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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].

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