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- W4297782624 abstract "The {it{Polaron measure}} is defined as the transformed path measure $$widehat{mathbb P}_{epsilon,T}= Z_{epsilon,T}^{-1},, expbigg{frac{1}{2}int_{-T}^Tint_{-T}^Tfrac{epsilone^{-epsilon|t-s|}}{|omega(t)-omega(s)|} ,d s ,d tbigg}dmathbb P$$ with respect to the law $mathbb P$ of three dimensional 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 $$ g_0=lim_{epsilon to 0}bigg[lim_{Ttoinfty}frac{log Z_{eps,T}}{2T}bigg]=sup_{heap{psiin H^1(R^3)}{|psi|_2=1}} bigg{int_{mathbb R^3}int_{mathbb R^3}d xd y,frac {psi^2(x) psi^2(y)}{|x-y|} -frac 12big|nabla psibig|_2^2bigg}. $$ In cite{MV18} we analyzed the actual path measures and showed that the limit ${widehat {mathbb P}}_{eps}=lim_{Ttoinfty}widehat{mathbb P}_{eps,T}$ exists and identified this limit explicitly, and as a corollary, we also deduced the central limit theorem for $(2T)^{-1/2}(omega(T)-omega(-T))$ under $widehat{mathbb P}_{eps,T}$ and obtained an expression for the limiting variance $sigma^2(eps)$. In the present article, we investigate the {it{strong coupling limit}} $lim_{epsto 0} lim_{Ttoinfty} widehat{mathbb P}_{eps,T}=lim_{epsto 0} widehat {mathbb P}_eps$ and show that this limit coincides with the increments of the stationary Pekar process with generator $$ frac 12 Delta+ bigg(frac{nablapsi}psibigg)cdot nabla $$ 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.}" @default.
- W4297782624 created "2022-10-01" @default.
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- W4297782624 date "2018-06-18" @default.
- W4297782624 modified "2023-09-25" @default.
- W4297782624 title "Strong coupling limit of the Polaron measure and the Pekar process" @default.
- W4297782624 doi "https://doi.org/10.48550/arxiv.1806.06865" @default.
- W4297782624 hasPublicationYear "2018" @default.
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