On the Cauchy problem for integro-differential equations with space-dependent operators in generalized H\"{o}lder classes

TL;DR

This paper studies parabolic integro-differential Kolmogorov equations with space-dependent operators in generalized Hölder spaces, proving existence, uniqueness, and regularity of solutions using probabilistic methods.

Contribution

It introduces a framework for analyzing such equations in Hölder spaces defined by scalable Lévy measures, providing new regularity results and solution properties.

Findings

Proved continuity of the operator using probabilistic representations.

Established existence and uniqueness of solutions.

Derived regularity estimates for solutions.

Abstract

Parabolic integro-differential Kolmogorov equations with different space-dependent operators are considered in H\"{o}lder-type spaces defined by a scalable L\'{e}vy measure. Probabilistic representations are used to prove continuity of the operator. Existence and uniqueness of the solution are established and some regularity estimates are obtained.