On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

Abstract

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov–Bogolyubov procedure and compactness fails.

Appendix: Existence of Invariant Measures by a Limiting Procedure

We now present some absotract results which are used above to infer the existence of an invariant measure via an approximation procedure relying on invariant measures for a collection of regularized systems. It was used in Sect. 3.4.3 to prove the existence of an invariant measure.

Let $(H, \Vert \cdot \Vert _H)$, $(V, \Vert \cdot \Vert _V)$ be two separable Banach spaces. The associated Borel $\sigma$-algebras are denoted as $\mathcal {B}(H)$ and $\mathcal {B}(V)$ respectively. We suppose that V is continuously and compactly embedded in $H$. Moreover we assume that there exists continuous functions $\rho _n: H \rightarrow V$ for $n \ge 1$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } \Vert \rho _n(u)\Vert _V = {\left\{ \begin{array}{ll} \Vert u\Vert _V &{} \text { for } u \in V\\ \infty &{} \text { for } u \in H \setminus V. \end{array}\right. } \end{aligned}$$

Notice that, under these circumstances, $\mathcal {B}(V) \subset \mathcal {B}(H)$ and moreover that $A \cap V\in \mathcal {B}(V)$ for any $A \in \mathcal {B}(H)$. We can therefore extend any Borel measure $\mu$ on V to a measure $\mu _E$ on H by setting $\mu _E(A) = \mu (A \cap V)$ and hence we identify $Pr(V) \subset Pr(H)$. This natural extension will be made without further comment in what follows.

By appropriately restricting the domain of elements $\phi \in C_b(H)$ to V we have that $C_b(H) \subset C_b(V)$. Similarly $\mathrm {Lip}(H) \subset \mathrm {Lip}(V)$, etc. Furthermore, under the given conditions on H and V, $C_b(H) \cap \mathrm {Lip}(H)$ determines measures in Pr(V) namely if $\int _V \phi \, d\mu = \int _V \phi \, d \nu$ for all $\phi \in C_b(H) \cap \mathrm {Lip}(H)$ then $\mu = \nu$.

On $V$ we consider a Markov transition kernel P, which is assumed to be Feller in H, that is to say P maps $C_b(H)$ to itself. We also suppose that $\{P^\epsilon \}_{\epsilon > 0}$ is a sequence of Markov transition kernels (again defined on V) such that, for any $\phi \in C_b(H) \cap \mathrm {Lip}(H)$, and $R>0$,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \sup _{u \in B_{R}(V)} | P^\epsilon \phi (u) - P\phi (u)| = 0, \end{aligned}$$

(4.1)

where $B_R(V)$ is the ball of radius R in V.

Lemma 4.1

In the above setting, let $\{\mu ^\epsilon \}_{\epsilon > 0}$ be a sequence of probability measures on $V$. Assume that there is an increasing continuous function $\psi :[0,\infty ) \rightarrow [0,\infty )$ with $\psi (r) \rightarrow \infty$ as $r\rightarrow \infty$ and a finite constant $C_0 >0$ so that

$$\begin{aligned} \sup _{\epsilon > 0} \int \psi (\Vert u\Vert _{V}) d \mu ^\epsilon \le C_0. \end{aligned}$$

(4.2)

Then there exists a probability measure $\mu$, supported on V, with $\int \psi (\Vert u\Vert _{V}) d \mu (u) \le C_0$ such that (up to a subsequence) $\mu ^\epsilon P^\epsilon$ converges weakly in $H$ to $\mu P$ that is, for all $\phi \in C_b(H)$,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \mu ^\epsilon P^\epsilon \phi = \mu P\phi \end{aligned}$$

(4.3)

Proof of Lemma 4.1

From our assumption we know that

$$\begin{aligned} \mu ^\epsilon (\psi (\Vert u\Vert _{V}) \ge R) \le C_0/\psi (R) \end{aligned}$$

(4.4)

for all $\epsilon > 0$. We infer that the family of measures $\{\mu ^\epsilon \}_{\epsilon >0}$ is tight on $H$ and thus that there exists a measure $\mu$ on $H$ such that $\mu ^{\epsilon _n}$ converges weakly in H to $\mu$ for some decreasing subsequence $\epsilon _n \rightarrow 0$. For $k, m \ge 1$ define $f_{k,m} \in C_b(H)$ as $f_{k,m}(u) := \psi (\Vert \rho _m (u)\Vert _{V}) \wedge k$. Weak convergence in H implies that $\int f_{k,m} d \mu ^{\epsilon _n} \rightarrow \int f_{k,m} d \mu \le C_0$ as $n \rightarrow \infty$ for each fixed k, m. Fatou’s lemma then implies that

$$\begin{aligned} \int \psi (\Vert u\Vert _{V}) d\mu (u) \le \lim _{k, m \rightarrow \infty } \int f_{k,m}(u) d\mu (u) \le C_0 \end{aligned}$$

and in particular that $\mu (V)=1$.

We now turn to demonstrate (4.3). Observe that, for any $\phi \in C_b(H)$ and any $\epsilon >0$,

$$\begin{aligned} \big | \mu ^{\epsilon } P^{\epsilon }\phi - \mu P\phi \big |&\le \left| \mu ^{\epsilon } P^{\epsilon }\phi - \mu ^{\epsilon } P\phi \right| + \left| \mu ^{\epsilon } P\phi \ - \mu P\phi \right| \end{aligned}$$

(4.5)

Taking $\epsilon =\epsilon _n$, the first term is bounded as

$$\begin{aligned} \left| \mu ^{\epsilon _n} P^{\epsilon _n}\phi - \mu ^{\epsilon _n} P\phi \right|&\le \sup _{u \in B_{S}(V)}| P^{\epsilon _n} \phi (u)- P\phi (u) | + 2 \sup _{u} | \phi (u)| \, \mu ^{\epsilon _n} ( B_{S}(V)^c) \end{aligned}$$

(4.6)

for any $S > 0$. Combining (4.5), (4.6) with (4.1), (4.4), using that $\mu ^{\epsilon _n}$ converges weakly in $H$ and that $P^\epsilon \phi \in C_b(H)$ we infer (4.3), completing the proof. $\square$

This produces the following corollary.

Corollary 4.2

In the above setting, if in addition we assume that, for every $\epsilon >0$, $\mu ^\epsilon$ is an invariant measure for $P^\epsilon$ then the limiting measure $\mu$ is an invariant measure of P.

Proof

By the above result we may pick $\epsilon _n \rightarrow 0$ such that $\mu ^{\epsilon _n}$ and $\mu ^{\epsilon _n} P^{\epsilon _n}$ converge weakly in $H$ to $\mu$ and $\mu P$ respectively. However since $\mu ^{\epsilon _n} P^{\epsilon _n}=\mu ^{\epsilon _n}$ we also have that $\mu ^{\epsilon _n} P^{\epsilon _n}$ converges weakly in $H$ to $\mu$. Hence we conclude that $\mu P=\mu$ which is means the $\mu$ is an invariant measure for P. $\square$