Partial smoothing of delay transition semigroups acting on special functions

TL;DR

This paper investigates the regularizing effects of delay transition semigroups on special functions, enabling the proof of existence and uniqueness of solutions to certain Kolmogorov equations and applications to stochastic control.

Contribution

It introduces a partial smoothing approach for delay transition semigroups acting on special functions, advancing the analysis of regularity and control problems.

Findings

Established regularizing properties on special functions

Proved existence and uniqueness of mild solutions for semilinear Kolmogorov equations

Applied results to stochastic optimal control problems

Abstract

It is well known that the transition semigroup of an Ornstein Uhlenbeck process with delay is not strong Feller for small times, so it has no regularizing effects when acting on bounded and continuous functions. In this paper we study regularizing properties of this transition semigroup when acting on special functions of the past trajectory. With this regularizing property, we are able to prove existence and uniqueness of a mild solution for a special class of semilinear Kolmogorov equations; we apply these results to a stochastic optimal control problem.