Heat Kernels for a Class of Hybrid Evolution Equations

Abstract

AbstractThe 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 ${\mathscr{L}}_{1} + {\mathscr{L}}_{2} - \partial _{t}$ L 1 + L 2 − ∂ 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 ${\mathscr{L}}_{1} - \partial _{t}$ L 1 − ∂ t and ${\mathscr{L}}_{2} - \partial _{t}$ L 2 − ∂ t . Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander’s 1967 groundbreaking paper on hypoellipticity.