Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator LJf(x) = ∫(f(x) − f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (−LJ, D) acts in L2(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel pJ (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (−LJ, D) extends in L2(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel pJ(t, x, y), and the function pJ(t, x, y) is uniformly comparable in t, x, y to the function pJ(t, x, y), the heat kernel related to the operator (−LJ,D). © 2018, Pleiades Publishing, Ltd.