Pointwise Gradient Bounds for Degenerate Semigroups (of UFG type)

TL;DR

This paper investigates the long-term behavior of gradients of diffusion semigroups generated by degenerate operators satisfying the UFG condition, extending understanding beyond the classical Hörmander framework.

Contribution

It provides new results on exponential decay of gradients over time for semigroups under the UFG condition, a weaker assumption than Hörmander.

Findings

Gradients decay exponentially in certain directions under specific conditions.

Extended the analysis of semigroup behavior to large time scales.

Identified conditions for long-term smoothing effects in degenerate diffusions.

Abstract

In this paper we consider diffusion semigroups generated by second order differential operators of degenerate type. The operators that we consider do not, in general, satisfy the Hormander condition and are not hypoelliptic. In particular, instead of working under the Hormander paradigm, we consider the so-called UFG condition, introduced by Kusuoka and Strook in the eighties. The UFG condition is weaker than the uniform Hormander condition, the smoothing effect taking place only in certain directions (rather than in every direction, as it is the case when the Hormander condition is assumed). Under the UFG condition, Kusuoka and Strook deduced sharp small time asymptotic bounds for the derivatives of the semigroup in the directions where smoothing occurs. In this paper, we study the large time asymptotics for the gradients of the diffusion semigroup in the same set of directions and under the same UFG condition. In particular, we identify conditions under which the derivatives of the diffusion semigroup in the smoothing directions decay exponentially in time. This paper constitutes therefore a stepping stone in the analysis of the long time behaviour of diffusions which do not satisfy the Hormander condition.