On the transition density function of the diffusion process generated by the Grushin operator

TL;DR

This paper investigates the short-time asymptotic behavior of the transition density function for diffusion processes generated by the Grushin operator, revealing how these asymptotics depend on various parameters using explicit expectation-based expressions.

Contribution

It provides a detailed analysis of the short-time asymptotics of the transition density for Grushin-generated diffusions, including explicit expressions and parameter dependence.

Findings

Explicit short-time asymptotic formulas derived

Dependence of asymptotics on process parameters identified

Enhanced understanding of Grushin diffusion behavior

Abstract

The short-time asymptotic behavior of the transition density function of the diffusion process generated by the general Grushin operator will be investigated, by using its explicit expression in terms of expectation. Further the dependence on of the asymptotics will be seen.