Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey
- Authors: Bendikov A.1
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Affiliations:
- Institut Matematyczny
- Issue: Vol 10, No 1 (2018)
- Pages: 1-11
- Section: Review Articles
- URL: https://journals.rcsi.science/2070-0466/article/view/200917
- DOI: https://doi.org/10.1134/S2070046618010016
- ID: 200917
Cite item
Abstract
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).
About the authors
Alexander Bendikov
Institut Matematyczny
Author for correspondence.
Email: bendikov@math.uni.wroc.pl
Poland, Pl. Grunwaldzki 2/4, Wroclaw, 50-384