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An application of Gelfand pairs to a problem of diffusion in compact ultrametric spaces. (English) Zbl 0985.60004
Taylor, J. C. (ed.), Topics in probability and Lie groups: boundary theory. Providence, RI: American Mathematical Society (AMS). CRM Proc. Lect. Notes. 28, 51-67 (2001).
Let $$X$$ be a compact ultrametric space whose group of isometries $$G$$ acts on $$X$$ transitively. If $$x_0\in X$$ and $$K=\{ g\in G:\;gx_0=x_0\}$$, then $$X\cong G/K$$ is endowed with a unique $$G$$-invariant probability measure, and $$(G,K)$$ is a Gelfand pair, that is, the convolution algebra of $$K$$-biinvariant integrable functions on $$G$$ is commutative. The author finds corresponding spherical functions and uses them for the construction of a $$K$$-invariant Markov process on $$X$$. The idea of the construction is to interpret $$X$$ as the set of ends of an infinite tree, to approximate it by finite trees, and to introduce on each of them an appropriate Markov chain. The process on $$X$$ is then obtained as their renormalized limit.
An expression for the generator $$-\Delta$$ is given. If $$1=r_1>r_2>\ldots >r_n>\ldots$$ are the values attained by the distance on $$X$$, and $$q_j$$ is the number of balls of the radius $$r_{j+1}$$ contained in a ball of the radius $$r_j$$, then $-\Delta f=\sum\limits_{j=1}^nq^{j-1}(E_jf-E_{j-1}f),\quad f\in L_2(X).$ Here $$E_jf$$ is the function which on each ball of the radius $$r_{j+1}$$ takes as the value the average of $$f$$ on that ball.
For the entire collection see [Zbl 0970.00015].

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 43A90 Harmonic analysis and spherical functions 60J60 Diffusion processes