Celebratio Mathematica — Varadhan

Large deviations

Works connected to Mátyás Arató

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M. D. Donsker and S. R. S. Varadhan: “Some problems of large deviations,” pp. 41–46 in Stochastic differential systems (Visegrád, Hungary, September 15–20, 1980). Edited by M. Arató, D. Vermes, and A. V. Balakrishnan. Lecture Notes in Control and Information Sciences 36. Springer (Berlin), 1981. MR 653644 Zbl 0472.60028 incollection

Let \( E_t \) refer to the expectation with respect to a three dimensional Brownian path \( \beta(\,\cdot\,) \) tied down at both ends with \( \beta(0) = \beta(t) = 0 \). Let \[ G(\alpha,t)=E_t\Bigl\{\exp\Bigl[\alpha\int_0^t\int_0^t\frac{e^{-|\sigma-s|}}{|\beta(\sigma)-\beta(s)|}d\sigma\,ds\Bigr]\Bigr\} \] show that, if \[ \lim_{t\to\infty}\frac{1}{t}\log G(\alpha,t)=g(\alpha) \] exists, \[ \lim_{\alpha\to\infty} \frac{g(\alpha)}{\alpha^2}=g_0 \] exists with \[ g_0 =\!\!\!\sup_{\varphi\in L_2(R^3), \,\|\varphi\|_2=1} \Bigl[2\iint\frac{\varphi^2(x)\,\varphi^2(y)}{|x-y|}\,dx\,dy-\frac{1}{2}\int|\nabla\varphi|^2\,dx\Bigr] \] The problem comes up in statistical mechanics. See for instance the book by Feynman [1972]. The formula for \( g_0 \) has been conjectured by Pekar [1949]. We shall outline a theory that allows us to prove these formulae.

Monroe D. Donsker	Related
Mátyás Arató	Related
A. V. Balakrishnan	Related
D. Vermes	Related