Measure-valued equations for Kolmogorov operators with unbounded coefficients

TL;DR

This paper develops measure-valued equations for Kolmogorov operators with unbounded coefficients in Hilbert spaces, proving existence and uniqueness of solutions by linking to stochastic differential equations and extending to reaction-diffusion operators.

Contribution

It introduces a framework for measure-valued equations with unbounded coefficients and establishes core properties of the Kolmogorov operator in infinite-dimensional settings.

Findings

Proved existence and uniqueness of solutions for measure-valued equations.

Established that the Kolmogorov operator is a core of the associated generator.

Extended results to reaction-diffusion operators with polynomial nonlinearities.

Abstract

Given a real and separable Hilbert space H we consider the measure-valued equation \begin{equation*} \int_H\phi(x)\mu_t(dx)- \int_H\phi(x)\mu(dx)= \int_0^t(\int_HK_0\phi(x)\mu_s(dx))ds, \end{equation*} where K_0 is the Kolmogorov differential operator \[ K_0\phi(x)=\frac12\textrm{Trace}\big[BB^*D^2\phi(x)\big]+< x,A^*D\phi(x)>+< D\phi(x),F(x)>, \] $x\in H$, $\phi:H\to \Rset$ 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.