Regularity properties for solutions of infinite dimensional Kolmogorov equations in Hilbert spaces

TL;DR

This paper proves that solutions to infinite dimensional Kolmogorov equations in Hilbert spaces inherit smoothness properties from their coefficients, with enhanced regularity of derivatives due to the linear operator, aiding numerical approximation analysis.

Contribution

It establishes regularity and enhanced derivative regularity for solutions of infinite dimensional Kolmogorov equations with smooth coefficients, crucial for numerical methods.

Findings

Solutions are n-times Fréchet differentiable if coefficients are.

Enhanced regularity of derivatives due to the linear operator.

Results facilitate weak convergence analysis for SPDE approximations.

Abstract

In this article we establish regularity properties for solutions of infinite dimensional Kolmogorov equations. We prove that if the nonlinear drift coefficients, the nonlinear diffusion coefficients, and the initial conditions of the considered Kolmogorov equations are $n$-times continuously Fr\'{e}chet differentiable, then so are the generalized solutions at every positive time. In addition, a key contribution of this work is to prove suitable enhanced regularity properties for the derivatives of the generalized solutions of the Kolmogorov equations in the sense that the dominating linear operator in the drift coefficient of the Kolmogorov equation regularizes the higher order derivatives of the solutions. Such enhanced regularity properties are of major importance for establishing weak convergence rates for spatial and temporal numerical approximations of stochastic partial differential equations.