  The path measure corresponding to the Fr\"ohlich Polaron appearing in quantum
statistical mechanics is defined as the tilted measure $$ \widehat{\mathbb
P}_{\varepsilon,T}=
\frac{1}{Z(\varepsilon,T)}\exp\bigg(\frac{1}{2}\int_{-T}^T\int_{-T}^T
\frac{\varepsilon\mathrm e^{-\varepsilon |t-s|}}{|\omega(t)-\omega(s)|} \mathrm
d s \,\mathrm d t\bigg)\mathrm d\mathbb P. $$ Here $\varepsilon>0$ is the Kac
parameter (or the inverse-coupling), and $\mathbb P$ is the law of $3d$
Brownian increments. In [13] it was shown that the (thermodynamic) limit
$\lim_{T\to\infty}\widehat{\mathbb P}_{\varepsilon,T}=\widehat{\mathbb
P}_\varepsilon$ exists as a process with stationary increments and this limit
was identified explicitly as a mixture of Gaussian processes. In the present
article, the strong coupling limit or the vanishing Kac parameter limit
$\lim_{\varepsilon\to 0} \widehat{\mathbb P}_\varepsilon$ is investigated. It
is shown that this limit exists and coincides with the increments of the Pekar
process, which is a stationary diffusion process with generator $\frac 12
\Delta+ (\nabla\psi/\psi)\cdot \nabla$, where $\psi$ is the unique (modulo
shifts) maximizer of the Pekar variational problem $$ g_0=\sup_{\|\psi\|_2=1}
\Big\{\int_{\mathbb R^3}\int_{\mathbb R^3}\,\psi^2(x)
\psi^2(y)|x-y|^{-1}\mathrm d x\mathrm d y -\frac 12\|\nabla \psi\|_2^2\Big\}.
$$ As shown in [12,6,1], the Pekar process is itself approximated by the
limiting "mean-field Polaron measures", and thus, the present identification of
the strong coupling Polaron is a rigorous justification of the "mean-field
approximation" (on the level of path measures) conjectured by Spohn in [15].
This approximation in the vanishing Kac limit ($\varepsilon\to 0$) is also
shown to hold for a general class of Kac-Interaction of the form
$H(t,x)=\varepsilon \mathrm e^{-\varepsilon|t|} V(x)$ where $V$ is any
continuous function vanishing at infinity.
