Upper and lower bounds for Dunkl heat kernel
Year: 2021
Type: preprint
Abstract: On $\mathbb R^N$ equipped with a normalized root system $R$, a multiplicity function $k(α) > 0$, and the associated measure $$ dw(\mathbf x)=\prod_{α\in R}|\langle \mathbf x,α\rangle|^{k(α)}\, d\mathbf x, $$ let $h_t(\mathbf x,\mathbf y)$ denote the heat kernel of the semigroup generated by the Dunkl Laplace operator $Δ_k$. Let $d(\mathbf x,\mathbf y)=\min_{σ\in G} \| \mathbf x-σ(\mathbf y)\|$, where $G$ is the reflection group associated with $R$. We derive the following upper and lower bounds for $h_t(\mathbf x,\mathbf y)$: for all $c_l>1/4$ and $00$ such that $$ C_{l}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{l}\frac{d(\mathbf{x},\mathbf{y})^2}{t}} Λ(\mathbf x,\mathbf y,t) \leq h_t(\mathbf{x},\mathbf{y}) \leq C_{u}w(B(\mathbf{x},\sqrt{t}))^{-1}e^{-c_{u}\frac{d(\mathbf{x},\mathbf{y})^2}{t}} Λ(\mathbf x,\mathbf y,t), $$ where $Λ(\mathbf x,\mathbf y,t)$ can be expressed by means of some rational functions of $\| \mathbf x-σ(\mathbf y)\|/\sqrt{t}$. An exact formula for $Λ(\mathbf x,\mathbf y,t)$ is provided.
Source: arXiv (Cornell University)
Authors: Dziubański, Jacek, Hejna, Agnieszka
Institution: University of Wrocław
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Narodowe Centrum Nauki
OpenAlex ID: W4225619453