Ergodic means for large values of~$T$ and exact asymptotics of small deviations for a~multi-dimensional Wiener process

V. R. Fatalov

Izvestiya. Mathematics , Tome 77 (2013) no. 6, pp. 1224-1259

Résumé

We prove results on exact asymptotics as $T\to\infty$ for the means $\mathsf{E}_{a,c}\exp\bigl\{-\int_0^T g(\mathbf{w}(t))\,dt\bigr\}$ and probabilities $\mathsf{P}_{a,c}\bigl\{\frac1T\int_0^Tg(\mathbf{w}(t))\,dt$, where $\mathbf{w}(t)=(w_1(t),\dots,w_n(t))$, $t\geqslant 0$, is an $n$-dimensional Wiener process, $g(x)$ is a positive continuous function (potential) satisfying certain conditions, $d>0$, and $a,c\in\mathbb{R}^n$ are prescribed vectors. The results are obtained by a new method developed in this paper, the Laplace method for the occupation time of a multi-dimensional Wiener process. We consider examples of monomial and radial potentials and prove results on exact asymptotics of small deviations for the probabilities $\mathsf{P}_0\bigl\{\int_0^1\sum_{j=1}^n|w_j(t)|^p\,dt\varepsilon^p\bigr\}$ as $\varepsilon\to 0$ with a fixed $p>0$.

Détail

Keywords: large deviations, Markov processes, Laplace method, action functional, occupation time, multi-dimensional Schrödinger operator.