Log-Harnack Inequality for Gruschin Type Semigroups

TL;DR

This paper establishes a log-Harnack inequality for a broad class of Gruschin type semigroups, including degenerate and non-Lipschitzian cases, using a novel coupling method and Girsanov theorem.

Contribution

It introduces a new coupling approach combined with Girsanov theorem to prove inequalities for complex semigroups with degenerate generators.

Findings

Log-Harnack inequality proven for Gruschin type semigroups

Applicable to degenerate and non-Lipschitzian generators

Method extends previous results to more general semigroups

Abstract

By constructing a coupling in two steps and using the Girsanov theorem under a regular conditional probability, the log-Harnack inequality is established for a large class of Gruschin type semigroups whose generator might be both degenerate and non-Lipschitzian.