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

TL;DR

This paper investigates the controllability and qualitative properties of solutions to stochastic partial differential equations driven by boundary Lévy noise, establishing conditions for positivity of transition measures and uniqueness of invariant measures.

Contribution

It provides new conditions linking approximate controllability to the positivity of transition measures and analyzes when solutions are asymptotically strong Feller with unique invariant measures.

Findings

Approximate controllability implies positivity of transition measures under certain conditions.

Conditions are identified for solutions to be asymptotically strong Feller.

Results are applied to the damped wave equation with 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 \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.