Measure-valued equations for Kolmogorov operators with unbounded coefficients
Abstract
Description
Given a real and separable Hilbert space H we consider the measure-valued equation \begin{equation*} \int_Hϕ(x)μ_t(dx)- \int_Hϕ(x)μ(dx)=
\int_0^t(\int_HK_0ϕ(x)μ_s(dx))ds, \end{equation*} where K_0 is the Kolmogorov differential operator \[
K_0ϕ(x)=\frac12\textrm{Trace}\big[BB^*D^2ϕ(x)\big]+< x,A^*Dϕ(x)>+< Dϕ(x),F(x)>, \] $x\in H$, $ϕ:H\to \Rset$ is a suitable smooth function, $A:D(A)\subset H\to H $ is linear, $F:H\to H$ is a globally Lipschitz function and $B:H\to H$ is linear and continuous. In order prove existence and uniqueness of a solution for the above equation, we show that $K_0$ is a core, in a suitable way, of the infinitesimal generator associated to the solution of a certain stochastic differential equation in H.
We also extend the above results to a reaction-diffusion operator with polinomial nonlinearities.
38 pages
38 pages