Differentiable perturbations of Ornstein-Uhlenbeck operators

TL;DR

This paper extends the Ornstein-Uhlenbeck operator to include bounded, differentiable perturbations, proving the resulting operator remains m-dissipative and generates a diffusion semigroup in a Hilbert space.

Contribution

It introduces a method to handle small perturbations of Ornstein-Uhlenbeck operators in infinite-dimensional spaces, establishing their m-dissipativity and connection to stochastic differential equations.

Findings

The perturbed operator is m-dissipative.

Its closure generates a diffusion semigroup.

Connection to stochastic differential equations in Hilbert space.

Abstract

We prove an extension theorem for a small perturbation of the Ornstein-Uhlenbeck operator $(L,D(L))$ in the space of all uniformly continuous and bounded functions $f:H\to \Rset$, where $H$ is a separable Hilbert space. We consider a perturbation of the form $N_0\phi=L\phi+< D\phi,F>$ where $F:H\to H$ is bounded and Fr\'echet differentiable with uniformly continuous and bounded differential. Hence, we prove that $N_0$ is $m$-dissipative and its closure in $C_b(H)$ coincides with the infinitesimal generator of a diffusion semigroup associated to a stochastic differential equation in $H$.