### 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 language | English |
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Journal | Forum of Mathematics, Sigma |

Volume | 5 |

DOIs | |

Publication status | Published - Jan 1 2017 |

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### 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

**Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus.** / Inahama, Yuzuru; Taniguchi, Setsuo.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Short time full asymptotic expansion of hypoelliptic heat kernel at the cut locus

AU - Inahama, Yuzuru

AU - Taniguchi, Setsuo

PY - 2017/1/1

Y1 - 2017/1/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85053826353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053826353&partnerID=8YFLogxK

U2 - 10.1017/fms.2017.14

DO - 10.1017/fms.2017.14

M3 - Article

VL - 5

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

SN - 2050-5094

ER -