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Abstract  We consider the Fröhlich model of the  polaron , whose path integral formulation leads to the transformed path measure    ℙ   α  ,  T    =   Z   α  ,  T    −  1    expα  2    ∫   −  T   T   ‍   ∫   −  T   T   ‍    e   −  ∣  t  −  s  ∣∣  ω   t   −  ω   s   ∣d  s    d  t    d  ℙ   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   α ,  T    is the partition function or the normalizing constant and   α  > 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   α  →  ∞1   α  2lim   T  →  ∞log   Z   α  ,  T2  Texists and has a variational formula. In this article we show that when   α  > 0  is either sufficiently small or sufficiently large, the limitℙ  ⌢   α   =    lim   T→  ∞ℙ  ⌢    α  ,  Texists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for    1    2  Tω   T   −  ω    −  Tunderℙ  ⌢    α  ,  Tand obtain an expression for the limiting variance. © 2019 the Authors.  Communications on Pure and Applied Mathematics  is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, LLC.

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