Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus

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Abstract

In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on a Euclidean space and a compact manifold. We study the 'cut locus' case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe's distributional Malliavin calculus and T. Lyons' rough path theory.

Original languageEnglish
JournalForum of Mathematics, Sigma
Volume5
DOIs
Publication statusPublished - Jan 1 2017

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Cut Locus
Heat Kernel
Asymptotic Expansion
Compact Manifold
Rough Paths
Malliavin Calculus
Unique Solution
Diffusion Process
Join
Stochastic Equations
Euclidean space
Finite Set
Differential equations
Differential equation
Path
Energy
Hot Temperature

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Analysis
  • Computational Mathematics
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Mathematical Physics
  • Statistics and Probability
  • Theoretical Computer Science

Cite this

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AB - In this paper we prove a short time asymptotic expansion of a hypoelliptic heat kernel on a Euclidean space and a compact manifold. We study the 'cut locus' case, namely, the case where energy-minimizing paths which join the two points under consideration form not a finite set, but a compact manifold. Under mild assumptions we obtain an asymptotic expansion of the heat kernel up to any order. Our approach is probabilistic and the heat kernel is regarded as the density of the law of a hypoelliptic diffusion process, which is realized as a unique solution of the corresponding stochastic differential equation. Our main tools are S. Watanabe's distributional Malliavin calculus and T. Lyons' rough path theory.

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