On the dichotomy in the heat kernel two sided estimates

We study the off-diagonal estimates for transition densities of diffusions and jump processes in a setting when they depend essentially only on the time and distance. We state and prove the dichotomy for the tail of the transition density. 1. Preliminaries Let (M, d) be a locally compact separable metric space and μ be a Radon measure on M with full support. Definition 1.1. A family {pt}t>0 of measurable functions pt(x, y) on M ×M is called a heat kernel if the following conditions are satisfied, for μ-almost all x, y ∈ M and all s, t > 0: (i) Positivity: pt (x, y) ≥ 0. (ii) The total mass inequality (1.1) ∫ M pt(x, y)dμ(y) ≤ 1. (iii) Symmetry: pt(x, y) = pt(y, x). (iv) Semigroup property: (1.2) ps+t(x, y) = ∫ M ps(x, z)pt(z, y)dμ(z). (v) Approximation of identity: for any f ∈ L := L (M, μ), (1.3) ∫ M pt(x, y)f(y)dμ(y) L −→ f(x) as t → 0 + . Any heat kernel gives rise to the heat semigroup {Pt}t>0 where Pt is the operator on functions defined by (1.4) Ptu(x) = ∫ M pt(x, y)u(y)dμ(y). 2000 Mathematics Subject Classification. 47D07, 60G52, 60J35, 60J60. Supported by SFB 701 of the German Research Council (DFG). Supported by the Grant-in-Aid for Scientific Research (B) 18340027 (Japan). c ©0000 (copyright holder) 1 2 ALEXANDER GRIGOR’YAN AND TAKASHI KUMAGAI The conditions (i) − (iii) of Definition 1.1 imply that Pt is a bounded self-adjoint operator in L and, moreover, is a contraction (see, for example, [10, Page 28]). The semigroup identity (1.2) implies that PtPs = Pt+s, that is, the family {Pt}t>0 is a semigroup. It follows from (1.3) that slim t→0 Pt = I, where I is the identity operator in L and s-lim stands for strong limit. Hence, {Pt}t>0 is a strongly continuous, self-adjoint, contraction semigroup in L . Given the semigroup {Pt}t>0, define the infinitesimal generator L of the semigroup by (1.5) Lf := lim t→0 f − Ptf t , where the limit is understood in the L-norm. The domain dom(L) of the generator L is the space of functions f ∈ L for which the limit in (1.5) exists. By the Hille– Yosida theorem, dom(L) is dense in L. Furthermore, L is a self-adjoint, positive definite operator, which immediately follows from the fact that the semigroup {Pt} is self-adjoint and contractive. Moreover, we have (1.6) Pt = exp (−tL) , where the right hand side is understood in the sense of spectral theory. The notion of the heat kernel is closely linked to Markov processes. Let