The Poisson boundary of Teichmüller space
Let \(T_g\) be the Teichmüller space of conformal structures on a compact Riemann surface \(M\) of genus \(g\geq 2\) and \(\Gamma\) the corresponding mapping class group. In analogy to the case \(g=1\), when \(T_1\) is given by the hyperbolic plane \(H^2\), the authors prove an integral representation Poisson formula for bounded harmonic functions on \(T_g\) in terms of their boundary values on PMF, the space of projective measured foliations on \(M\) which is the boundary of \(T_g\) in the Thurston \(\Gamma\)-compactification of \(T_g\). Based on anterior papers, they construct the Poisson boundary of a Markov operator \(P\) on \(T_g\) and identify it with PMF. Given a family of \(\Gamma\)-invariant probability measures \(\pi_x\), \(x\in T_g\), the authors consider the Markov chain with transition probabilities \(\pi_x\), the Markov operator \(P\), \(Pf(x)= \langle f,\pi_x\rangle\), and the \(P\)-harmonic functions \(f= Pf\) on \(T_g\). The Poisson boundary of \(P\) is given by a measure \(\Gamma\)-space \((\partial P,\nu)\) with a family of probability measures \(\nu_x\sim\nu\) on \(\partial P\), \(x\in T_g\), such that the Poisson formula \(f(x)= \langle\widehat f,\nu_x\rangle\) gives an isometry between the space \(H^\infty(T_g, P)\) of bounded \(P\)-harmonic functions and \(L^\infty(\partial P,\nu)\). Existence and uniqueness of \((\partial P,\nu)\) under assumptions on \(\pi_x\) and \(\nu_x\) have been established by \textit{V. A. Kaimanovich} in 1991 [Isr. J. Math. 89, No. 1-3, 77-134 (1995; Zbl 0843.43001)]. Under additional conditions on \(P\) and if \(\overline P\) (the quotient operator on the moduli space) is Harris recurrent (H.r.), the authors prove that there exists a unique family of \(\Gamma\)-invariant probability measures \(\lambda_x\), \(x\in T_g\), on PMF, stationary, pairwise equivalent, concentrated on the set of the uniquely ergodic foliations \(\text{UE}\subset \text{PMF}\) (Theorem 4.1). If in addition \(\overline P\) is positively H.r., then for any starting point \(x\) a.e. sample path of the Markov chain converges to a point of UE in Thurston compactification of \(T_g\), the limit distribution coincides with \(\lambda_x\) and \((\text{PMF},\lambda_x)\) is isomorphic to \((\partial P,\nu)\) (Theorem 4.3). Proofs are based on a Harnack inequality for discrete time Markov operators on \(T_g\), which allows a discretization assuring that under conditions of Theorem 4.1 for any point \(o\) in \(T_g\) with trivial stabilizer in \(\Gamma\), there is a probability measure \(\mu\) on \(\Gamma\) such that \((\partial P,\nu_0)\) is isomorphic as a measure \(\Gamma\)-space to the Poisson boundary \(\partial P_\mu\) of the random walk \((\Gamma,\mu)\), \(P_\mu\) being the Markov operator of \((\Gamma,\mu)\), and the restriction to the orbit \(\Gamma o\) gives a bijection between \(H^\infty(T_g,P)\) and the bounded \(P_\mu\)-harmonic functions. The coincidence of \(\partial P_\mu\) and PMF was already proved by the authors in [Invent. Math. 125, 221-264 (1996; Zbl 0864.57014)]. Conditions in Theorem 4.3 are satisfied for a class of geodesic random walk on \(T_g\), whose Poisson boundary coincides with \((\text{PMF},\lambda_x)\). Case \(g=2\) implies a special treatment due to the hyperelliptic involution in \(\Gamma\). Different explanations of applied methods, comments on possible generalizations, and open problems complete this important paper.
scientific article; zbMATH DE number 1214155
scientific article