Invariant measures for stochastic Cauchy problems with asymptotically unstable drift semigroup

Onno Gaans,Jan Neerven

doi:10.1214/ecp.v11-1184

Abstract

We investigate existence and permanence properties of invariant measures for abstract stochastic Cauchy problems of the form $$dU(t) = (AU(t)+f)\,dt + B\,dW_H(t), \ \ t\ge 0,$$ governed by the generator $A$ of an asymptotically unstable $C_0$-semigroup on a Banach space $E$. Here $f \in E$ is fixed, $W_H$ is a cylindrical Brownian motion over a separable real Hilbert space $H$, and $B$ is a bounded operator from $H$ to $E$. We show that if $E$ does not contain a copy of $c_0$, such invariant measures fail to exist generically but may exist for a dense set of operators $B$. It turns out that many results on invariant measures which hold under the assumption of uniform exponential stability of $S$ break down without this assumption.

Highlights

Let A be the infinitesimal generator of a C0-semigroup S = {S(t)}t 0 on a real Banach space E and let WH = {WH (t)}t 0 be a cylindrical Brownian motion over a separable real Hilbert space H
In this note we study invariant measures for the stochastic abstract Cauchy problem of the form dU (t) = (AU (t) + f ) dt + B dWH (t), t 0, where f ∈ E is a fixed vector and B ∈ L (H, E) is a bounded operator
We are interested in the situation where the semigroup S fails to be uniformly exponentially stable and intend to answer such questions as for ‘how many’ operators B an invariant measure exists and what can be said about its properties

Summary

Introduction

As is well known [3, 4, 11] a unique invariant measure μ exists if a weak solution U x exists for some (all) x ∈ E and the semigroup generated by A is uniformly exponentially stable It is obtained as the weak limit μ = limt→∞ μ(t), where μ(t) is the distribution of U (t) := U 0(t) as given by (1.2) with initial value x = 0. If the operators S(t) are compact for all t > 0, the existence of a nondegenerate invariant measure for the problem (1.1) with f = 0 implies that the semigroup S is uniformly exponentially stable [8, Theorem 2.6]. By standard results on Gaussian measures this implies that μ∞ is nondegenerate

Linear equations with additive noise

The inhomogeneous problem with additive noise