Identification of the Polaron measure I: Fixed coupling regime and the central limit theorem for large times

We consider the Fr\"ohlich model of the Polaron whose path integral formulation leads to the transformed path measure $$ \widehat{\mathbb P}_{\alpha,T}(\mathrm d\omega)= Z_{\alpha,T}^{-1}\,\, \exp\bigg\{\frac{\alpha}{2}\int_{-T}^T\int_{-T}^T\frac{e^{-|t-s|}}{|\omega(t)-\omega(s)|} \, d s \, d t\bigg\}\,\mathbb P(\mathrm d\omega) $$ with respect to $\mathbb P$ which 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. The Polaron measure reflects a self attractive interaction. According to a conjecture of Pekar that was proved in [DV83] $$ g_0=\lim_{\alpha \to\infty}\frac{1}{\alpha^2}\bigg[\lim_{T\to\infty}\frac{\log Z_{\alpha,T}}{2T}\bigg] $$ exists and has a variational formula. In this article we show that for any $\alpha>0$, the infinite-volume limit $\widehat{\mathbb P}_{\alpha}=\lim_{T\to\infty}\widehat{\mathbb P}_{\alpha,T}$ exists which is also identified explicitly. As a corollary, we deduce the central limit theorem (for any $\alpha>0$ and as $T\to\infty$) for the distribution of $\frac{\omega(T)-\omega(-T)}{\sqrt{2T}}$ both under the finite-volume Polaron measure $\widehat{\mathbb P}_{\alpha,T}$ and its infinite-volume counterpart $\widehat{\mathbb P}_\alpha$, and obtain an expression for the limiting variance.