Strong coupling limit of the Polaron measure and the Pekar process

The {\it{Polaron measure}} is defined as the transformed path measure $${\widehat{\mathbb{P}}}_{ϵ,T}=Z_{ϵ,T}^{-1}\exp \left\{\frac{1}{2}{\int }_{-T}^T{\int }_{-T}^T\frac{ϵ{\text{e}}^{-ϵ\mathrm{\mid }t-s\mathrm{\mid }}}{\mathrm{\mid }\omega (t)-\omega (s)\mathrm{\mid }}\text{d}s\text{d}t\right\}\text{d}\mathbb{P}$$ with respect to the law $\mathbb{P}$ of three dimensional Brownian increments on a finite interval $[-T,T]$, and $Z_{ϵ,T}$ is the partition function with $ϵ>0$ being a constant. The logarithmic asymptotic behavior of the partition function $Z_{ϵ,T}$ was analyzed in \cite{DV83} showing that $$g_0=\underset{ϵ\to 0}{\lim }\left[\underset{T\to \mathrm{\infty }}{\lim }\frac{\log Z_{\text{eps},T}}{2T}\right]=\underset{\text{heap}\psi \in H^1({\mathbb{R}}^3)\mathrm{\parallel }\psi {\mathrm{\parallel }}_2=1}{\sup }\{{\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}\text{d}x\text{d}y\frac{{\psi }^2(x){\psi }^2(y)}{\mathrm{\mid }x-y\mathrm{\mid }}-\frac{1}{2}\parallel \mathrm{\nabla }\psi {\parallel }_2^2\}\mathrm{.}$$ In \cite{MV18} we analyzed the actual path measures and showed that the limit ${\widehat {\mathbb P}}{\eps}=\lim{T\to\infty}\widehat{\mathbb P}{\eps,T}$ exists and identified this limit explicitly, and as a corollary, we also deduced the central limit theorem for $(2T{)}^{-1\mathrm{/}2}(\omega (T)-\omega (-T))$ under ${\widehat{\mathbb{P}}}_{\text{eps},T}$ and obtained an expression for the limiting variance ${\sigma }^2(\text{eps})$. In the present article, we investigate the {\it{strong coupling limit}} $\lim{\eps\to 0} \lim_{T\to\infty} \widehat{\mathbb P}{\eps,T}=\lim{\eps\to 0} \widehat {\mathbb P}_\eps$ and show that this limit coincides with the increments of the stationary Pekar process with generator $$\frac{1}{2}\mathrm{\Delta }+\left(\frac{\mathrm{\nabla }\psi }{\psi }\right)\cdot \mathrm{\nabla }$$ for any maximizer $\psi$ of the free enrgy $g_0$. The Pekar process was also earlier identified in \cite{MV14}, \cite{KM15} and \cite{BKM15} as the limiting object of the {\it{mean-field Polaron}} measures.}