Heat Kernel Analysis for Ornstein-Uhlenbeck Operators with Quadratic Potentials

TL;DR

This paper analyzes Ornstein-Uhlenbeck operators with quadratic potentials using Hamiltonian formalism to explicitly characterize singularities and derive closed-form heat kernels through a probabilistic approach.

Contribution

It introduces a Hamiltonian formalism to explicitly characterize singularities and derives closed-form heat kernels for Ornstein-Uhlenbeck operators with quadratic potentials.

Findings

Explicit geodesics of the operators are obtained.

Closed-form heat kernel formulas are derived.

Singularities produced by the potentials are characterized.

Abstract

In this paper, we study Ornstein-Uhlenbeck operators with quadratic potentials. We use Hamiltonian formalism to characterise the singularities produced by the potentials by finding explicit geodesics of the operators, and obtain the heat kernels via a probabilistic ansatz. All the formulae are closed.