Ergodic Properties of Quasi-Markovian Generalized Langevin Equations with Configuration Dependent Noise and Non-conservative Force

Abstract

We discuss the ergodic properties of quasi-Markovian stochastic differential equations, providing general conditions that ensure existence and uniqueness of a smooth invariant distribution and exponential convergence of the evolution operator in suitably weighted $L^{\infty }$ spaces, which implies the validity of central limit theorem for the respective solution processes. The main new result is an ergodicity condition for the generalized Langevin equation with configuration-dependent noise and (non-)conservative force.

Appendices

A Auxiliary Material on Linear Algebra

The following Lemma A.1 is repeatedly used in the proofs of Proposition 3 and Lemma 3, as well as in Example 3 to show the positive (semi-)definiteness of symmetric matrices.

Lemma A.1

Let A be a symmetric block structured matrix of the form

$$ {\varvec{A}}:=\begin{pmatrix} {\varvec{A}}_{1,1} &{} {\varvec{A}}_{1,2} \\ {\varvec{A}}_{1,2}^{{T}} &{} {\varvec{A}}_{2,2} \\ \end{pmatrix} \in \mathbb {R}^{n+m \times n+m} $$

• (i) If ${\varvec{A}}_{2,2}$ is positive definite, then ${\varvec{A}}$ is positive (semi-)definite if and only if

  $$ {\varvec{A}}_{1,1} - {\varvec{A}}_{1,2}{\varvec{A}}_{2,2}^{-1} {\varvec{A}}_{1,2}^{{T}} $$

  is positive (semi-)definite

• (ii) If ${\varvec{A}}_{1,1}$ is positive definite, then ${\varvec{A}}$ is positive (semi-)definite if and only if

  $$ {\varvec{A}}_{2,2} - {\varvec{A}}_{1,2}^{{T}}{\varvec{A}}_{1,1}^{-1} {\varvec{A}}_{1,2} $$

  is positive (semi-)definite

• (iii) Let ${\varvec{A}}_{2,2}^{g}$ denote a generalised inverse of ${\varvec{A}}_{2,2}$, i.e., ${\varvec{A}}_{2,2}^{g}$ is a $m\times m$ matrix which satisfies

  $$ {\varvec{A}}_{2,2}{\varvec{A}}_{2,2}^{g}{\varvec{A}}_{2,2} = {\varvec{A}}_{2,2}. $$

  The matrix ${\varvec{A}}$ is positive semi-definite if and only if the matrices ${\varvec{A}}_{2,2}$ and ${{\varvec{A}}_{1,1} - {\varvec{A}}_{1,2}{\varvec{A}}_{2,2}^{g} {\varvec{A}}_{1,2}^{{T}}}$ are positive semi-definite, and

  $$ ({\varvec{I}} - {\varvec{A}}_{2,2} {\varvec{A}}_{2,2}^{g}){\varvec{A}}_{1,2}^{{T}} = \mathbf{0}, $$

  i.e., the span of the column vectors of ${\varvec{A}}_{1,2}$ is contained in the span of the column vectors of ${\varvec{A}}_{1,1}$.

Proof

The statements (i) and (ii) follow from Theorem 1.12 in [61]. Statement (iii) corresponds to Theorem 1.20 in the same reference.    $\square$

B Auxiliary Material on Stochastic Analysis

In this section we provide a brief overview of the general framework used in the ergodicity proofs and derivation of convergence rate in Sect. 3. For a comprehensive overview we refer to the review articles [32, 36, 50].

Consider an SDE defined on the domain ${\varOmega _{{\varvec{x}}}}= \mathbb {T}^{n_{1}} \times \mathbb {R}^{n_{2}}, n = n_{1}+n_{2}\in \mathbb {N}$ which is of the form

$$\begin{aligned} \mathrm{d}X = {\varvec{a}}(X) \mathrm{d}t+ {\varvec{b}}(X) \mathrm{d}{\varvec{W}}, ~ X(0) \sim \mu _{0}, \end{aligned}$$

(88)

with smooth coefficients ${\varvec{a}}\in \mathcal {C}^{\infty }( {\varOmega _{{\varvec{x}}}}, {\mathbb R}^{n}), {\varvec{b}}= [{\varvec{b}}_{i}]_{1\le i \le n} \in \mathcal {C}^{\infty }({\varOmega _{{\varvec{x}}}},{\mathbb R}^{n\times n})$, and initial distribution $\mu _{0}$. In order to simplify the presentation we further assume that the diffusion coefficient ${\varvec{b}}$ is such that the Itô and Stratonovich interpretation of (88) coincide, i.e.,

$$ \nabla \cdot \left( {\varvec{b}}\, {\varvec{b}}^{{T}} \right) - {\varvec{b}}\, \nabla \cdot {\varvec{b}}^{{T}} \equiv \mathbf{0}. $$

Let further $\mathcal {L}$ denote the associated infinitesimal generator of (88), i.e.,

$$\begin{aligned} \mathcal {L}= {\varvec{a}}(X) \cdot \nabla + {\varvec{b}}(X) : \nabla ^{2}, \end{aligned}$$

(89)

when considered as an operator on the core $\mathcal {C}^{\infty }({\varOmega _{{\varvec{x}}}},{\mathbb R})$, and let $\mathcal {L}^{\dagger }$ denote the formal adjoint of $\mathcal {L}$, i.e., the Fokker-Planck operator associated with the SDE (88). Furthermore, let $e^{t\mathcal {L}},e^{t\mathcal {L}^{\dagger }}$ denote the associated semigroup operators of $\mathcal {L}$, and $\mathcal {L}^{\dagger }$, respectively, i.e.,⁶

$$\begin{aligned} \forall \varphi \in \mathcal {C}^{\infty }({\varOmega _{{\varvec{x}}}},{\mathbb R}) : e^{t\mathcal {L}} \varphi (x) = { \mathbb {E}}[ \varphi (X(t)) \,\vert \,X(0) = x], \end{aligned}$$

(90)

for (Lebesgue-)almost all $x \in {\mathbb R}^{n}$, and

$$ \int \left( e^{t\mathcal {L}} \varphi \right) (x) \mu _{0}(\mathrm{d}x) = \int \varphi (x) \left( e^{t\mathcal {L}^{\dagger }} \mu _{0}\right) (\mathrm{d}x). $$

Definition 1

For a given function $\mathcal {K}\in \mathcal {C}^{\infty }({\varOmega _{{\varvec{x}}}}, [1,\infty ))$ which is such that $\mathcal {K}({\varvec{x}}) \rightarrow \infty$ as $||{\varvec{x}}||\rightarrow \infty$, define

$$\begin{aligned} ||\varphi ||_{L^{\infty }_{\mathcal {K}}} := \bigg \Vert { \frac{\varphi }{\mathcal {K}}}\bigg \Vert _{\infty }, ~\varphi : {\varOmega _{{\varvec{x}}}}\rightarrow {\mathbb R}~\text {measureable}. \end{aligned}$$

(91)

We denote by

$$\begin{aligned} L^{\infty }_{\mathcal {K}}({\varOmega _{{\varvec{x}}}}) := \left\{ \varphi \text { measurable } :||\varphi ||_{L^{\infty }_{\mathcal {K}}}<\infty \right\} \end{aligned}$$

(92)

the set of measurable functions for which the ratio $\frac{\varphi }{\mathcal {K}}$ is bounded.

It can be easily verified that $||\varphi ||_{L^{\infty }_{\mathcal {K}}}$ defines a norm and that $L^{\infty }_{\mathcal {K}}({\varOmega _{{\varvec{x}}}})$ equipped with the norm $||\varphi ||_{L^{\infty }_{\mathcal {K}}}$ can be associated with a Banach space, which we denote by $\left( L^{\infty }_{\mathcal {K}}({\varOmega _{{\varvec{x}}}}), ||\cdot ||_{L^{\infty }_{\mathcal {K}}} \right)$.

Throughout this article we use Lyapunov function techniques to show (geometric) ergodicity of SDEs of the generic form (88). More specifically, we follow the standard recipe for proofs of exponential convergences of the semigroup operator $e^{t\mathcal {L}}$ in weighted $L^{\infty }$ spaces as outlined, e.g., in [32, 36, 38, 50], that is we show that a suitable Lyapunov condition (Assumption B.1) and a minorization condition (Assumption B.2) are satisfied:

Assumption B.1

(Infinitesimal Lyapunov condition). There is a function $\mathcal {K}\in \mathcal {C}^{\infty }({\varOmega _{{\varvec{x}}}}, [1,\infty ))$ with $\lim _{\left\| {\varvec{x}}\right\| \rightarrow \infty }\mathcal {K}(x) = \infty$, and real numbers $a\in (0,\infty ), b \in {\mathbb R}$ such that,

$$\begin{aligned} \mathcal {L} \mathcal {K} \le - a \mathcal {K}+b. \end{aligned}$$

(93)

Assumption B.2

(Minorization condition). For some $t^{\prime }>0$ there exists a constant $\eta \in (0,1)$ and a probability measure $\nu$ such that

$$ \inf _{x\in \mathcal {C}} e^{t^{\prime }\mathcal {L}^{\dagger }}\delta _{x}(\mathrm{d}y) \ge \eta \nu (\mathrm{d}y) $$

where $\mathcal {C} = \{ x \in {\varOmega _{{\varvec{x}}}}: \; \mathcal {K}(x) \le \mathcal {K}_\mathrm{max} \}$ for some $\mathcal {K}_\mathrm{max} > 1+ 2b/a,$ where a, b are the same constants as in (93).

If the above assumptions are satisfied, then the following proposition, which follows from the arguments in [32] (see also the other above mentioned references), allows to derive exponential decay estimates in the respective weighted $L^{\infty }$ space associated with the Lyapunov function $\mathcal {K}$.

Proposition B.1

(Geometric ergodicity, [32]). Let Assumptions B.1 and B.2 hold. The solution of the SDE (88) admits a unique invariant probability measure $\pi$ such that

• (i) there exist positive constant $\lambda , \widetilde{C}$ so that for any $\varphi \in L^{\infty }_{\mathcal {K}}({\varOmega _{{\varvec{x}}}})$

  $$\begin{aligned} \big \Vert { e^{t\mathcal {L}} \varphi -{ \mathbb {E}}_{\pi }\varphi }\big \Vert _{L^{\infty }_{\mathcal {K}}} \le \widetilde{C}e^{-t\lambda } \big \Vert {\varphi - { \mathbb {E}}_{\pi }\varphi }\big \Vert _{L^{\infty }_{\mathcal {K}}}. \end{aligned}$$

  (94)
• (ii)
  $$\begin{aligned} \int _{{\varOmega _{{\varvec{x}}}}} \mathcal {K}\mathrm{d}\pi < \infty . \end{aligned}$$

  (95)

If for the solution of (88) the implications of Proposition B.1 hold we also say that the solution X of (88) is geometrically ergodic. In the main body of this article we use Proposition B.1 to derive exponential decay estimates of the form (46) in Theorems 1 to 4. In these theorems Assumption B.1 can be directly shown to hold by explicitly constructing a suitable Lyapunov function $\mathcal {K}$ satisfying (93) (see Lemmas 1, 3 and 10). A very common way to show Assumption B.2 is by showing (i) that the transition kernel associated with the SDE (88) is smooth as specified in Assumption B.3, and (ii) that the SDE (88) is controllable as specified in Assumption B.4. By virtue Lemma B.1 it then follows that a minorization condition holds.

Assumption B.3

For any $t>0$ the transition kernel associated with the SDE (88) possesses a density $p_{t} (x, y)$, i.e.,

$$ \forall \,x \in {\varOmega _{{\varvec{x}}}}:~(e^{t\mathcal {L}^{\dagger }}\delta _{x})(A)= \int _{A} p_{t}(x,y)dy, ~ A \subset {\varOmega _{{\varvec{x}}}}, \; A \text { measurable}. $$

and $p_{t}(x,y)$ is jointly continuous in $(x,y)\in {\varOmega _{{\varvec{x}}}}\times {\varOmega _{{\varvec{x}}}}$.

Assumption B.4

There is a $t_{\max }>0$ so that for any $x^{-}, x^{+} \in {\varOmega _{{\varvec{x}}}}$, there is a $t>0$, with $t \le t_{\max }$, so that the control problem

$$\begin{aligned} \begin{aligned} \dot{\tilde{X}}&= {\varvec{a}}(\tilde{X}) + {\varvec{b}}(\tilde{X}) u, \end{aligned} \end{aligned}$$

(96)

subject to

$$ \tilde{X}(0)= x^{-}, \text { and } \tilde{X}(t)=x^{+}, $$

has a smooth solution $u \in \mathcal {C}^{1}([0,t_{\max }],{\varOmega _{{\varvec{x}}}})$.

Lemma B.1

([36]). If Assumptions B.3 and B.4 are satisfied, then also Assumption B.2 holds.

Assumption B.3 follows directly from hypoellipticity of the operator $\partial _{t} - \mathcal {L}^{\dagger }$ (see e.g. [47, 50], for a precise definition of hypoellipticity). A common way to establish hypoellipticity of a differential operators is via Hörmander’s theorem ([20], Theorem 22.2.1, on p. 353). The following proposition is an adaption of Hörmander’s theorem to the parabolic differential operator $\partial _{t} -\mathcal {L}^{\dagger }$:

Proposition B.2

Let ${\varvec{a}}$ and ${\varvec{b}}$ be the drift coefficient and the diffusion coefficient of the SDE (88), respectively. Let ${\varvec{b}}_{0} := {\varvec{a}}$. Iteratively define a collection of vector fields by

$$\begin{aligned} \mathscr {V}_{0} = \{ {\varvec{b}}_{i} : i \ge 1 \}, \mathscr {V}_{k+1} =\mathscr {V}_{k} \cup \{ [{\varvec{v}},{\varvec{b}}_{i}] : {\varvec{v}} \in \mathscr {V}_{k}, 0 \le i \le n\}. \end{aligned}$$

(97)

where

$$ [{\varvec{X}}, {\varvec{Y}}] = (\nabla {\varvec{Y}} ) {\varvec{X}} -( \nabla {\varvec{X}}) {\varvec{Y}}, $$

denotes the commutator of vector fields ${\varvec{X}},{\varvec{Y}} \in \mathcal {C}^{\infty }({\varOmega _{{\varvec{x}}}},{\mathbb R}^{n})$ and $(\nabla {\varvec{X}}),(\nabla {\varvec{Y}})$ their Jacobian matrices. If

$$\begin{aligned} \forall {\varvec{x}}\in \mathbb {R}^{n},~~ \mathrm {lin}\left\{ {\varvec{v}}({\varvec{x}}) : {\varvec{v}} \in \bigcup _{k \in \mathbb {N}}\mathscr {V}_{k} \right\} = \mathbb {R}^{n}, \end{aligned}$$

(98)

we say that the SDE (88) satisfies the parabolic Hörmander condition, and it follows that the operator $\partial _{t} - \mathcal {L}^{\dagger }$ is hypoelliptic.

We use Lemma B.1 in the proof of Lemma 4 in Theorem 2. For some instances of (20) it is not easy to construct a suitable control u such that Assumption B.4 is satisfied. In these cases we either show a minorization condition by explicitly constructing the minorizing measure $\nu$ in Assumption B.2 if the right hand side of (20) can be decomposed into a linear and a bounded part (see Theorem 1), or by inferring the existence of a suitable minorizing measure by showing that the support of the SDE under consideration is equivalent to the support of another SDE satisfying a minorization condition via Girsanov’s theorem (Lemmas 5 and 7). Girsanov’s theorem provides conditions under which the path measures of two Itô processes are mutually absolutely continuous, which in particular implies that at any time $t \ge 0$ the laws of these Itô processes are equivalent. We will use Girsanov’s theorem in Sect. 3 in order to prove the minorization condition for GLEs which in a Markovian representation possess coefficients which depend on the configurational variable. Here we provide a version of Girsanov’s theorem which is adapted to Itô-diffusion processes.

Proposition B.3

(Girsanov’s theorem, [45]). Consider the two Itô diffusion processes

$$\begin{aligned} \mathrm{d}X(t)&= {\varvec{a}}_{x}(X) \mathrm{d}t + {\varvec{b}}(X)\mathrm{d}{\varvec{W}}(t); ~ X(0) = x_{0}, \end{aligned}$$

(99)

$$\begin{aligned} \mathrm{d}Y(t)&= {\varvec{a}}_{y}(Y) \mathrm{d}t + {\varvec{b}}(Y)\mathrm{d}{\varvec{W}}(t); ~ Y(0) = x_{0}, \end{aligned}$$

(100)

where $x_{0} \in {\varOmega _{{\varvec{x}}}}$, ${\varvec{W}}$ is a standard Wiener process in ${\mathbb R}^{n}$, and ${\varvec{a}}_{x},{\varvec{a}}_{y} : {\varOmega _{{\varvec{x}}}}\rightarrow {\mathbb R}^{n}$ and ${\varvec{b}}: {\varOmega _{{\varvec{x}}}}\rightarrow {\mathbb R}^{n \times m}, m \in \mathbb {N}$, are such that there exist unique strong solutions X, Y for (99) and (100), respectively. If there is a function ${\varvec{u}} \in \mathcal {C}({\varOmega _{{\varvec{x}}}},{\mathbb R}^{n})$ such that

$$ {\varvec{a}}_{x} -{\varvec{a}}_{y} = {\varvec{b}}{\varvec{u}} $$

and ${\varvec{u}}$ satisfies Novikov’s condition

$$\begin{aligned} { \mathbb {E}}\left[ \exp \left( \frac{1}{2} \int _{0}^{T} ||{\varvec{u}} (X(t))||_{2}^{2} \mathrm{d}t \right) \right] < \infty . \end{aligned}$$

(101)

then the path measures of X and Y on any finite time interval are equivalent. In particular, the support of the law of X(t) and the support of the law of Y(t) coincide for any $t>0$.