Deformation of the heat kernel and the Brownian motion from the perspective of the Ben Sa\"id--Kobayashi--{\O}rsted $(k,a)$-generalized Laguerre semigroup theory

TL;DR

This paper introduces a new harmonic analysis framework called $(k,a)$-generalized Fourier analysis, constructs associated heat kernels and Brownian motions, and studies their properties, providing insights into deformations of classical analysis on $R^N$.

Contribution

It develops the $a$-deformed heat kernel and Brownian motion within the $(k,a)$-generalized Fourier analysis framework, a novel approach from representation theory.

Findings

Constructed the $a$-deformed heat kernel and Brownian motion.

Proved polynomial growth of Fourier integral kernels when $k=0$.

Explored basic properties of the deformed stochastic processes.

Abstract

We deform the heat kernel and the Brownian motion on $\mathbb{R}^{N}$ from the perspective of "$(k,a)$-generalized Fourier analysis" with $k=0$. This is a new type of harmonic analysis proposed by S.Ben Sa\"id--T.Kobayashi--B.{\O}rsted from the representation theoretic viewpoint. In this paper, we construct the $a$-deformed heat kernel and $a$-deformed Brownian motion, and explore their some basic properties. We also prove that the $(k,a)$-generalized Fourier integral kernels are polynomial growth when $k=0$, for a justification of some discussions.