Random walks on groups and discrete subordination (Q2883880)

Subordination of Lévy processes, resp., of random walks, is a well-known tool in classical probability theory as well as in probabilities on groups. Let \(\left(\mu_t\right)_{t\geq 0}\) be a continuous convolution semigroup of probabilities on a locally compact group \(\mathbb{G}\), i.e., the distributions of a Lévy process \(\left(X_t\right)\), and let \(T\) be a \(\mathbb{R}_+\)-valued random variable (a random time) with distribution \(\nu\), \(T\) independent of \(\left(X_t\right)\). Then the random time substitution \(X_T\) is distributed as \(\mu_\nu:=\int_{\mathbb{R}_+} \mu_t d\nu(t)\). \(\mu_\nu\) is subordinated to \(\left(\mu_t\right)\) with subordinator \(T\). If \(T\) itself is embeddable into a Lévy process, i.e., if \(\nu\) is embeddable into a continuous convolution semigroup \(\left(\nu_t\right)\subseteq M^1(\mathbb{R}_+)\), then we obtain a subordinated continuous convolution semigroup \(\left(\mu_{\nu_t}\right)_{t\geq 0} \subseteq M^1(\mathbb{G})\).NEWLINENEWLINEA natural generalization is the subordination of (discrete time) random walks. Let \(\mu\in M^1(\mathbb{G})\), let \(\left(\mu^k\right)_{k\geq 0}\) denote the corresponding random walk, and let \(\nu\in M^1(\mathbb{Z}_+)\). Then \(\left(\mu^{\nu^k} := \sum_{l=0}^\infty\nu^k(l)\mu^l \right)_{k\geq 0}\) is the subordinated random walk (which can again be interpreted by random time substitutions). (The notations are similar to the ones used by \textit{J. T. Kent}, [Math. Proc. Camb. Philos. Soc. 90, 141--153 (1981; Zbl 0467.60023)]). Put \(c_l:=\nu(l)\), \(c_l^k:=\nu^k(l)\), hence \(\mu^\nu=\sum_{l=0}^\infty c_l\mu^l,\;\mu^{\nu^k} = \sum_{l=0}^\infty c_l^k\mu^l\).NEWLINENEWLINEIn the continuous time model, let \(-A\) and \(-B\) denote the infinitesimal generators of the \(C_0\)-semigroups of convolution operators \(\left(R_{\mu_t}\right)\) and \(\left(R_{\mu_{\nu_t}}\right)\), respectively, then \(B = \psi(A)\) for some function \(\psi\) (defined by the Lévy Khinchine representation of \(\left(\nu_t\right)\)). In the case of discrete times, replace \(A\) and \(B\) by \(I-R_\mu\) and \(I- R_{\mu^\nu}\), respectively, and define \(\psi\) via \(\psi(s):= 1-c_0 +\sum_1^\infty c_n s^n\) \((|s|\leq 1)\). Then \(T_\psi := I-\psi(I-R_\mu) \) is defined by a power series representation, and \(I-T_\psi = \psi(I-R_\mu) = \psi(I-R_{\mu^\nu})\).NEWLINENEWLINEIn the paper under review, the authors consider unimodular locally compact groups and absolutely continuous symmetric probabilities \(\mu\) with density \(\phi\). Of particular interest is the behaviour of \(\phi^{(2n)}(e)\), \(n\geq 1\), \(\phi^{(k)}\) denoting the convolution power. The determination of the rate of decay of \(\phi^{(2n)}(e)\) is a challenging problem in the theory of random walks on groups (cf., e.g., [\textit{N. Th. Varopoulos, L. Saloff-Coste} and \textit{T. Coulhon}, Analysis on Lie groups. Cambridge etc.: Cambridge University Press (1992; Zbl 0813.22003); \textit{W. Woess}, Random walks on infinite graphs and groups. Cambridge: Cambridge University Press (2000; Zbl 0951.60002)]). Here, in the paper under review, the emphasis is laid on densities with non-compact supports (if \(\mathbb{G}\) is discrete, then \(\phi^{(2n)}(e)\) can be interpreted as the probability of return to the unit after time \(2n\)).NEWLINENEWLINELet \(\phi_\psi\) denote the density of the subordinated measure, \(\phi_\psi =\sum c_n\phi^{(n)}\). \(\psi\) is supposed to be a Bernstein function, or a complete Bernstein function, i.e., representable as \(\psi(s) = s^2\mathcal{L}(g)(s)\), \(\mathcal{L}(g)\) denoting the Laplace transform of a Bernstein function \(g\). If \(\psi\) is given in the Lévy Khinchine representation with drift \(b\) and Lévy measure \(\nu\), then \(c_n = (1/n!)\int_{\mathbb{R}_+}t^ne^{-t}d\nu(t)\).NEWLINENEWLINEThe authors determine the behaviour of \(\phi_\psi^{(2n)}(e)\) if the behaviours of \(\phi^{(2n)}(e)\) (for \(n\to\infty\)) and of \(\psi\) (near \(0\)) are known. The investigations rely on the functional calculus for self adjoint elements in the von Neumann (group) algebra \(V(\mathbb{G})\), a tool which replaces Fourier analysis in the Abelian case. Note that \(V(\mathbb{G})\) is endowed with a faithful semi-finite normal trace \(\tau\), and that, for square-integrable \(\phi\), we have \(\tau\left(R_{\phi^{(2n)}}\right) = \phi^{(2n)}(e)\).NEWLINENEWLINEThis allows in various cases of densities and Bernstein functions to describe the behaviour of \(\phi_\psi^{(2n)}(e)\), depending on the behaviour of \(\phi^{(2n)}(e)\) and of \(\psi(s)\) for \(n\to\infty\) and \(s\to 0\), respectively.