Celebratio Mathematica — Varadhan

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J. Quastel and S. R. S. Varadhan: “Diffusion semigroups and diffusion processes corresponding to degenerate divergence form operators,” Comm. Pure Appl. Math. 50 : 7 (July 1997), pp. 667–706. MR 1447057 Zbl 0907.47040 article

In this article we study existence and uniqueness problems for semigroups and processes determined by time-dependent, second-order linear partial differential operators of the form \[ L_tu = \nabla\cdot a\nabla u + \sigma c\cdot\nabla u .\] For simplicity we will take the space to be the \( d \)-dimensional torus \( \mathbb{T}^d \). The matrix \[ a:[0,T]\times\mathbb{T}^d\to\mathbb{R}^d\otimes\mathbb{R}^d \] is assumed to be symmetric and nonnegative definite. We define \( \sigma \) to be the symmetric, nonnegative definite square root of \( a \), and \( c:[0,T]\times\mathbb{T}^d\to\mathbb{R}^d \) to be a function about which assumptions will be made later. The focus of our study is on cases where the coefficients \( a \) and \( c \) have minimal regularity, and \( a \) is allowed to degenerate.

J. Quastel, F. Rezakhanlou, and S. R. S. Varadhan: “Large deviations for the symmetric simple exclusion process in dimensions \( d\geq 3 \),” Probab. Theory Related Fields 113 : 1 (1999), pp. 1–84. MR 1670733 Zbl 0928.60087 article

We consider symmetric simple exclusion processes with \( L=\overline{\rho}N^d \) particles in a periodic \( d \)-dimensional lattice of width \( N \). We perform the diffusive hydrodynamic scaling of space and time. The initial condition is arbitrary and is typically far away form equilibrium. It specifies in the scaling limit a density profile on the \( d \)-dimensional torus. We are interested in the large deviations of the empirical process, \[ N^{-d}\Bigl[\sum_1^L\delta_{x_i(\,\cdot\,)}\Bigr] \] as random variables taking values in the space of measures on \( D[0,1] \). We prove a large deviation principle, with a rate function that is more or less universal, involving explicitly besides the initial profile, only such canonical objects as bulk and self diffusion coefficients.

Fraydoun Rezakhanlou	Related
Jeremy Quastel	Related