The heat semigroup associated with the Jacobi--Cherednik operator and its applications
Anirudha Poria, Ramakrishnan Radha
TL;DR
The paper studies the heat equation tied to the Jacobi--Cherednik operator on $\mathbb{R}$, constructing a strictly positive heat kernel $p_t^{\alpha,\beta}(x,y)$ and a strongly continuous, contraction semigroup $P_t^{\alpha,\beta}$ via the Opdam--Cherednik transform. It provides an explicit spectral representation $p_t^{\alpha,\beta}(x,y)=\int_{\mathbb{R}} e^{-{t}/{2}(\lambda^2+\rho^2)} G^{\alpha,\beta}_\lambda(x) G^{\alpha,\beta}_\lambda(-y) d\sigma_{\alpha,\beta}(\lambda)$, proves positivity and analyticity, and identifies the image of $L^2(\mathbb{R},A_{\alpha,\beta})$ under the heat semigroup as a reproducing kernel Hilbert space with kernel $K_t^{\alpha,\beta}(z,u)=p^{\alpha,\beta}_{2t}(-z,u)$. Applications include the construction of Jacobi--Cherednik Markov processes, analysis of the diffusion generator for $|X_t|$, and solving the modified Poisson equation via a Green operator with explicit spectral integral form. These results connect harmonic analysis on root systems with probabilistic processes and potential theory in one dimension.
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 $L^2(\mathbb R, A_{α, β})$ 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.