The heat semigroup associated with the Jacobi--Cherednik operator and its applications

TL;DR

This paper investigates the properties of the heat kernel and semigroup related to the Jacobi--Cherednik operator on the real line, providing solutions to associated PDEs and characterizing the transformed function spaces.

Contribution

It introduces new analysis of the Jacobi--Cherednik heat kernel, solves the Cauchy problem, and characterizes the heat semigroup's image on $L^2$ spaces, with applications to Poisson equations and Markov processes.

Findings

Heat kernel is strictly positive.

Solved the Cauchy problem for the Jacobi--Cherednik heat operator.

Characterized the image of $L^2$ under the heat semigroup.

Abstract

In this paper, we study the heat equation associated with the Jacobi--Cherednik operator on the real line. We establish some basic properties of the Jacobi--Cherednik heat kernel and heat semigroup. We also provide a solution to the Cauchy problem for the Jacobi--Cherednik heat operator and prove that the heat kernel is strictly positive. Then, we characterize the image of the space $L^2(\mathbb R, A_{\alpha, \beta})$ under the Jacobi--Cherednik heat semigroup as a reproducing kernel Hilbert space. As an application, we solve the modified Poisson equation and present the Jacobi--Cherednik--Markov processes.