The global behavior of heat kernels in exterior domains

We establish the global boundary behavior of Dirichlet heat kernels on some unbounded domains. These, combined with the interior estimate in the recent paper by Grigoryan and Saloff-Coste (Comm. Pure Appl. Math 55(1) (2002) 93), provide a complete and qualitatively sharp description of heat kernels G of Dirichlet Laplacians on some unbounded C1,1 domains D⊂M. Here M is certain complete noncompact manifold with nonnegative Ricci curvature. For example, there exist positive constants c1,c2 depending on the unbounded domain D, M such that, for ρ(x)=dist(x,∂D), ρ(x)t∧1 ∧1ρ(y)t∧1 ∧1c1e−c2d(x,y)2t|B(x,t)|⩽G(x,t;y,0)⩽ρ(x)t∧1 ∧1ρ(y)t∧1 ∧1e−d(x,y)2c2tc1|B(x,t)|for all x,y∈D and t>0. Existence of these global bounds had been an unresolved problem even in the Euclidean case.