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The {\it{Polaron measure}} is defined as the transformed path measure  $$\widehat{\mathbb P}_{\epsilon,T}= Z_{\epsilon,T}^{-1}\,\,\exp\bigg\{\frac{1}{2}\int_{-T}^T\int_{-T}^T\frac{\epsilon\e^{-\epsilon|t-s|}}{|\omega(t)-\omega(s)|}\,\d s \,\d t\bigg\}\d\mathbb P$$ with respect to the law $\mathbb P$ of threedimensional Brownian increments on a finite interval $[-T,T]$, and $Z_{\epsilon,T}$ is the partition function with $\epsilon>0$ being a constant.The logarithmic asymptotic behavior of the partition function $Z_{\epsilon,T}$was analyzed in \cite{DV83} showing that $$ g_0=\lim_{\epsilon \to0}\bigg[\lim_{T\to\infty}\frac{\log Z_{\eps,T}}{2T}\bigg]=\sup_{\heap{\psi\inH^1(\R^3)}{\|\psi\|_2=1}} \bigg\{\int_{\mathbb R^3}\int_{\mathbb R^3}\d x\dy\,\frac {\psi^2(x) \psi^2(y)}{|x-y|} -\frac 12\big\|\nabla\psi\big\|_2^2\bigg\}. $$ In \cite{MV18} we analyzed the actual path measuresand showed that the limit ${\widehat {\mathbbP}}_{\eps}=\lim_{T\to\infty}\widehat{\mathbb P}_{\eps,T}$ exists and identifiedthis limit explicitly, and as a corollary, we also deduced the central limittheorem for  $(2T)^{-1/2}(\omega(T)-\omega(-T))$ under $\widehat{\mathbb P}_{\eps,T}$ andobtained an expression for the limiting variance $\sigma^2(\eps)$.  In the present article, we investigate the {\it{strong coupling limit}}$\lim_{\eps\to 0} \lim_{T\to\infty} \widehat{\mathbb P}_{\eps,T}=\lim_{\eps\to0} \widehat {\mathbb P}_\eps$ and show that this limit coincides with theincrements of the stationary Pekar process with generator  $$  \frac 12 \Delta+ \bigg(\frac{\nabla\psi}\psi\bigg)\cdot \nabla  $$ for any maximizer $\psi$ of the free enrgy $g_0$. The Pekar process wasalso earlier identified in \cite{MV14}, \cite{KM15} and \cite{BKM15} as thelimiting object of the {\it{mean-field Polaron}} measures.}

Strong coupling limit of the Polaron measure and the Pekar process