Relative heat content asymptotics for sub-Riemannian manifolds

The relative heat content associated with a subset Ω⊂M of a sub-Riemannian manifold is defined as the total amount of heat contained in Ω at time t, with uniform initial condition on Ω, allowing the heat to flow outside the domain. We obtain a fourth-order asymptotic expansion in the square root of t of the relative heat content associated with relatively compact noncharacteristic domains. Compared to the classical heat content that was studied by Rizzi and Rossi (J. Math. Pures Appl. (9)148 (2021), 267–307), several difficulties emerge due to the absence of Dirichlet conditions at the boundary of the domain. To overcome this lack of information, we combine a rough asymptotics for the temperature function at the boundary, coupled with stochastic completeness of the heat semigroup. Our technique applies to any (possibly rank-varying) sub-Riemannian manifold that is globally doubling and satisfies a global weak Poincaré inequality, including in particular sub-Riemannian structures on compact manifolds and Carnot groups.