Long-time behaviour of nonautonomous SPDE's.
The asymptotic behaviour of the time-inhomogeneous Markov process \((X,\mathbf P_ {s,x})\) defined by a stochastic evolution equation \(dX =\{ AX + f(t,X)\}\,dt + Q^ {1/2}\,dW\) in a Hilbert space \(H\) is studied. Here, \(W\) is a standard cylindrical Wiener process on \(H\), \(Q\) is a nonnegative self-adjoint bounded linear operator in \(H\), \(A\) is an infinitesimal generator of a strongly continuous semigroup on \(H\) and \(f:\mathbb R_ {+}\times H\to H\) is a Borel mapping. Define \(P_ {s,t} \psi (x) = \mathbf E_ {s,x}\psi (X_ {t})\) for all \(t\geq s\geq 0\), \(x\in H\) and all bounded Borel real functions \(\psi \) on \(H\), denote by \(P_ {s,t}^ {*}\) the adjoint operators on the space of finite (signed) Borel measures on \(H\) and by \(\| \cdot \| _ {\text{var}}\) the total variation norm on this space. The main objective of the paper is to find sufficient conditions for \(\lim _ {t\to \infty }\| P^ {*}_ {s,t}\mu - P^ {*}_ {s,t}\varrho \| _ {\text{var}} =0\) to hold for any \(s\geq 0\) and all probability measures \(\mu \), \(\varrho \) on \(H\). Some interesting examples of stochastic second-order parabolic equations are covered by the obtained result. The proof is based on the method of lower measures; lower bounds for the Radon-Nikodym derivative of the transition function \(P(t,x,t+1,\cdot )\) with respect to a suitable Gaussian measure, which might be of independent interest, are derived in the course of the proof.
nonautonomous stochastic partial differential equations
scientific article; zbMATH DE number 2138899