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**Large deviations for the symmetric simple exclusion process in dimensions \(d\geqq 3\).**
*(English)*
Zbl 0928.60087

Let \(L\) particles perform a simple exclusion process on the \(d\)-dimensional lattice torus \(Z_N^d\) (where \(Z_N=\{0,\dots,N-1\}\) with addition modulo \(N\)) as follows. Each particle waits an exponential time with mean one and chooses a random site according to a given jump probability \(p(\cdot)\). If this site is not occupied by an other particle, then it jumps there, otherwise it does not jump and waits another exponential time. All particles do this simultaneously and independently of each other. There is never more than one particle at any site, and the particle number stays \(L\) for all times. The jump probability distribution \(p(\cdot)\) is assumed to be symmetric, irreducible and of finite range. The position of the \(i\)th particle at time \(t\) is denoted by \(x_i(t)\).

Introduce a time-space scaling by putting \(y_i(t)=x_i(N^2 t)/N\) and view \((y_1,\dots,y_L)\) as an element of the function space \(D=D([0,T],\mathbb T^d)\) for some fixed \(T>0\) where \(\mathbb T\) denotes the unit torus in \(\mathbb R\). The authors study large deviations for the empirical measure \(R_{N,\omega}=N^{-d}(\delta_{y_1}+\dots+\delta_{y_L})\) which is a random probability measure on \(D\). The particle number \(L\) and the scaling parameter are coupled by the assumption that \(L\sim N^d\overline \rho\) as \(L\) resp. \(N\) goes to infinity, for some \(\overline \rho\in(0,\infty)\). The initial configuration \((x_1(0),\dots,x_L(0))\) is chosen such that \(N^{-d}[\delta_{y_1(0)}+\dots+\delta_{y_L(0)}]\) converges weakly on \(\mathbb T^d\) toward a measure \(\rho_0(\theta) d\theta\).

In the authors’ earlier works it was proved that the empirical marginal distributions \(R_{N,\omega,t}=N^{-d}[\delta_{y_1(t)}+\dots+\delta_{y_L(t)}]\) satisfy a law of large numbers for fixed \(t> 0\) and converge weakly toward a non-random measure \(\rho(t,\theta) d\theta\) which satisfies the linear heat equation \(\partial_t\rho=\frac 12 \nabla\cdot D\nabla\rho\), where the diffusion matrix \(D\) is the covariance matrix of \(p(\cdot)\). Also it is known that for the empirical processes \(R_{N,\omega}\) one has a law of large numbers toward an inhomogeneous diffusion process on \(\mathbb T^d\) with explicitly given generator.

A large-deviation principle was established by Kipnis, Olla and Varadhan for \(R_{N,\omega,t}\) for fixed \(t>0\). The rate function consists of an initial part describing the deviations of the initial profile and a dynamical part which is defined in terms of an infimum of a certain Dirichlet form on a class of functions on \([0,T]\times\mathbb T^d\) that satisfy a certain linear heat equation. The result of the paper under consideration is a large-deviation principle for the empirical processes \(R_{N,\omega}\) with explicit rate function which again consists of an initial part (the same as above) and a dynamical part. The latter one is defined in terms of the relative entropy w.r.t. an inhomogeneous process that has the minimizer of the above-mentioned variational problem as its marginal distribution. The rate function is lower semicontinuous and has compact level sets.

Introduce a time-space scaling by putting \(y_i(t)=x_i(N^2 t)/N\) and view \((y_1,\dots,y_L)\) as an element of the function space \(D=D([0,T],\mathbb T^d)\) for some fixed \(T>0\) where \(\mathbb T\) denotes the unit torus in \(\mathbb R\). The authors study large deviations for the empirical measure \(R_{N,\omega}=N^{-d}(\delta_{y_1}+\dots+\delta_{y_L})\) which is a random probability measure on \(D\). The particle number \(L\) and the scaling parameter are coupled by the assumption that \(L\sim N^d\overline \rho\) as \(L\) resp. \(N\) goes to infinity, for some \(\overline \rho\in(0,\infty)\). The initial configuration \((x_1(0),\dots,x_L(0))\) is chosen such that \(N^{-d}[\delta_{y_1(0)}+\dots+\delta_{y_L(0)}]\) converges weakly on \(\mathbb T^d\) toward a measure \(\rho_0(\theta) d\theta\).

In the authors’ earlier works it was proved that the empirical marginal distributions \(R_{N,\omega,t}=N^{-d}[\delta_{y_1(t)}+\dots+\delta_{y_L(t)}]\) satisfy a law of large numbers for fixed \(t> 0\) and converge weakly toward a non-random measure \(\rho(t,\theta) d\theta\) which satisfies the linear heat equation \(\partial_t\rho=\frac 12 \nabla\cdot D\nabla\rho\), where the diffusion matrix \(D\) is the covariance matrix of \(p(\cdot)\). Also it is known that for the empirical processes \(R_{N,\omega}\) one has a law of large numbers toward an inhomogeneous diffusion process on \(\mathbb T^d\) with explicitly given generator.

A large-deviation principle was established by Kipnis, Olla and Varadhan for \(R_{N,\omega,t}\) for fixed \(t>0\). The rate function consists of an initial part describing the deviations of the initial profile and a dynamical part which is defined in terms of an infimum of a certain Dirichlet form on a class of functions on \([0,T]\times\mathbb T^d\) that satisfy a certain linear heat equation. The result of the paper under consideration is a large-deviation principle for the empirical processes \(R_{N,\omega}\) with explicit rate function which again consists of an initial part (the same as above) and a dynamical part. The latter one is defined in terms of the relative entropy w.r.t. an inhomogeneous process that has the minimizer of the above-mentioned variational problem as its marginal distribution. The rate function is lower semicontinuous and has compact level sets.

Reviewer: W.König (Berlin)