Heat kernels for a class of hybrid evolution equations

TL;DR

This paper develops explicit heat kernels for hybrid evolution equations involving coupled operators, extending classical methods and leveraging generalized Ornstein-Uhlenbeck operators to address complex PDEs in physics and geometry.

Contribution

It introduces a novel approach to construct heat kernels for coupled PDE operators that cannot be separated, based on generalized Ornstein-Uhlenbeck operators.

Findings

Explicit heat kernels for hybrid PDEs are constructed.

The method applies to equations in physics, geometry, and subelliptic PDEs.

The approach advances the understanding of hypoelliptic operators.

Abstract

The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form $\mathscr L_1 + \mathscr L_2 - \partial_t$, but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators $\mathscr L_1 - \partial_t$ and $\mathscr L_2 - \partial_t$. Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of H\"ormander's 1967 groundbreaking paper on hypoellipticity.