Pointwise gradient bounds for degenerate semigroups (of UFG type)

DOI10.1098/RSPA.2016.0442zbMATH Open1371.60132arXiv1606.01027OpenAlexW2412973687WikidataQ59767471 ScholiaQ59767471MaRDI QIDQ5363695

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Publication date: 29 September 2017

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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.

Full work available at URL: https://arxiv.org/abs/1606.01027

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