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Heat kernel estimates for Markov processes of direction-dependent type

Jaehoon Kang, Moritz Kassmann

Abstract

We prove sharp pointwise heat kernel estimates for symmetric Markov processes associated with symmetric Dirichlet forms that are local with respect to some coordinates and nonlocal with respect to the remaining coordinates. The main theorem is a robustness result like the famous estimate for the fundamental solution of second order differential operators, obtained by Donald G. Aronson. Analogous to his result, we show that the corresponding translation-invariant process and the one given by the general Dirichlet form share the same pointwise points.

Heat kernel estimates for Markov processes of direction-dependent type

Abstract

We prove sharp pointwise heat kernel estimates for symmetric Markov processes associated with symmetric Dirichlet forms that are local with respect to some coordinates and nonlocal with respect to the remaining coordinates. The main theorem is a robustness result like the famous estimate for the fundamental solution of second order differential operators, obtained by Donald G. Aronson. Analogous to his result, we show that the corresponding translation-invariant process and the one given by the general Dirichlet form share the same pointwise points.

Paper Structure

This paper contains 7 sections, 21 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Near-diagonal estimates and auxiliary results
  3. Off-diagonal lower bound
  4. Off-diagonal upper bound
  5. The case $d=n=1$
  6. Strategy in the general case
  7. Proof of \ref{['prop:main_general']}, \ref{['lem:Gl-gen']} and \ref{['lem:Fl-gen']}

Key Result

Theorem \oldthetheorem

Let $d,n\in\mathbb{N}$. Suppose $J$ satisfies J_comp and the functions $a_{ij}$ satisfy (H). Let $(\mathcal{E}, \mathcal{F})$ be the Dirichlet form given by def:DF. Then, there is a conservative Hunt process $X=(X_t, \mathbb{P}^x, x\in \mathbb{R}^{d+n}, t \ge 0)$ associated with $(\mathcal{E}, \math

Theorems & Definitions (26)

  • Remark
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  • Proposition \oldthetheorem
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  • ...and 16 more