Abstract
We study functions of bounded variation (and sets of finite perimeter) on a convex open set \({\varOmega }\subseteq X\), X being an infinite-dimensional separable real Hilbert space. We relate the total variation of such functions, defined through an integration by parts formula, to the short-time behaviour of the semigroup associated with a perturbation of the Ornstein–Uhlenbeck operator.
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1 Introduction
In this paper we study some properties of functions of bounded variation (BV functions, for short) defined on an open convex subset of a real separable Hilbert space, endowed with a weighted Gaussian measure.
In finite dimension the theory of BV functions is widely developed (see e.g. [3] and the references therein), whereas in the infinite-dimensional setting the analysis is still at the initial stage and many basic properties are unexplored. Besides the interest on its own, the study of BV functions in infinite-dimensional spaces is motivated by problems arising in calculus of variations, stochastic analysis and it is connected with the applications in information technology (see, for example, [19, 22,23,24, 26]).
BV functions for Gaussian measures in separable Banach spaces were introduced in [17] using Dirichlet forms. Inspired by the results in finite dimension, which connect the theory of functions of bounded variation to that of semigroups of bounded operators, the authors of [18] have proved an elegant characterisation of BV functions in terms of the short-time behaviour of the Ornstein–Uhlenbeck semigroup. More precisely, in a separable Banach space X, if \(\gamma \) is a centred and nondegenerate Gaussian measure on X and u belongs to the Orlicz space \(L(\log L)^{1/2}(X,\gamma )\), then \(u\in BV(X, \gamma )\) if, and only if,
where \(D_H\) is the gradient operator along the Cameron–Martin space H (see Sect. 2) and S(t) is the classical Ornstein–Uhlenbeck semigroup defined via the Mehler formula (see (1)). The latter is the analogous, in the Gaussian setting, of the heat semigroup used by De Giorgi in [11] to provide the original definition of BV functions in the Euclidean case. An analytic approach based on geometric measure theory is proposed in [4] to prove, as in the finite-dimensional case, the equivalence of different definitions of \(BV(X,\gamma )\) functions also, as in [18], in terms of the Ornstein–Uhlenbeck semigroup S(t) near \(t=0\). Similar De Giorgi-type characterisations of BV functions have been obtained for weighted Gaussian measures and more recently for general Fomin differentiable measures in Hilbert spaces, see [12] and the reference therein.
Beside the difficulty of considering general measures, another difficulty of different nature comes from the consideration of functions defined in domains rather than in the whole space. These difficulties come from the lack of factorisation of the underlying measure (that is lost even for Gaussian measures in domains) and the unavailability of decomposition of the domain through the classical method of local charts. Therefore, the easiest interesting case seems to be that of convex domains that are possible to deal with through global penalisation techniques. This is the approach we followed in [6] (see also [21]), and in this paper we take advantage of the results proved there. We start from a weighted Gaussian measure \(\nu :=e^{-U}\gamma \) in a Hilbert space X, where \(U:X\rightarrow {{\mathbb {R}}}\) is convex and sufficiently regular, and consider an open convex domain \(\varOmega \subseteq X\). After introducing the Cameron–Martin space H and the Malliavin gradient \(D_H\) along it, we define the form \((u,v)\mapsto \int _\varOmega \langle D_Hu,D_Hv\rangle _H\mathrm{d}\nu \) on the appropriate Sobolev spaces. The perturbed Ornstein–Uhlenbeck operator \(L_\varOmega \) is then defined in the usual variational way, and it is the generator of an analytic, strongly continuous and contraction semigroup \(T_\varOmega (t)\) in \(L^p(\varOmega ,\nu )\), for \(1<p<\infty \).
For the latter, differently from the Ornstein–Uhlenbeck semigroup in the whole space, no explicit integral representation which allows for direct computations is known. In this direction, in [21] the authors consider the restrictions to an open convex set \({\varOmega }\subseteq X\) of \(BV(X,\gamma )\) functions and they characterise the finiteness of their total variation in \({\varOmega }\) in terms of the Neumann Ornstein–Uhlenbeck semigroup defined in \({\varOmega }\).
Following the ideas in [2], we define the \(BV(\varOmega ,\nu )\) space through an integration by parts formula against suitable Lipschitz functions. Then, we show that the functions u of bounded variation in \({\varOmega }\) with respect to \(\nu \) can be characterised by the finiteness of the limit of \(\Vert D_HT_\varOmega (t)u\Vert _{L^1(\varOmega ,\nu ;H)}\) as \(t\rightarrow 0^+\). The proof of this result relies on a commutation formula between the semigroup \(T_{\varOmega }(t)\) and the gradient operator along H (see Proposition 5). This result was already known in the case of the whole space (see [12]). Here, by means of the crucial pointwise gradient estimate (7) and suitable penalisations \(\varPhi _\varepsilon \) of U outside \(\varOmega \) based on the distance function from \(\varOmega \) along H (here is a first point where the convexity of \(\varOmega \) comes into the play) and the penalisation \(\nu _\varepsilon :=e^{-\varPhi _\varepsilon }\gamma \) of the measure \(\nu \), see Sect. 2.1, we are able to let \(\varepsilon \) to \(0^+\) and to come back to \(\varOmega \).
Finally, we provide a necessary condition in order that a set E is of finite perimeter in \({\varOmega }\) with respect to \(\nu \) (i.e. \(\chi _E\in BV(\varOmega ,\nu )\)). This condition is given in terms of the short-time behaviour of the Ornstein–Uhlenbeck content \(\Vert T_\varOmega (t)\chi _E-\chi _E\Vert _{L^1(\varOmega ,\nu )}\) as \(t\rightarrow 0^+\). Further, a sufficient condition in terms of a related quantity is also shown. This circle of ideas goes back to [20], which originated several researches. Among these, the only infinite-dimensional result, proved for BV functions in space endowed with a Gaussian measure, is in [5].
2 Hypotheses and preliminaries
Let \(H_1\) and \(H_2\) be two real Hilbert spaces with inner products \({\left\langle \cdot ,\cdot \right\rangle }_{H_1}\) and \({\left\langle \cdot ,\cdot \right\rangle }_{H_2},\) respectively. We denote by \({\mathcal {B}}(H_1)\) the \(\sigma \)-algebra of Borel subsets of \(H_1\) and by \(C^k_b(H_1;H_2)\), \(k\in {{\mathbb {N}}}\cup \{\infty \}\) the set of k-times Fréchet differentiable functions from \(H_1\) to \(H_2\) with bounded derivatives up to order k (\(C_b^k(H_1)\) if \(H_2={{\mathbb {R}}}\)). For \(\varPhi \in C_b^1(H_1;H_2)\) we denote by \({{D}} \varPhi (x)\) the derivative of \(\varPhi \) at \(x\in H_1\): if \(f\in C_b^1(H_1)\), for every \(x\in H_1\) there exists a unique \(k\in H_1\) such that \(D f(x)(h)={\left\langle h,k\right\rangle }_{H_1}\), \(h\in H_1\) and we set \(D f(x):=k\). Let X be a separable Hilbert space, with inner product \(\langle \cdot ,\cdot \rangle \) and norm \(|\cdot |\). Let \(B\in {\mathcal {L}}(X)\) (the set of bounded linear operators from X to itself). We say that B is non-negative if \({\left\langle Bx,x\right\rangle }\ge 0\) for every \(x\in X\) and positive if \({\left\langle Bx,x\right\rangle }>0\) for every \(x\in X\setminus {\left\{ 0\right\} }\). We recall that a non-negative and self-adjoint operator \(B\in {\mathcal {L}}(X)\) is a trace class operator whenever \(\mathrm{Tr}(B):=\sum _{n=1}^{\infty }\langle Be_n,e_n\rangle <\infty \) for some (and hence, every) orthonormal basis \((e_n)_{n\in {{\mathbb {N}}}}\) of X.
Let \(\gamma \) be a nondegenerate Gaussian measure on X with mean zero and covariance operator \(Q_\infty :=-QA^{-1}\), where the operators Q and A satisfy the following assumptions.
Hypotheses 1
-
(i)
\(Q\in {\mathcal {L}}(X)\) is a self-adjoint and non-negative operator with \(\mathrm{Ker}\, Q=\{0\}\);
-
(ii)
\(A:D(A)\subseteq X\rightarrow X\) is a self-adjoint operator satisfying \({\left\langle Ax,x\right\rangle }\le -\omega {\left| x\right| }^2\) for every \(x\in D(A)\) and some positive \(\omega \);
-
(iii)
\(Qe^{tA}=e^{tA}Q\) for any \(t\ge 0\);
-
(iv)
\(\mathrm{Tr}(-QA^{-1})<\infty \).
Under Hypotheses 1(i)–(iii), the measure \(\gamma \) is well defined and the Ornstein–Uhlenbeck semigroup defined via the Mehler formula
is symmetric in \(L^2(X,\gamma )\). We fix an orthonormal basis \((v_k)_{k\in {{\mathbb {N}}}}\) of X such that
where \((\lambda _k)_{k\in {{\mathbb {N}}}}\) is the decreasing sequence of eigenvalues of \(Q_\infty \). Under Hypothesis 1(iv), the Cameron–Martin space \((H, {\left| \cdot \right| }_H)\)
where \({\left| \cdot \right| }_H\) is induced by the inner product \({\left\langle h,k\right\rangle }_H:=\langle Q_\infty ^{-1/2}h,Q_\infty ^{-1/2}k\rangle \), is a Hilbert space compactly and densely embedded in X (see [8] and [13] for further details). The sequence \((e_k)_{k\in {{\mathbb {N}}}}\), where \(e_k=\sqrt{\lambda _k}v_k\) for any \(k\in {{\mathbb {N}}}\), is an orthonormal basis of H. By Hypotheses 1, the operator \(-Q^{-1}_\infty :D(Q^{-1}_\infty )\subseteq X\rightarrow X\) (\(-Q^{-1}_\infty :D(Q^{-1}_\infty )\subseteq H\rightarrow H\), respectively) is the generator of a contractive and strongly continuous semigroup \(e^{-tQ_\infty ^{-1}}\) on X (on H, respectively), see [14, Proposition p. 84]). If Y is a Banach space with norm \({\left\| \cdot \right\| }_Y\), a function \(F:X\rightarrow Y\) is said to be H-Lipschitz continuous if there exists a positive constant C such that
for every \(h\in H\) and \(\gamma \)-a.e. \(x\in X\). We denote by \([F]_{H\text {-Lip}}\) the best constant C in (3). For more information see [8, Sections 4.5 and 5.11]. We denote by \({\mathcal {H}}_2\) the space of the Hilbert–Schmidt operators in H that is the space of the bounded linear operators \(B:H\rightarrow H\) such that \({\left\| B\right\| }_{{\mathcal {H}}_2}^2:=\sum _{i=1}^{\infty }{\left| Bg_i\right| }^2_H\) is finite, where \(\{g_n\,|\,n\in {{\mathbb {N}}}\}\) is any orthonormal basis of H. We say that \(f:X\rightarrow {{\mathbb {R}}}\) is H-differentiable at \(x_0\in X\) if there exists \(\ell \in H\) such that
In such a case we set \(D_H f(x_0):=\ell \) and \(D_i f(x_0):= \langle D_H f(x_0), e_i\rangle _H\) for any \(i\in {{\mathbb {N}}}\). The derivative \(D_H f(x_0)\) is called the Malliavin derivative of f at \(x_0\). In a similar way we say that f is twice H-differentiable at \(x_0\) if f is H-differentiable near \(x_0\) and there exists \({\mathcal {B}}\in {\mathcal {H}}_2\) such that
In such a case we set \(D^2_H f(x_0):={\mathcal {B}}\) and \(D_{ij} f(x_0):= \langle D^2_H f(x_0)e_j, e_i\rangle _H\) for any \(i,j\in {{\mathbb {N}}}\). If f is twice H-differentiable at \(x_0\), then \(D_{ij}f(x_0)=D_{ji}f(x_0)\) for every \(i,j\in {{\mathbb {N}}}\). Notice that if \(f:X\rightarrow {{\mathbb {R}}}\) is once or twice Fréchet differentiable at \(x_0\), then it is once or twice H-differentiable at \(x_0\) and it holds \(D_Hf(x_0)=Q_{\infty }Df(x_0)\), and \(D^2_Hf(x_0)=Q_{\infty }D^2 f(x_0)Q_{\infty }\), where the equality must be understood as holding in H. For any \(k\in {{\mathbb {N}}}\cup {\left\{ \infty \right\} }\), we denote by \({\mathcal {F}}C_b^k(X)\), the space of cylindrical \(C^k_b\) functions, i.e. the set of functions \(f:X\rightarrow {{\mathbb {R}}}\) such that \(f(x)=\varphi (\langle x, h_1\rangle ,\ldots , \langle x, h_N\rangle )\) for some \(\varphi \in C_b^k({{\mathbb {R}}}^N)\), \(h_1,\ldots , h_N\in H\) and \(N\in {{\mathbb {N}}}\). By \({\mathcal {F}}C_b^k(X,H)\) we denote H-valued cylindrical \(C_b^k\) functions with finite rank. The Sobolev spaces in the sense of Malliavin \(D^{1,p}(X,\gamma )\) and \(D^{2,p}(X,\gamma )\) with \(p\in [1,\infty )\) are defined as the completions of the smooth cylindrical functions \({\mathcal {F}}C_b^\infty (X)\) in the norms
This is equivalent to considering the domain of the closure of the gradient operator, defined on smooth cylindrical functions, in \(L^p(X,\gamma )\) (see [8, Section 5.2]). Let \(U:X\rightarrow {{\mathbb {R}}}\) satisfy the following assumptions.
Hypotheses 2
U is a convex function which belongs to \(C^2(X)\cap D^{1,q}(X,\gamma )\) for all \(q\in [1,\infty )\) with H-Lipschitz gradient.
The convexity of the function U guarantees that U is bounded from below by a linear function, and therefore, it decreases at most linearly and by Fernique theorem (see [8, Theorem 2.8.5]) \(e^{-U}\) belongs to \(L^1(X,\gamma )\). Then, we can consider the finite log-concave measure
It is obvious that \(\gamma \) and \(\nu \) are equivalent measures, hence saying that a statement holds \(\gamma \)-a.e. is the same as saying that it holds \(\nu \)-a.e. Moreover as \(U\in \cap _{q\ge 1}D^{1,q}(X,\gamma )\), the operator \(D_H:{\mathcal {F}}C^1_b(X)\rightarrow L^p(X,\nu ;H)\) is closable in \(L^p(X,\nu )\), \(p\in (1,\infty )\) and the space \(D^{1,p}(X,\nu )\), \(p>1\) can be defined as the domain of its closure (still denoted by \(D_H\)). In a similar way we may define \(D^{2,p}(X,\nu )\), \(p\in (1,\infty )\) (for more details see [1, 9, 16]). The Gaussian integration by parts formula \(\int _X D_i f \mathrm{d}\gamma =\frac{1}{\sqrt{\lambda _i}}\int _X \langle x,v_i\rangle f \mathrm{d}\gamma \), which holds true for any \(f\in {\mathcal {F}}C^1_b(X)\) and \(i\in {{\mathbb {N}}}\), yields
for any \(\varphi \in D^{1,p}(X,\nu )\) (\(p>1\)) and \(\psi \in {\mathcal {F}}C^1_b(X)\).
In what follows \(\varOmega \) denotes an open subset of X. In this case, the spaces \(D^{1,p}(\varOmega ,\nu )\) and \(D^{2,p}(\varOmega ,\nu )\), \(p\in (1,\infty )\), can be defined in a similar way as in the whole space, thanks to the following result (see [6, Proposition 1.4]).
Proposition 1
Assume that Hypotheses 1 and 2 are satisfied. Let \(p\in (1,\infty )\) and let \({\varOmega }\) be an open subset of X. The operators \(D_H:{\mathcal {F}}C_b^\infty (\varOmega )\rightarrow L^p(\varOmega ,\nu ; H)\) and
are closable in \(L^p(\varOmega ,\nu )\) and \(L^p(\varOmega ,\nu )\times L^p(\varOmega ,\nu )\), respectively. Here \({\mathcal {F}}C_b^\infty (\varOmega )\) is the space of the restrictions to \(\varOmega \) of functions in \({\mathcal {F}}C_b^\infty (X)\).
The spaces \(D^{1,p}(\varOmega ,\nu ;H)\), \(p\in (1,\infty )\), are defined in a similar way, replacing smooth cylindrical functions with H-valued smooth cylindrical functions with finite rank. We recall that if \(F\in D^{1,p}(\varOmega ,\nu ;H)\), then \(D_H F(x)\) belongs to \({\mathcal {H}}_2\) for a.e. \(x\in {\varOmega }\). We denote by \(p'\) the conjugate exponent to \(p\in (1,\infty )\).
2.1 Perturbed Ornstein–Uhlenbeck semigroup on convex domains
In order to consider the initial boundary value problems defined in \({\varOmega }\) we define the distance function along H
for \(\ x\in X\), and we recall some useful regularity results, (see, for instance, [8, Theorems 2.8.5 and 5.11.2] and [10, Section 3]).
Proposition 2
If \(\varOmega \subseteq X\) is an open convex set, then \(d_\varOmega ^2\) is H-differentiable and its Malliavin derivative is H-Lipschitz with H-Lipschitz constant less than or equal to 2, i.e.
for any \(h\in H\) and for \(\nu \)-a.e \(x\in X\). Moreover, \(D_H^2d_\varOmega ^2\) exists \(\nu \)-a.e. in X and \(d_\varOmega ^2\) belongs to \(D^{2,p}(X,\nu )\) for every \(p\in [1,\infty )\).
We require some further regularity on \(d_\varOmega ^2\).
Hypotheses 3
Let \({\varOmega }\) be an open convex subset of X such that \(\nu (\partial \varOmega )=0\) and \(D_H^2d_\varOmega ^2\) is H-continuous \(\gamma \)-a.e. in X, i.e. for \(\gamma \)-a.e. \(x\in X\) we have
Remark 1
As stated in [6, Remark 1.7] there is a rather large class of subsets of X satisfying Hypothesis 3. For instance, if \(\partial \varOmega \) is (locally) a \(C^2\)-embedding in X of an open subset of a hyperplane in X and \(\nu (\partial \varOmega )=0\), then Hypothesis 3 is satisfied. Easy examples are open balls and open ellipsoids of X, open hyperplanes of X and every set of the form \(\varOmega ={\left\{ x\in X\, |\,G(x)< 0\right\} }\), where \(G:X\rightarrow {{\mathbb {R}}}\) is a \(C^2\)-convex function such that \(D_H G\) is nonzero at every point of \(\partial \varOmega \).
We consider the semigroup \(T_{\varOmega }(t)\) on \(L^2({\varOmega },\nu )\) and its generator \(L_\varOmega \):
with \(L_{\varOmega }u:=v\) if \(u\in D(L_{\varOmega })\). We recall (see [6, Section 2]) an approximation procedure of \(T_{\varOmega }(t)f\), when \(f\in L^2({\varOmega }, \nu )\), through \({\mathcal {F}}C^3_b(X)\) functions that relies on a reduction to a finite (say n-) dimensional space and on a \(\varepsilon \)-penalisation argument. Accordingly, the approximation depends on two parameters n and \(\varepsilon \). More precisely, we consider the function \(\varPhi _\varepsilon :X\rightarrow {{\mathbb {R}}}\) defined by
and the measure \(\nu _\varepsilon \) given by \(e^{-\varPhi _\varepsilon }\gamma \). Next, we consider the operator \(L_\varepsilon \) on the whole X defined as
with \(L_{\varepsilon }u:=v\) if \(u\in D(L_{\varepsilon })\), and the semigroup \(T_\varepsilon (t)\) generated by \(L_\varepsilon \) in \(L^2(X, \nu _\varepsilon )\). We point out that \(L_\varepsilon \) acts on smooth cylindrical functions \(\varphi \) as follows
Now we recall a useful approximation result whose proof can be found in [6, Theorem 2.8].
Theorem 1
Under Hypotheses 1, 2 and 3 the following statements hold true.
-
(i)
For any \(\varepsilon >0\) and \(f\in L^2(X,\nu _\varepsilon )\), there exists a sequence \((f_n)_{n\in {{\mathbb {N}}}}\subseteq L^2(X,\nu _\varepsilon )\) converging to f in \(L^2(X, \nu _\varepsilon )\) such that \(T_\varepsilon (t)f_n\) is in \({\mathcal {F}}C^3_b(X)\) and
$$\begin{aligned} \lim _{n\rightarrow \infty }{\left\| T_{\varepsilon }(t)f_n-T_\varepsilon (t) f\right\| }_{D^{1,2}(X,\nu _\varepsilon )}=0,\qquad \;\, t>0. \end{aligned}$$In addition, if \(f\in D^{1,2}(X,\nu _\varepsilon )\) then the sequence \((f_n)\) can be chosen in a way that \(D_Hf_n\) converges to \(D_Hf\) in \(L^1(X, \nu _\varepsilon ; H)\), as \(n\rightarrow \infty \).
-
(i)
For any \(f\in L^2(\varOmega ,\nu )\) there exists an infinitesimal sequence \((\varepsilon _n)_{n\in {{\mathbb {N}}}}\) such that \(T_{\varepsilon _n}(t) {\widetilde{f}}\) weakly converges to \(T_\varOmega (t) f\) in \(D^{1,2}(\varOmega ,\nu )\), where \({\widetilde{f}}\) is any \(L^2\)-extension of f to X.
We collect some properties of \(T_{\varOmega }(t)\), see [6, Proposition 1.10, Theorems 3.1 & 3.3].
Proposition 3
If Hypotheses 1, 2 and 3 hold true, then
-
(i)
the semigroup \(T_\varOmega (t)\) generated in \(L^2({\varOmega },\nu )\) can be extended to a positivity preserving contraction semigroup in \(L^p(\varOmega ,\nu )\) for every \(1\le p\le \infty \) and \(t\ge 0\), still denoted by \(T_\varOmega (t)\). It is strongly continuous in \(L^p({\varOmega }, \nu )\) for any \(p \in [1,\infty )\) and consistent;
-
(ii)
for any \(p\in [1,\infty )\), \(f\in L^p(\varOmega ,\nu )\) and \(g\in L^\infty (\varOmega ,\nu )\) it holds
$$\begin{aligned} \int _\varOmega f T_\varOmega (t)g\mathrm{d}\nu =\int _\varOmega gT_\varOmega (t)f\mathrm{d}\nu ,\qquad t>0; \end{aligned}$$(6) -
(iii)
for any \(p\in (1,\infty )\), \(f\in L^p({\varOmega },\nu )\) and \(t>0\) there is \(K_p>0\) such that
$$\begin{aligned} |D_HT_{\varOmega }(t)f|_H^p\le K_pt^{-p/2}T_{\varOmega }(t)|f|^p\qquad \nu \text {-a.e. in }{\varOmega }; \end{aligned}$$(7) -
(iv)
if \(f\in D^{1,p}({\varOmega },\nu )\), \(t>0\) and \(p\in [1,\infty )\) it holds
$$\begin{aligned} |D_H T_{\varOmega }(t) f|^p\le e^{-p\lambda _1^{-1}t}T_{\varOmega }(t)|D_H f|^p_H\qquad \nu \text {-a.e. in }{\varOmega }. \end{aligned}$$(8)
We point out that the results in Proposition 3 continue to hold if we replace \({\varOmega }\), \(\nu \) and \(T_{\varOmega }(t)\) by X, \(\nu _\varepsilon \) and \(T_\varepsilon (t)\), respectively.
2.2 BV functions in Hilbert spaces: definitions and some known facts
We introduce BV functions in the Wiener space setting. Let Y be a separable Hilbert space with norm \({\left| \cdot \right| }_Y\). We recall that in a separable space X the \(\sigma \)-algebra \({\mathcal {B}}(X)\) is generated by the family of the cylindrical sets (see e.g. [25]). Denote by \({\mathcal {M}}({\varOmega };Y)\) the set of Borel Y-valued measures on \({\varOmega }\). If \(Y={{\mathbb {R}}}\) then we write \({\mathcal {M}}({\varOmega })\). The total variation of \(\mu \in {\mathcal {M}}({\varOmega };Y)\) is the positive Borel measure
Let \(\mathrm{Lip}_c(\varOmega ;Y)\) be the set of bounded Lipschitz continuous Y-valued functions \(g:\varOmega \rightarrow Y\) such that \(\mathrm{dist}(\mathrm{supp}\, g,X\smallsetminus \varOmega )>0\) and define the space \(BV({\varOmega },\nu )\) as follows.
Definition 1
Let \(\varOmega \) be an open subset of X. We say that a function \(f\in L^2(\varOmega ,\nu )\) is of bounded variation in \(\varOmega \), and we write \(f\in BV(\varOmega ,\nu )\), if there exists a measure \(\mu \in {\mathcal {M}}(\varOmega ;H)\) such that
for every \(g\in \mathrm{Lip}_c(\varOmega )\) and \(h\in H\), where \(\partial _h^*\) denotes, up to the sign, the adjoint in \(L^2({\varOmega },\nu )\) of the partial derivative along \(h\in H\). In this case we set \(D_\nu f:=\mu \).
As in the finite-dimensional case, one can characterise functions of bounded variation by their total variation.
Definition 2
Let \(\varOmega \) be an open subset of X and \(u\in L^2(\varOmega ,\nu )\). We define the variation of u in \(\varOmega \) by
Here \(\mathrm{div}_\nu \, g= \sum _{i=1}^N \partial _{k_i}^*g_i(x)\) if \(g(x)=\sum _{i=1}^N g_i(x)k_i\) and \(F=\mathrm{span}\{k_1,\ldots , k_N\}\) for some \(N\in {{\mathbb {N}}}\).
When \({\varOmega }=X\), in the two definitions above we can consider \(\mathrm{Lip}_b(X)\) and \(\mathrm{Lip}_b(X;F)\), respectively, as test functions spaces.
As announced, in [2, Theorem 5.7] it has been proved that \(u\in BV(\varOmega ,\nu )\) if and only if \(V_\nu (u,\varOmega )\) is finite. Moreover, in this case
Finally we say that a Borel subset E of X is of finite perimeter in \(\varOmega \) with respect to \(\nu \), whenever the function \(\chi _E\) belongs to \(BV({\varOmega },\nu )\). In this case we denote by \(P_\nu (E,\varOmega )\) the total variation of \(\chi _E\) in \({\varOmega }\).
3 A De Giorgi type characterisation
The main result of this section is the De Giorgi type characterisation of \(BV({\varOmega }, \nu )\) functions in Theorem 3, which relies on a “quasi-commutative” formula between the semigroup \(T_\varOmega (t)\) and the H-gradient operator \(D_H\); here estimate (8) plays a crucial role. This formula is inspired by an analogous formula proved in [12]. We first define the Sobolev spaces \(D^{1,2}(X,\nu _\varepsilon ;H)\).
Definition 3
We denote by \(D^{1,2}(X,\nu _\varepsilon ;H)\) the domain of the closure of the operator \(D_H:{\mathcal {F}}C_b^1(X,H)\rightarrow L^2(X,\nu _\varepsilon ;{\mathcal {H}}_2)\) in the \(L^2(X,\nu _\varepsilon ;H)\) norm (see [7, Section 8.1]). \(D_H\) is defined as
where \(\{e_i\,|\,i\in {{\mathbb {N}}}\}\) is an orthonormal basis of H and
for some \(n\in {{\mathbb {N}}}\), \(k(i)\in {{\mathbb {N}}}\), \(x_1,\ldots , x_{k(i)}\in X\) and \(\varphi _i\in C_b^1({{\mathbb {R}}}^{k(i)})\) for every \(i=1,\ldots ,n\). In an analogous way we define the space \(D^{1,2}(\varOmega ,\nu ;H)\).
We first show a vector-valued version of Theorem 1. Let \(\mathbf{L}_{\varepsilon }\) in \(L^2(X,\nu _\varepsilon ;H)\) be the operator defined via the quadratic form by
In the same way we define the operator \(\mathbf{L}_\varOmega \) in in \(L^2({\varOmega },\nu ;H)\). We recall that by [14, p. 84] ( [14, Corollary 3.17 and Proposition 3.23] and [14, Corollary 4.8], respectively), the operators \(\mathbf{L}_{\varepsilon }\) and \(\mathbf{L}_{\varOmega }\) generate strongly continuous semigroups \(\mathbf{T}_\varepsilon (t)\) and \(\mathbf{T}_\varOmega (t)\) (contractive and analytic, respectively).
Proposition 4
The operators \(\mathbf{L}_\varepsilon \), \(\mathbf{L}_\varOmega \) and the semigroups \(\mathbf{T}_\varepsilon (t)\) and \(\mathbf{T}_\varOmega (t)\) act component by component, i.e. if \(F\in D(\mathbf{L}_\varepsilon )\) \((D(\mathbf{L}_\varOmega )\), respectively), and it is such that \(F=\sum _{i=1}^{\infty }f_ie_i\) for some basis \(\{e_n\,|\,n\in {{\mathbb {N}}}\}\) of H, then \(f_i\in D(L_\varepsilon )\) \((D(L_\varOmega )\), respectively) and
Moreover, for every \(t>0\), if \(F\in L^2(X,\nu _\varepsilon ;H)\) \((L^2(\varOmega ,\nu ;H)\), respectively), and it is such that \(F=\sum _{i=1}^{\infty }f_ie_i\) for some basis \(\{e_n\,|\,n\in {{\mathbb {N}}}\}\) of H and \(f_i\in L^2(X,\nu _\varepsilon )\) (\(L^2(\varOmega ,\nu )\), respectively) then
The above identities hold \(\nu _\varepsilon \)-a.e. in X (\(\nu _{\varOmega }\)-a.e. in \({\varOmega }\), respectively).
Proof
We only show the results for \(\mathbf{L}_\varepsilon \) and \(\mathbf{T}_\varepsilon (t)\). Let \(F=\sum _{i=1}^{\infty }f_ie_i \in D(\mathbf{L}_\varepsilon )\) and let \(G=ge_j\) for some \(j\in {{\mathbb {N}}}\) and \(g\in D^{1,2}(X,\nu _\varepsilon )\); then
This shows that \(f_j\in D(L_\varepsilon )\) (see (5)) and \(L_\varepsilon f_j=(\mathbf{L}_\varepsilon F)_j\). Now observe that
Thus, by the uniqueness of the solution of the Cauchy problem associated with \(D_t-L_\varepsilon \) in \(L^2(X, \nu _\varepsilon )\), it follows that \((\mathbf{T}_\varepsilon (t)F)_j=T_\varepsilon (t)f_j\) for any \(t>0\). The arbitrariness of \(j\in {{\mathbb {N}}}\) concludes the proof. \(\square \)
Remark 2
According to the definition of \(\mathbf{T}_\varepsilon (t)\) and \(\mathbf{T}_\varOmega (t)\) it is immediately seen that for every \(F\in L^2(X, \nu _\varepsilon ;H)\) and \(G\in L^2({\varOmega },\nu ;H)\)
and
Moreover, taking into account that the semigroups \(\mathbf{T}_\varOmega (t)\) and \(\mathbf{T}_\varepsilon (t)\) act component by component, we can obtain a vector-valued version of Theorem 1.
Theorem 2
Under Hypotheses 1, 2 and 3, the following statements hold true.
-
(i)
For any \(\varepsilon >0\) and \(F\in L^2(X,\nu _\varepsilon ;H)\), there exists a sequence \((F_n)_{n}\subseteq L^2(X,\nu _\varepsilon ;H)\) such that \(\mathbf{T}_\varepsilon (t)F_n\) belongs to \({\mathcal {F}}C^3_b(X;H)\) and
$$\begin{aligned}&\lim _{n\rightarrow \infty }{\left\| F_n-F\right\| }_{L^{2}(X,\nu _\varepsilon ;H)}=0,\nonumber \\&\lim _{n\rightarrow \infty }{\left\| \mathbf{T}_{\varepsilon }(t)F_n-\mathbf{T}_\varepsilon (t) F\right\| }_{D^{1,2}(X,\nu _\varepsilon ;H)}=0,\qquad \;\, t>0. \end{aligned}$$(12)If, in addition, \(F\in D^{1,2}(X,\nu _\varepsilon ;H)\) then \(D_H F_n\) converges to \(D_H F\) in \(L^1(X, \nu _\varepsilon ; {\mathcal {H}}_2)\), as \(n\rightarrow \infty \).
-
(ii)
For any \(F\in L^2(\varOmega ,\nu ;H)\) there exists an infinitesimal sequence \((\varepsilon _n)_{n\in {{\mathbb {N}}}}\) such that \(\mathbf{T}_{\varepsilon _n}(t) {\widetilde{F}}\) weakly converges to \(\mathbf{T}_\varOmega (t) F\) in \(D^{1,2}(\varOmega ,\nu ;H)\), where \({\widetilde{F}}\) is any \(L^2\)-extension of F to X.
Proof
-
(i)
Let \(F=\sum _{i=1}^{\infty }f^{(i)}e_i\) where \(f^{(i)}\in L^2(X,\nu _\varepsilon )\), \(i\in {{\mathbb {N}}}\). For every \(i\in {{\mathbb {N}}}\), by Theorem 1(i), there exists \((f_k^{(i)})_{k\in {{\mathbb {N}}}}\subseteq L^2(X,\nu _\varepsilon )\) converging to \(f^{(i)}\) in \(L^2(X,\nu _\varepsilon )\) such that \(T_\varepsilon (t)f_k^{(i)}\) belongs to \({\mathcal {F}}C^3_b(X)\) and
$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert T_\varepsilon (t)f_k^{(i)}-T_\varepsilon (t)f^{(i)}\Vert _{D^{1,2}(X,\nu _\varepsilon )}=0,\qquad t, \varepsilon >0. \end{aligned}$$(13)Observe that (12) follows immediately from (13). Now fix \(i,n\in {{\mathbb {N}}}\) and consider \(k_i\in {{\mathbb {N}}}\) such that for every \(k\ge k_i\) it holds
$$\begin{aligned} \int _X|f_k^{(i)}-f^{(i)}|^2\mathrm{d}\nu _\varepsilon <\frac{1}{n2^i}. \end{aligned}$$Consider the vector field \(F_n:=\sum _{i=1}^n f_{k_i}^{(i)}e_i\). We claim that \((F_n)\) is the sequence we are looking for. Indeed \(F_n\) belongs to \(L^2(X,\nu _\varepsilon ;H)\) for any \(n\in {{\mathbb {N}}}\). Let \(n_0\in {{\mathbb {N}}}\) be such that \(\sum _{i=n_0+1}^{\infty }\Vert f^{(i)}\Vert _{L^2(X,\nu _\varepsilon )}^2\le \eta /2\) and let \(n\ge n_0\) such that \(1/n<\eta /2\). We have
$$\begin{aligned} {\left\| F_n-F\right\| }_{L^2(X,\nu _\varepsilon ;H)}^2&\le \sum _{i=1}^n\int _X|f_{k_i}^{(i)}-f^{(i)}|^2\mathrm{d}\nu _\varepsilon +\sum _{i=n+1}^{\infty }\Vert f^{(i)}\Vert _{L^2(X,\nu _\varepsilon )}^2\\&\le \frac{1}{n}+\frac{\eta }{2}\le \eta . \end{aligned}$$In a similar way we can prove the other statements.
-
(ii)
is an immediate consequence of Proposition 4 and Theorem 1(ii). \(\square \)
Before going on, recall that usually in the characterisation of functions of bounded variation in terms of the short-time behaviour of suitable semigroups a crucial tool is an appropriate commutation formula between the semigroup and the gradient operator. For instance, for the Wiener space and the Ornstein–Uhlenbeck semigroup the equality \(D_H S(t)f=e^{-t}{\mathbf {S}}(t)D_H f\) holds true for any \(t \ge 0\). Let us prove a (quasi) commutation formula between \(T_{\varOmega }(t)\) and \(D_H\), under the following additional assumption.
Hypotheses 4
The map \((d_{\varOmega })^{-2}\Vert D_H^2d_{\varOmega }^2\Vert _{{{\mathcal {H}}}_{2}}\) belongs to \(L^2(X, \nu )\).
Remark 3
It is not difficult to show that every open ball and every open ellipsoid of X as well as every open hyperplane of X satisfy Hypothesis 4. We show that Hypothesis 4 is satisfied when \({\varOmega }\) is the unit ball \(B_X\) centred at zero. The other examples can be discussed in a similar fashion. Observe that, by Proposition 2, \(\Vert D_H^2d_{\varOmega }^2\Vert _{{\mathcal H}_{2}}\le 2\) and \(\Vert D_H^2d_{\varOmega }^2(x)\Vert _{{{\mathcal {H}}}_{2}}=0\) if \(x\in B_X\). Moreover, there exists a constant \(C>0\) such that
where \(\mathrm{dist}(x,B_X)\) is the distance of x from \(B_X\). So
where K is a positive constant and \(\gamma _n\) denotes the n-dimensional Gaussian measure, image of \(\gamma \) under the projection on \(\mathrm{span}\,\{v_1,\ldots ,v_n\}\). To conclude, observe that there exists \(n\in {{\mathbb {N}}}\) such that the right-hand side of (14) is finite.
Proposition 5
Under Hypotheses 1, 2, 3 and 4, the formula
holds true \(\nu \)-a.e. in \({\varOmega }\), for any \(f\in \mathrm{Lip}_c({\varOmega })\) and \(t>0\).
Proof
In order to prove (15) we show that
for any \(f\in \mathrm{Lip}_c({\varOmega })\), \(G \in C_b({\varOmega };H)\) and \(t>0\). By performing slight changes in [12, Appendix A] we get
\(\nu _\varepsilon \)-a.e. in X for any \(g\in \mathrm{\mathrm{Lip}}_b(X)\) and \(\varepsilon >0\), where \(T_\varepsilon (t)\) is the semigroup introduced in Sect. 2.1. Now, let \(f\in \mathrm{Lip}_c({\varOmega })\) and \({\widetilde{f}}\) be the trivial extension to zero of f in the whole space X. Clearly, \({\widetilde{f}}\) belongs to \(\mathrm{\mathrm{Lip}}_b(X)\) and (17) holds true with g replaced by \({\widetilde{f}}\). Consequently, multiplying (17) by the function G and integrating on \({\varOmega }\) with respect to \(\nu \) yield
where in the last line we used the Fubini–Tonelli theorem.
The proof of (16) is split in two steps.
Step 1. We argue by approximation on the last terms in (18) and (16).
For every \(\varepsilon ,s>0\) we fix a Borel measurable version of \(D_HT_{\varOmega }(s)f\) and \(D_HT_\varepsilon (s){\widetilde{f}}\) in \(L^2(\varOmega ,\nu ;H)\) and \(L^2(X,\nu _\varepsilon ;H)\), respectively. Consider the function
Observe that the map \(x\mapsto \varGamma _\varepsilon (s,x)\) is an extension of \(D_HT_{\varOmega }(s)f\) to the whole X. Thus, by Theorem 2 there is a sequence \(\varepsilon _n\downarrow 0\) such that for every \(\eta >0\) the function \(\mathbf{T}_{\varepsilon _n}(t-s)(D_H^2 U\varGamma _\eta (s,\cdot ))\) weakly converges to \(\mathbf{T}_{{\varOmega }}(t-s)(D_H^2 UD_HT_{\varOmega }(s)f)\) in \(D^{1,2}(\varOmega ,\nu ;H)\). Observe that the set
is bounded in \(L^2(\varOmega ,\nu ;H)\). Indeed by the contractivity of \(\mathbf{T}_{\varepsilon _n}(t)\) in the space \(L^2(X,\nu _{\varepsilon _n};H)\), the fact that \(U\equiv \varPhi _\varepsilon \) on \({\varOmega }\) and estimate (8) we have
where in the last line we used the contractivity of \(T_\eta (t)\) and \(T_{\varOmega }(t)\) in \(L^\infty \) and the fact that \(\nu _{\varepsilon _n}(X)\le \nu (X)\) for any \(n\in {{\mathbb {N}}}\). So there exists \(M>0\) large enough so that the family in (19) is contained in B(0, M), the ball of \(L^2(\varOmega ,\nu ;H)\) with centre 0 and radius M.
Recall that every bounded subset of \(L^2(\varOmega ,\nu ;H)\) is weakly metrisable (see [15, Proposition 3.106]) and let \(\rho :B(0,M)\times B(0,M)\rightarrow {{\mathbb {R}}}\) be a metric such that the topology generated by \(\rho \) and the weak topology in B(0, M) coincide. Now we use a diagonal argument to pass to the limit in (18). Let \(n_1\in {{\mathbb {N}}}\) be such that for every \(n\ge n_1\) it holds
where \({\overline{\varGamma }}_j(s,x)= \varGamma _{j^{-1}}(s,x)\) for any \(s>0\) and \(x\in X\). Now assume that \(n_1,\ldots , n_k\) are already constructed and consider \(n_{k+1}>n_k\)such that for every \(n\ge n_{k+1}\)
Consider now the sequence \((\mathbf{T}_{\varepsilon _{n_k}}(t-s)(D_H^2 U{\overline{\varGamma }}_{n_k}(s,\cdot )))_{k\in {{\mathbb {N}}}}\) and observe that it weakly converges to \(\mathbf{T}_{{\varOmega }}(t-s)(D_H^2 UD_HT_{\varOmega }(s)f)\) in \(L^2(\varOmega ,\nu ;H)\) as \(k\rightarrow \infty \).
Step 2. To complete the proof, we replace \(\varepsilon \) in (18) by a sequence \(\varepsilon _{m}\downarrow 0\) such that step 1 and Theorems 1, 2 apply. Let us show that we can take the limit as \(m\rightarrow \infty \). Indeed, from Theorem 1 it follows that for any \(f \in L^2({\varOmega }, \nu )\), \(T_{\varepsilon _m}(t) {\widetilde{f}}\) weakly converges (up to a subsequence) to \(T_\varOmega (t)f\) in \(D^{2,2}(\varOmega ,\nu )\) as \(m \rightarrow \infty \), hence writing \(T_m, \mathbf{T}_m, \varPhi _m, \varGamma _m\) in place of \(T_{\varepsilon _m}, \mathbf{T}_{\varepsilon _m}, \varPhi _{\varepsilon _m}, \varGamma _{\varepsilon _m}\) we obtain
and by the analogous vector-valued result (see Theorem 2, (10) and again (7))
To conclude we have to prove that the last term in the right-hand side of (18) converges to the last term in the right-hand side of (16).
Let us estimate \(I_1\). Using that \(\varPhi _m\equiv U\) on \({\varOmega }\) for every \(m\in {{\mathbb {N}}}\), formula (10) and the invariance property of \(T_m\) with respect to \(\nu _m:=\nu _{\varepsilon _m}\) we have
The right-hand side of (20) converges to zero as \(m\rightarrow \infty \): indeed, \(D_H T_m(s){\widetilde{f}}\) converges pointwise \(\nu \)-almost everywhere in \(\varOmega \) to \(D_H T_\varOmega (s)f\). Furthermore, by Proposition 3 we have that \(\nu \)-a.e. in \(\varOmega \)
So by the dominated convergence theorem we get that \(I_1(m)\) vanishes as \(m\rightarrow \infty \). Now, using similar arguments we can estimate \(I_2(m)\) as follows
Now observe that the right-hand side of (21) vanishes as \(m\rightarrow \infty \). Indeed the function \(\frac{1}{4\varepsilon _m}\Vert D_H^2 d_\varOmega ^2\Vert _{{\mathcal {H}}_2}^2 e^{-U-\frac{1}{2\varepsilon _m}d^2_\varOmega }\) identically vanishes in \({\varOmega }\) and converges pointwise to 0 \(\nu \)-almost everywhere in \(X{\setminus } \varOmega \) as \(m\rightarrow \infty \). Furthermore, observe that the function \((0,\infty )\ni \varepsilon \mapsto R(\varepsilon ):=\frac{1}{4\varepsilon ^2}{\left| D^2_Hd_\varOmega ^2\right| }^2e^{-\frac{1}{2{\varepsilon }}d_\varOmega ^2}\) attains its maximum in \(\varepsilon =d_\varOmega ^2/4\) where it equals to \(4d_\varOmega ^{-4}\Vert D_H^2 d_\varOmega ^2|_{{\mathcal {H}}_2}^2\). Thus, using Hypothesis 4 and applying the dominated convergence theorem we infer that also \(I_2(m)\) converges to zero as m goes to infinity.
Finally \(I_3(m)\) converges to zero as m goes to infinity thanks to step 1 and this concludes the proof. \(\square \)
Corollary 1
Assume Hypotheses 1, 2, 3 and 4 hold true. For any \(t>0\) and \(p>1\) there exist two operators \(\mathbf{S}_1(t): L^p(\varOmega ,\nu ;H)\rightarrow L^1(\varOmega ,\nu ;H)\) and \(S_2(t): L^p(\varOmega ,\nu )\rightarrow L^1(\varOmega ,\nu ;H)\) such that for every continuous and H-differentiable function \(\varphi :\varOmega \rightarrow {{\mathbb {R}}}\) with H-Lipschitz gradient
Moreover, the adjoint operator \((\mathbf{S}_1(t))^*\) maps \(\mathrm{\mathrm{Lip}}_c(\varOmega ;H)\) into \(L^{\infty }(\varOmega ,\nu ;H)\) and verifies \({\left\| \mathbf{S}_1^*(t)F\right\| }_\infty \le C_1(t){\left\| F\right\| }_\infty \) for any \(F\in \mathrm{\mathrm{Lip}}_c(\varOmega ;H)\) with \(C_1(t)\rightarrow 1\) as \(t\rightarrow 0\) and the norm \(C_2(t):={\left\| S_2(t)\right\| }_{{\mathcal {L}}(L^p,L^1)}\rightarrow 0\) as \(t\rightarrow 0\).
Proof
Setting \(\mathbf{S}_1(t):=e^{-tQ_\infty ^{-1}}{} \mathbf{T}_\varOmega (t)\), [6, Proposition 1.10] yields that \(\mathbf{S}_1^*(t)=\mathbf{T}_\varOmega (t)e^{-tQ_\infty ^{-1}}\) maps \(\mathrm{Lip}_c(\varOmega ;H)\) into \(L^{\infty }(\varOmega ,\nu ;H)\) and
Moreover, setting \(S_2(t):=-\int _0^te^{(s-t)Q_\infty ^{-1}}\mathbf{T}_{\varOmega }(t-s)(D_H^2 U D_H T_{\varOmega }))\mathrm{d}s\), by the contractivity of \({\mathbf {T}}_\varOmega (t)\) in \(L^1(\varOmega ,\nu ; H)\), (7), Hypothesis 2, the contractivity of \(e^{-tQ^{-1}_\infty }\) in H, estimate (11) and the invariance property of \(T_\varOmega (t)\), we get
for any \(t>0\). By the assumption on U we deduce that the operator \(S_2(t)\) is bounded from \(L^p(\varOmega ,\nu )\) into \(L^1(\varOmega ,\nu ;H)\) for any \(t>0\). Finally, estimates (22) and (23) allow us to complete the proof. \(\square \)
Now, we are able to prove the main result of this section.
Theorem 3
Assume Hypotheses 1, 2, 3 and 4 hold true and let \(u\in L^2({\varOmega },\nu )\). The following statements are true:
-
(i)
if \(\liminf _{t \rightarrow 0^+}\Vert D_H T_{\varOmega }(t)u \Vert _{L^1({\varOmega },\nu ;H)}\) is finite, then \(u\in BV(\varOmega ,\nu )\);
-
(ii)
if \(u\in BV(\varOmega ,\nu )\), then \(\limsup _{t \rightarrow 0^+}\Vert D_H T_{\varOmega }(t)u \Vert _{L^1({\varOmega },\nu ;H)}\le |D_\nu u|(\varOmega )\).
Hence, \(u\in BV(\varOmega ,\nu )\) iff \(\displaystyle \lim _{t\rightarrow 0^+}\Vert D_H T_{\varOmega }(t)u \Vert _{L^1({\varOmega },\nu ;H)}<\infty \). In this case
Proof
(i) follows from the strong continuity of \(T_{{\varOmega }}(t)\) in \(L^1({\varOmega }, \nu )\), see Proposition 3(i), and the lower semicontinuity of the norm (9), which imply
To prove (ii) we write the \(L^1\)-norm of the gradient of \(T_{\varOmega }(t)u\) by duality, as
Taking into account that, for any \(F\in \mathrm{Lip}_b({\varOmega };H)\) we get
we deduce that
for any \(t>0\) where \(C_i\) (\(i=1,2\)) are the positive functions in Corollary 1. Thus, taking the limsup as \(t\rightarrow 0^+\) in (25) we get
and the proof is complete. \(\square \)
It follows from Theorem 3 that functions in \(BV({\varOmega },\nu )\) may be approximated in variation by smooth functions. This result was already known in infinite dimension when \({\varOmega }=X\) and \(T_{\varOmega }(t)\) is the Ornstein–Uhlenbeck semigroup and in a convex set, see [21], where the approximation is based on finite-dimensional reductions of the semigroup generated by the Neumann Ornstein–Uhlenbeck operator in \({\varOmega }\).
Proposition 6
Under Hypotheses 1, 2, 3 and 4, for any \(f\in BV({\varOmega },\nu )\) there exists a sequence \((f_n)_{n\in {{\mathbb {N}}}}\subseteq D^{1,2}({\varOmega },\nu )\) such that
If \(C\subseteq {\varOmega }\) is closed and \(|D_\nu f|(\partial C)=0\) then \(\displaystyle |D_\nu f|(C)=\lim _{n\rightarrow \infty }\int _{C}|D_H f_n|_H\mathrm{d}\nu .\)
Proof
Consider the semigroup \(T_{\varOmega }(t)\) generated in \(L^2({\varOmega },\nu )\) by the operator \(L_{\varOmega }\) defined in (4). It is known that for any \(f\in L^2({\varOmega }, \nu )\) the function \(T_{\varOmega }(t)f\) belongs to \(D^{1,2}({\varOmega },\nu )\) for any \(t>0\) and by the strong continuity of \(T_{\varOmega }(t)\), \(T_{\varOmega }(t)f\) converges to f in \(L^2({\varOmega },\nu )\) as \(t\rightarrow 0^+\). Moreover, Theorem 3 implies that \(\Vert D_H T_{\varOmega }(t)f\Vert _{L^1({\varOmega },\nu ;H)}\) converges to \(|D_\nu f|({\varOmega })\) as \(t \rightarrow 0^+\). Thus, (26) is proved. To complete the proof let us observe that, by the lower semicontinuity of the total variation, for any open set \(A\subseteq {\varOmega }\)
(see [21, Corollary 2.5]). Analogously we deduce that
for any closed subset \(C\subseteq {\varOmega }\). Indeed, by (27) we obtain
whence, using (26)(ii), estimate (28) follows. Now, using estimates (27), (28) and the fact that \(|D_\nu f|(\partial C)=0\) we obtain
where \(\mathring{C}\) denotes the interior of C. Estimate (29) yields the claim. \(\square \)
We conclude this section showing that estimate (8) and the previous approximation result allow to improve estimate (25) obtaining (30).
Theorem 4
Under Hypotheses 1, 2, 3 and 4, if \(f\in BV({\varOmega }, \nu )\) then
\(\lambda _1\) being the maximum eigenvalue of the covariance operator \(Q_\infty \), see (2). Moreover, for any open set \(A\subset {\varOmega }\) with \({\overline{A}}\subset \varOmega \),
Proof
Let \(f\in BV({\varOmega },\nu )\) and let \((f_n)_{n\in {{\mathbb {N}}}}\in D^{1,2}({\varOmega },\nu )\) be the sequence given by Proposition 6. By the contractivity of \(T_{\varOmega }(t)\) we deduce that \(T_{\varOmega }(t)f_n\) converges to \(T_{\varOmega }(t)f\) in \(L^2({\varOmega }, \nu )\) as \(n\rightarrow \infty \). This fact, together with the lower semicontinuity of the total variation, (8) and (6) yield
whence (30) is proved. The last assertion follows immediately from Proposition 6 taking into account that \(T_{\varOmega }(t)f\) satisfies (26). \(\square \)
4 Sets of finite perimeter in \({\varOmega }\)
This section is devoted to provide some sufficient and necessary conditions in order that a Borel set \(E \subseteq X\) have finite perimeter in \({\varOmega }\). We consider also the case of \(BV({\varOmega },\nu )\) functions and \({\varOmega }=X\). There are three semigroups involved: beside \(T_\varOmega (t)\), we consider the Ornstein–Uhlenbeck semigroup S(t) generated in \(L^2(X, \gamma )\) by the realisation of the operator
and the semigroup T(t) generated in \(L^2(X,\nu )\) by the realisation of the operator
Recall that S(t) admits a pointwise representation by means of the Mehler formula (1).
Theorem 5
Assume Hypotheses 1, 2, 3 and 4 hold true and let \(E\subseteq X\) be a Borel set such that \(P_\nu (E,{\varOmega })<\infty \). Then
More precisely
for any \(u \in BV({\varOmega }, \nu )\) and \(t>0\) where \(K_2\) is the constant in (7).
Proof
Clearly, once estimate (33) is proved, (32) follows at once choosing \(u=\chi _E\). Thus, let us prove (33). To this aim, we consider \(g\in L^\infty ({\varOmega }, \nu )\) and assume first that \(u \in D^{1,2}({\varOmega },\nu )\). By the self-adjointness of the operators \(L_{\varOmega }T_{\varOmega }(s)\) for \(s\ge 0\) in \(L^2({\varOmega },\nu )\) we have
The Cauchy–Schwarz inequality and (7) yield
From the contractivity of \(T_{\varOmega }(t)\) in \(L^\infty (\varOmega ,\nu )\), for any \(t>0\) we deduce
For \(u\in BV({\varOmega }, \nu )\), from Proposition 6 we get a sequence \(u_n\in D^{1,2}({\varOmega }, \nu )\) converging to u in \(L^2({\varOmega }, \nu )\) with \(\lim _{n \rightarrow \infty }\int _{\varOmega }|Du_n|_H \mathrm{d}\nu =|D_\nu u|({\varOmega })\). Thus, putting \(u_n\) in place of u in (34) and letting \(n \rightarrow \infty \) we get
Finally, taking the supremum on the \(g\in L^\infty (\varOmega ,\nu )\) with \(\Vert g\Vert _\infty \le 1\) we obtain
whence (33) follows. \(\square \)
Remark 4
Note that condition (32) is equivalent to
Indeed, \(|T_{\varOmega }(t)\chi _E-\chi _E|= (\chi _E-T_{\varOmega }(t)\chi _E)\chi _E+ (T_{\varOmega }(t)\chi _E-\chi _E)\chi _{E^c}\). The invariance of \(T_{\varOmega }(t)\) with respect to \(\nu \) in \(\varOmega \) yields
Consequently,
Now, we prove a quasi-converse of Theorem 5. We start with a preliminary result for bounded functions.
Proposition 7
Under Hypotheses 1, 2, 3 and 4, let \(u\in L^\infty (X,\nu )\) be such that
Then \(u \in BV({\varOmega },\nu )\) and \(|D_\nu u|({\varOmega })\le C\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}\sqrt{\pi }/2\).
Proof
We divide the proof into two steps.
Step 1. Here we prove that for any \(v \in C^1_b(X)\), it holds that
To this aim, we observe that
Now, for r fixed we perform the “Gaussian rotation”
to get, thanks to the invariance of \(\gamma \) under \(R_r\),
We claim that
Indeed, by the convexity of U there exist \(z\in X\) and \(a\in {{\mathbb {R}}}\) such that \(U(x)\ge {\left\langle x,z\right\rangle }+a\), hence
and, using De L’Hôpital’s rule, for almost every \((u,w)\in {\varOmega }\times X\)
So by the dominated convergence theorem we obtain
where we used that \(\int _X|{\left\langle D v(u),w\right\rangle }|\mathrm{d}\gamma (w) = \sqrt{2/\pi }|D v(u)|\). Indeed, using the factorisation \(\gamma = \gamma _1\otimes \gamma ^\perp \), where \(\gamma _1\) is the 1-dimensional standard Gaussian measure on \(E=\mathrm{span}Dv(u)\), we get
To conclude, consider the family of linear functionals \(L_t: C_b(X\times X)\rightarrow {{\mathbb {R}}}\), \(t \in (0,1)\)
By (37) we get \(\limsup _{t\rightarrow 0^+}{\left\| L_t\right\| }\le 2(\sqrt{\pi })^{-1}\Vert D v\Vert _{L^1({\varOmega },\nu )}\) and arguing as above
So \(L_t\) weakly\(^*\) converges to \(L_0\) as \(t \rightarrow 0^+\) and, by lower semicontinuity of the norm we get (36):
Step 2. For \(u\in L^\infty (X, \nu )\), let \((u_j)_{j\in {{\mathbb {N}}}}\subseteq C^1_b(X)\) be such that \(u_j\rightarrow u\) in \(L^2(X, \nu )\), almost everywhere in X and satisfying (35) (thanks to the dominated convergence theorem). Using (36), (6) and (8) we have
for any \(\sigma \in (0,1)\). Now, since the left-hand side of (38) is uniformly bounded from above by the constant C, the \(L^1\)-norm of \(D_HT_{\varOmega }(\sigma )u_j\) is bounded as well by the same constant for every \(j \in {{\mathbb {N}}}\) and \(\sigma \in (0,1)\), i.e.
Thus, recalling that \(D_H T_{\varOmega }(\sigma ) u_j\) converges to \(D_H T_{\varOmega }(\sigma ) u\) in \(L^1({\varOmega }, \nu )\) as \(j\rightarrow \infty \) (see (7)), letting first \(j \rightarrow \infty \) and then \(\sigma \rightarrow 0^+\) and using formula (24) we get that \(|D_\nu u|({\varOmega })\le C\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}\sqrt{\pi }/2\). \(\square \)
The following result is a quasi-converse of Theorem 5. In fact, we give a sufficient condition to have \(P_\nu (E,\varOmega )<\infty \) in terms of the short-time behaviour of T(t) and not of \(T_{\varOmega }(t)\), where T(t) is the semigroup generated by the operator L defined in (31) in \(L^2(X,\nu )\).
Theorem 6
Under Hypotheses 1, 2, 3 and 4, if \(E\in {{\mathcal {B}}}(X)\) and
then \(P_\nu (E, {\varOmega })\le C\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}\sqrt{\pi }/2\).
Proof
Choosing \(u=\chi _E\) in (35) and observing that
for any f with constant sign, from Proposition 7 we deduce that if
then \(P_\nu (E, {\varOmega })\le L\Vert Q_\infty ^{1/2}\Vert _{{\mathcal {L}}(X)}\sqrt{\pi }/2\). Here S(t) is the Ornstein–Uhlenbeck semigroup in (1). To conclude we prove that condition (40) is equivalent to (39). From the variation-of-constants formula we deduce
for every \(g\in {\mathcal {F}}C_b(X)\), \(\nu \)-a.e. \(x\in X\) and any \(t\ge 0\). To prove (41) it suffices that the map \(\sigma \mapsto S(t-\sigma )\langle D_H U, D_H T(\sigma )g\rangle _H\) belongs to \(L^1((0,t))\) for any \(t>0\). To this aim, let us observe that
for any \(g\in {\mathcal {F}}C_b(X)\). Indeed, the Hölder inequality and the contractivity of S(t) in \(L^2(X,\gamma )\) allow us to write
where in the last line we used estimate (7) which holds true even in the case \({\varOmega }=X\) and \(T_{\varOmega }(t)\) replaced by T(t). Hence, formula (41) follows.
Now, integrating (41) in \({\varOmega }\) with respect to \(\nu \) yields
for any \(t>0\) with
Using estimate (42) with \(g=\chi _E\) we infer that \(\limsup _{t\rightarrow 0^+}\frac{H(t)}{\sqrt{t}}<\infty \). This last estimate, together with (43), prove that (40) is equivalent to (39) and the proof is complete. \(\square \)
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20 May 2021
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Dedicated to Matthias Hieber on the occasion of his 60th-Birthday.
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The authors are members of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM). Work partially supported by the INdAM-GNAMPA Project 2019 “Metodi analitici per lo studio di PDE e problemi collegati in dimensione infinita” and by the Research project PRIN 2015 MIUR 2015233N5A “Deterministic and stochastic evolution equations” .
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Angiuli, L., Ferrari, S. & Pallara, D. On functions of bounded variation on convex domains in Hilbert spaces. J. Evol. Equ. 21, 3449–3475 (2021). https://doi.org/10.1007/s00028-021-00680-8
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DOI: https://doi.org/10.1007/s00028-021-00680-8