Invariance principle and empirical mean large deviations of the critical Ornstein-Uhlenbeck process
Let \(I=Z^ d\) be the d-dimensional lattice. The author considers a system of interacting diffusions on \({\mathbb{R}}^ I\) of the form \[ X_ t(i)=x(i)+2^{-1}\int^{t}_{0}(-X_ s(i)+\sum_{k\neq i}\alpha (i- k)X_ s(k))ds+W_ t(i),\quad i\in I, \] where \(\alpha\) is nonnegative and symmetric and (W(i), \(i\in I)\) is a collection of independent Wiener processes. In the subcritical case, \(\sum_{k\neq 0}\alpha (k)<1\), the system has a unique Gaussian invariant distribution and a standard rescaling yields a classical central limit theorem, the limit being an \({\mathcal S}'({\mathbb{R}}^ d)\)-valued Gaussian process (\({\mathcal S}'({\mathbb{R}}^ d)\) is the space of tempered distributions). In the critical case, \(\sum_{k\neq 0}\alpha (k)=1\), for \(d\geq 3\) the process has infinitely many invariant distributions and the author proves a nonstandard invariance principle, the limit being an \({\mathcal S}'({\mathbb{R}}^ d)\)-valued Gaussian process, and identifies a stochastic partial differential equation satisfied by the limit process. [This result is very similar to the one for a system of critical branching Brownian motions, see \textit{R. A. Holley} and \textit{D. W. Stroock}, Publ. Res. Inst. Math. Sci. 14, 741-788 (1978; Zbl 0412.60065)]. Next, in the critical case the author derives dimension dependent large deviation results for the empirical mean \(Z_ T\equiv (1/T)\int^{T}_{0}X_ s(0)ds\), namely, there exists \(s_{\alpha}(d)\in (0,\infty)\) such that \[ \lim_{T\to \infty}(1/r_ T(d))\log P(Z_ T>\beta)=-(1/2s_{\alpha}(d))\beta^ 2, \] where the rate \(r_ T(d)\) depends on the dimension: \(r_ T(d)=T^{1/2}\) if \(d=3\), \(=T/\log T\) if \(d=4\), \(=T\) if \(d\geq 5.\) [These results resemble those for the occupation times of critical branching Brownian motions, see \textit{J. T. Cox} and \textit{D. Griffeath}, Ann. Probab. 13, 1108-1132 (1985; Zbl 0582.60091)].
occupation times of critical branching Brownian motions
scientific article; zbMATH DE number 4106017