrm{a}} \) is the translation on \( {\textsf {S}}\) in (1.3).
Proof
Recall that \( T _{R+1}= S_{R+1}\backslash S_{R}\). Then we see \( \pi _{R}^c = \pi _{ T _{R+1}} + \pi _{R+ 1 }^c \). Hence,
This together with (5.40) yields \( {\mathcal {G}}_{R} \supset {\mathcal {G}}_{R+1} \).
From \( T _{R}= S_{R}\backslash S_{R-1 }\), we find \( \pi _{ T _{R}}^c = \pi _{R-1 } + \pi _{R}^{c}\). Hence, we see
This together with (5.40) yields \( {\mathcal {H}}_{R} \supset {\mathcal {G}}_{R} \). We thus obtain (1).
Without loss of generality, we set \( |\textrm{a}|< 1 \). Let \( S_{R}(\textrm{a}) =\vartheta _{\textrm{a}} (S_{R}) \). Then,
Replacing \( S_{R}\) with \( S_{R}(\textrm{a}) \) in (5.39), we define \( \textrm{N}_{R, a}\) and \( \textrm{M} _{R, a }\). Let \( \pi _{ R, \textrm{a}}^c = \pi _{ S_{R}(\textrm{a}) ^c}\). Then from (5.42), we find that
From (5.40), we find that \( \vartheta _{\textrm{a}} ({\mathcal {G}}_{R} ) = \sigma [\textrm{N}_{R, a}, \textrm{M} _{R, a }, \pi _{ R, \textrm{a}}^c ] \). From this and (5.43),
Using (5.40) and (5.44) and noting \( \bigcap _{R=1}^{\infty } \vartheta _{\textrm{a}} ({\mathcal {G}}_{R} ) = \vartheta _{\textrm{a}} ( \bigcap _{R=1}^{\infty } {\mathcal {G}}_{R} ) \), we obtain \( {\mathcal {G}}_{\infty } \supset \vartheta _{\textrm{a}} ({\mathcal {G}}_{\infty } ) \supset {\mathcal {G}}_{\infty } \), which yields (2). \(\square \)
Let \( \mu _{\textrm{a}} \) be the reduced Palm measure of \( \mu \) conditioned at \( \textrm{a}\in {\mathbb {R}} ^d\) as before. Let \( \mu _{\textrm{a}} (\cdot \vert {\mathcal {G}}_{R} )\), \( R\in {\mathbb {N}} \cup \{ \infty \} \), be the regular conditional probabilities.
Lemma 5.9
\( \mu _{0}( \cdot \vert {\mathcal {G}}_{\infty } ) = \mu _{\textrm{a}} (\, \cdot \, \vert {\mathcal {G}}_{\infty } ) \circ \vartheta _{\textrm{a}}^{-1} \) for each \( \textrm{a}\in {\mathbb {R}} ^d\).
Proof
From the martingale convergence theorem, (5.41), and Lemma 5.8(1),
Hence, the translation invariance of \( \mu \) and Lemma 5.8(2) yield the claim. \(\square \)
Lemma 5.10
Assume that f is \( {\mathcal {G}}_{\infty } \)-measurable. Then \( f \in {\mathscr {D}}_{\bullet }^{\perp }\) and \( {\mathbb {D}}^{\perp }[f,f] = 0 \).
Proof
We write \( {\textsf {s}}= \sum _i \delta _{s_i}\) as before. Let \( \Upsilon _{R,1}^{m}, \Upsilon _{R,2 } ^{ m , n }, \Upsilon _{R,3 } ^{ m , n }\) be as in Sect. 2.3.
Suppose \( \upsilon \in \Upsilon _{R,1}^{m}\). Then for \( {\textsf {s}}\in {\textsf {T}}_{R}^m \),
We note \( {\textsf {s}}( T _{R}) = \textrm{N}_{ T _{R}} ({\textsf {s}}) \) and \( \sum _{s_i \in T _{R}}s_i= \textrm{M}_{ T _{R}} ({\textsf {s}})\). Note that f is \( {\mathcal {G}}_{\infty } \)-measurable by assumption. Then, f is measurable with respect to \( {\mathcal {H}}_{R} = \sigma [\textrm{N}_{ T _{R}}, \textrm{M}_{ T _{R}}, \pi _{ T _{R}}^c ] \) from (5.40) and Lemma 5.8(1). Hence, we see that f is a function of \( \textrm{N}_{ T _{R}}({\textsf {s}}) \), \( \textrm{M}_{ T _{R}}({\textsf {s}}) \), and \( \pi _{ T _{R}}^c ({\textsf {s}})\). Note that \( \textrm{N}_{ T _{R}}({\textsf {s}}) = m \) on \( {\textsf {T}}_{R}^m \). Let \( f _{ T _{R}, {\textsf {s}}}^m \) be the function defined after (1.27) for f and \( A = T _{R}\). Then \( f _{ T _{R}, {\textsf {s}}}^m = f_{ T _{R}, \pi _{ T _{R}}^c ({\textsf {s}})}^m \) from (1.25). Let \( {\tilde{f}} _{ T _{R}, {\textsf {s}}}^m \) be the function defined on \( \{ x \in {\mathbb {R}} ^d; x = x_1+\cdots + x_m, (x_i)_{i=1}^m \in ( T _{R})^m \} \) such that
The vector \( \upsilon \) is perpendicular to \( {\partial }/{\partial \Gamma ( T _{R})} \) as we see in (2.16). Hence from (5.45),
Suppose \( \upsilon \in \Upsilon _{R,2 } ^{ m , n }\). Then \( {\textsf {s}}({\textsf {S}}_{ R- 1 }) = m \) and \( {\textsf {s}}(S_{R}) = m + n \). We find
From (5.41), f is \( {\mathcal {G}}_{R} \)-measurable. Hence, f is a function of \( \textrm{N}_{R} ({\textsf {s}}) \), \( \textrm{M}_{R} ({\textsf {s}}) \), and \( \pi _{R}^{c}({\textsf {s}})\). Note that \( \textrm{N}_{R}({\textsf {s}}) = m+n\) and \( \textrm{M}_{R} ({\textsf {s}}) = s_1 + \cdots + s_{m+n} \) on \( {\textsf {S}}_{ R- 1 }^m \cap {\textsf {T}}_{ R}^n \). Then we have a function \( {\tilde{f}} _{R, {\textsf {s}}}^{m+n} \) on \(\{ x \in {\mathbb {R}} ^d; x = x_1+\cdots + x_{m+n}, (x_i)_{i=1}^{m+n} \in S_{R}^{m+n} \}\) such that
From (2.17) and (5.47), \( \upsilon \) is perpendicular to \( \partial / \partial \Gamma (S_{R})\). Hence from this and (5.48),
Suppose \( \upsilon \in \Upsilon _{R,3 } ^{ m , n }\). Then \( {\textsf {s}}( T _{R}) = m \) and \( {\textsf {s}}(S_{R}) = n \). Hence, we see
From (5.41), f is \( {\mathcal {G}}_{R} \)-measurable. Hence, f is a function of \( \textrm{N}_{R} ({\textsf {s}}) \), \( \textrm{M}_{R} ({\textsf {s}}) \), and \( \pi _{R}^{c}({\textsf {s}})\). Note that \( \textrm{N}_{R}({\textsf {s}}) = n\) and \( \textrm{M}_{R} ({\textsf {s}}) = s_1 + \cdots + s_{n} \) on \( {\textsf {S}}_{R}^n \). Thus, we find a function \( {\tilde{f}} _{R, {\textsf {s}}}^{n} \) on \(\{ x \in {\mathbb {R}} ^d; x = x_1+\cdots + x_{n}, (x_i)_{i=1}^{n} \in S_{R}^n \}\) such that
From (2.17) and (5.50), \( \upsilon \) is perpendicular to \( \partial / \partial \Gamma (S_{R})\). Hence from this and (5.51),
Putting (5.46), (5.49), and (5.52) together and recalling (2.36), we obtain
Hence using (2.35), we conclude \( f \in {\mathscr {D}}_{\bullet }^{\perp }\) and \( {\mathbb {D}}^{\perp }[f,f] = \sum _{ \upsilon \in \Upsilon } {\mathbb {D}}_{\upsilon } [f,f] = 0 \). \(\square \)
6 Proof of the main theorems (Theorems 1.1–1.4)
In Sect. 6, we complete the proof of the main theorems.
Assume \( (\) A1 \()\)–\( (\) A3 \()\) and \( (\) A5 \()\). Thus, \( \mu \) is translation invariant and irreducibly one-decomposable with \( \{ {\textsf {S}}_{ 0 } ^{\diamond }, {\textsf {S}}_{ 1 } ^{\diamond }\}\). Let \( \mu _{0}\) be the reduced Palm measure conditioned at the origin. Let \( \mu _1^{\diamond }\) be the random point field introduced in (5.6) for \( m = 1 \). Because of (5.8), \( \mu _1^{\diamond }\approx \mu _{0}\).
For \( \mu _1^{\diamond }\)-a.s. \( {\textsf {y}}\in {\textsf {S}}_{ 1 } ^{\diamond }\), let \( \mu (\cdot \Vert {\textsf {y}})\) be the dual reduced Palm measure defined in Lemma 5.5 for \( m = 1 \). Using (5.31) and \( \mu _1^{\diamond }\approx \mu _{0}\), we have
From (6.1) and \( {\textsf {S}}_{1}= \{\delta _x; x \in {\mathbb {R}}^d\} \), set the probability measure \( \sigma _{{\textsf {y}}}\) on \( {\mathbb {R}} ^d\) by
For \( L \in {\mathbb {N}} \), let \( \xi _{L }\!:\!{\mathbb {R}} \!\rightarrow \!{\mathbb {R}} \) be a smooth non-decreasing function such that
Let \( {\mathcal {G}}_{\infty } \) be as in (5.41). Let \( x = (x _{ p }) _{ p =1 }^{d}\in {\mathbb {R}} ^d\). We set \( \chi _{ L }= (\chi _{ L , p }) _{ p =1 }^{d}\) by
Here \( \mu _{0}( \cdot \, \vert {\mathcal {G}}_{\infty } )({\textsf {s}}) \) is the regular conditional probability of \( \mu _{0}\) concerning \( {\mathcal {G}}_{\infty } \). We note that \( \chi _{ L , p }\) is \( {\mathcal {G}}_{\infty } \)-measurable by construction and that \( \chi _{ L , p }\) is neither a continuous nor local function on \( {\textsf {S}}\). Let \( D^{\textrm{trn}}_{ q } \) be as in (1.34). Let
Lemma 6.1
(1) \( \chi _{ L , p }\in {\mathscr {D}}^{\textrm{trn}}\) and \( |D^{\textrm{trn}}_{ q } \chi _{ L , p }({\textsf {s}}) |\le 1 \) for \( p, q = 1,\ldots , d \).
(2) \( \{ \chi _{ L , p }\} \) is an \( {\mathscr {E}}^{{\textsf {Y}},1}\)-Cauchy sequence as \( L \rightarrow \infty \) satisfying for \( p, q = 1,\ldots , d \)
Proof
From (6.2) and Lemma 5.6, we have for \( \mu _{0}\)-a.s. \( {\textsf {y}}\)
From Lemma 5.9, we deduce
Hence using (6.4), (6.5), and (6.6), we obtain for \( \mu _{0}\)-a.s. \( {\textsf {s}}\)
Using (6.3), (6.4), and (6.7), we find \( \chi _{ L , p }\in {\mathscr {D}}^{\textrm{trn}}\) and that for \( \mu _{0}\)-a.s. \( {\textsf {s}}\)
Hence, \( |D^{\textrm{trn}}_{ q } \chi _{ L , p }({\textsf {s}}) |\le 2 \) from (6.3) and (6.8). We have thus obtained (1).
From (6.3), (6.8), and the Lebesgue convergence theorem, we obtain (2). \(\square \)
Proof of Theorem 1.4
Let \( {\mathscr {D}}_{\bullet }^{\perp }\) be as in (2.37). By (6.4), \( \chi _{ L , p }\) is \( {\mathcal {G}}_{\infty } \)-measurable. Hence from Lemma 5.10, we have
From Lemma 6.1(1), we see
From (6.9) and (6.10), we obtain \( \chi _{ L , p }\in {\mathscr {D}}_{\bullet }^{{\textsf {Y}},\perp }\), where \( {\mathscr {D}}_{\bullet }^{{\textsf {Y}},\perp }\) is as in (3.28).
By Lemma 3.5(2), \( ({\mathscr {E}}^{{\textsf {Y}}, \perp }, {\mathscr {D}}^{{\textsf {Y}}, \perp } ) = ({\mathscr {E}}^{{\textsf {Y}}}, \underline{{\mathscr {D}}}^{{\textsf {Y}}})\). Thus, \( \underline{{\mathscr {D}}}^{{\textsf {Y}}}\) is the closure of \( {\mathscr {D}}_{\bullet }^{{\textsf {Y}},\perp }\) with respect to \( {\mathscr {E}}^{{\textsf {Y}}, \perp } \). This implies \( \chi _{ L , p }\in {\mathscr {D}}_{\bullet }^{{\textsf {Y}},\perp }\subset \underline{{\mathscr {D}}}^{{\textsf {Y}}}\). Hence for each \( \epsilon > 0 \) and \( L \in {\mathbb {N}} \), we find \( \chi _{ L , p ,\epsilon }\in {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\) such that
By Lemma 6.1(2) and (6.9), \( \{ \chi _{ L , p }\} \) is an \( {\mathscr {E}}^{{\textsf {Y}}}\)-Cauchy sequence such that
Using \( \chi _{ L , p ,\epsilon }\in {\mathscr {D}}_{\bullet }^{{\textsf {Y}}}\), (6.11), and (6.12), we see \( (\) A7 \()\). Thus, we obtain (1).
From Lemma 4.4, we have \( \lim _{\epsilon \rightarrow 0} \epsilon X _{t/ \epsilon ^2 } = 0 \) weakly in \( C([0,\infty ); {\mathbb {R}} ^d)\) in \( \mu _{0}\)-measure. Combining this with Theorem 3.7 in [24], we obtain (2). \(\square \)
Proof of Theorems 1.2 and 1.3
We obtain Theorem 1.2 from Theorem 1.4. Theorem 1.3 follows from Theorem 1.2 and Lemma 5.7. \(\square \)
The proof of the following lemma is available in the preprint "Ginibre interacting Brownian motion in infinite dimensions is sub-diffusive" on arXiv:2109.14833v3 [math.PR]. We will publish it elsewhere.
Lemma 6.2
The Ginibre random point field satisfies \( (\) A4 \()\).
Proof of Theorems 1.2 and 1.3
It is well known that \( \mu _{\textrm{Gin}}\) satisfies \( (\) A1 \()\) and \( (\) A2 \()\). \( (\) A3 \()\) was proved in [23, Theorem 2.3]. \( (\) A4 \()\) follows from Lemma 6.2. \( (\) A6 \()\) follows from Lemma 1.1. Thus, the Ginibre random point field satisfies all the assumptions of Theorem 1.3, which yields Theorem 1.1. \(\square \)
Data availibility
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Acknowledgements
The author thanks Shota Osada for his comment on dual reduced Palm measures and Stuart Jenkinson, PhD, from Edanz (https://jp.edanz.com/ac) for editing a draft of this manuscript. This work was supported by JSPS KAKENHI Grant Numbers JP16H06338, JP20K20885, JP21H04432, and JP21K13812.
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Osada, H. Vanishing self-diffusivity in Ginibre interacting Brownian motions in two dimensions. Probab. Theory Relat. Fields (2024). https://doi.org/10.1007/s00440-024-01303-2
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DOI: https://doi.org/10.1007/s00440-024-01303-2
Keywords
- Self-diffusion matrices
- Tagged particle problem
- Ginibre random point field
- Rigidity of random point fields
- Infinite particle systems