Controllability and Qualitative properties of the solutions to SPDEs driven by boundary Lévy noise

Abstract

Let $u$ be the solution to the following stochastic evolution equation (1) du(t,x)& = &A u(t,x) dt + B σ(u(t,x)) dL(t),\quad t>0; u(0,x) = x taking values in an Hilbert space $\HH$, where $L$ is a $\RR$ valued Lévy process, $A:H\to H$ an infinitesimal generator of a strongly continuous semigroup, $σ:H\to \RR$ bounded from below and Lipschitz continuous, and $B:\RR\to H$ a possible unbounded operator. A typical example of such an equation is a stochastic Partial differential equation with boundary Lévy noise. Let $\CP=(\CP_t)_{t\ge 0}$ %{\CP_t:0\le t<\infty}$thecorrespondingMarkoviansemigroup.Weshowthat,ifthesystem(2)du(t)=Au(t)dt+Bv(t),t>0u(0)=xisapproximatecontrollableintime$T>0$,thenundersomeadditionalconditionson$B$and$A$,forany$x\in H$theprobabilitymeasure$\CP_T^\star δ_x$ispositiveonopensetsof$H$.Secondly,asanapplication,weinvestigateunderwhichconditionon$\HH$andontheL\acute{e}vyprocess$L$andontheoperator$A$and$B$ the solution of Equation [1] is asymptotically strong Feller, respective, has a unique invariant measure. We apply these results to the damped wave equation driven by Lévy boundary noise.

References in corpus (3)

• Densities for Ornstein-Uhlenbeck processes with jumps
• Exponential mixing for some SPDEs with Lévy noise
• Linear Operator Inequality and Null Controllability with Vanishing Energy for unbounded control systems