Lifting partial smoothing to solve semilinear Kolmogorov equations

Gozzi Fausto, Università LUISS
Masiero Federica, Unibersità di Milano-Bicocca

We study a family of semilinear Kolmogorv equations that are motivated by stochastic control problems arising in typical applications (such as boundary control and control of delay equations with delay in the control). These problems are difficult to treat since the underlying transition semigroups do not possess good smoothing properties nor the so-called "structure condition" which typically allows to apply the backward equations approach. In the papers [1] and [2] such problems are studied by developing new partial smoothing techniques which allowed us to obtain the required regularity in the case when the nonlinear term in the semilinear Kolmogorov equation is independent of the state variable.This is a somehow strong restriction which is not verified in most applications. In this talk we present a new approach to overcome this restriction. We extend the partial smoothing result to a wider class of functions which depend on the whole trajectory of the underlying semigroup and we use this as a key tool to improve our regularity result for semilinear Kolmogorov equation. The fact that such class depends on trajectories requires a nontrivial technical work as we have to lift the original transition semigroup to a space of trajectories, defining a new "high-level" environment where our problems can be solved. The talk is mainly based on [3].

Area: CS4 - Kolmogorov equations and long time behaviour for SPDEs (Carlo Orrieri & Luca Scarpa)

Keywords: transitions semigroups, partial smoothing properties, Kolmogorov equations

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