Kolmogorov equations for stochastic convective Brinkman-Forchheimer equations forced by L\'evy Noise and its application to infinite horizon problems

Abstract

This article examines the Kolmogorov equation corresponding to the following stochastic two- and three-dimensional incompressible ($\nabla\cdot\boldsymbol{u}=0$) convective Brinkman-Forchheimer equations, also known as the damped Navier-Stokes equations, driven by L\'evy noise on the torus: \begin{align*} \mathrm{d}\boldsymbol{u}+[-\mu\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p]\mathrm{d} t =\sqrt{\mathrm{Q}}\mathrm{d}\mathrm{W}+\int_{Z}\sigma(t,z)\widetilde{\pi}(\mathrm{d} t,\mathrm{d} z), \end{align*} where $\mu,\alpha,\beta>0$ are physical constants; $\mathrm{Q}$ is a non-negative, trace-class operator; $\mathrm{W}$ is a cylindrical Wiener process on a Hilbert space; $\sigma$ represents the jump-noise coefficient; $(Z,\mathscr{B}(Z))$ is a measurable space; $\pi$ is a time-homogeneous Poisson random measure; and $\widetilde{\pi}$ denotes its compensator. The main contribution of this work is the establishment of the essential $m$-dissipativity of the corresponding Kolmogorov operator, a property that has received limited attention in the existing literature for systems driven by jump-type noise. \emph{Our main innovation is that, in contrast to traditional techniques which crucially depend on exponential moment estimates, we utilize the intrinsic structure of the absorption term $\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$ to dispense with these requirements. This allows us to establish the essential $m$-dissipativity of the Kolmogorov operator without the need for exponential moments.} We apply the developed framework to an infinite-horizon stochastic optimal control problem, demonstrating the solvability of the associated infinite-dimensional Hamilton-Jacobi-Bellman (integro-differential) equation.