Controllability and Qualitative properties of the solutions to SPDEs driven by boundary L\'evy noise

Abstract

Let $u$ be the solution to the following stochastic evolution equation (1) du(t,x)& = &A u(t,x) dt + B \sigma(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 L\'evy process, $A:H\to H$ an infinitesimal generator of a strongly continuous semigroup, $\sigma:H\to \RR$ 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 L\'evy noise. Let $\CP=(\CP_t)_{t\ge 0}$ %{\CP_t:0\le t<\infty}$ the corresponding Markovian semigroup. We show that, if the system (2) du(t) = A u(t)\: dt + B v(t),\quad t>0 u(0) = x is approximate controllable in time $T>0$, then under some additional conditions on $B$ and $A$, for any $x\in H$ the probability measure $\CP_T^\star \delta_x$ is positive on open sets of $H$. Secondly, as an application, we investigate under which condition on %$\HH$ and on the L\'evy process $L$ and on the operator $A$ 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 L\'evy boundary noise.