Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times

We consider the Frohlich model of the polaron, whose path integral formulation leads to the transformed path measure (P) over cap (alpha,T) = Z(alpha,T)(-1) exp {alpha/2 integral(T)(-T) integral(T)(-T) e(-vertical bar t-s vertical bar)/vertical bar omega(t) - omega(s)vertical bar ds dt}dP with respect to P that governs the law of the increments of the three-dimensional Brownian motion on a finite interval [-T, T], and Z(alpha,T) is the partition function or the normalizing constant and alpha > 0 is a constant, or the coupling parameter. The polaron measure reflects a self-attractive interaction. According to a conjecture of Pekar that was proved in [9], g(0) = lim(alpha ->infinity) 1/alpha(2) [lim(T ->infinity) log Z(alpha,T)/2T] exists and has a variational formula. In this article we show that when alpha > 0 is either sufficiently small or sufficiently large, the limit (P) over cap (alpha) = lim(T ->infinity) (P) over cap (alpha,T) exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for 1/root 2T(omega(T) - omega(-T)) under (P) over cap (alpha,T) and obtain an expression for the limiting variance. (C) 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, Inc.