Exponential asymptotics for time–space hamiltonians

Chen, Xia; Hu, Yaozhong; Song, Jian; Xing, Fei

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 4, pp. 1529-1561.

Dans ce papier, nous étudions le comportement en temps long du moment exponentiel du Hamiltonien dépendant du temps

$${\int }_0^t{\int }_0^t\frac{1}{{|r-s|}^{{\alpha }_0}}\gamma (B_r-B_s)\mathrm{d}s\mathrm{d}r,t\ge 0,$$

où $(B_s,s\ge 0)$ est un mouvement brownien de dimension $d$, le noyau $\gamma (·):{ℝ}^d\to [0,\infty )$ est une fonction homogène avec une singularité en zéro, ${\alpha }_0\in (0,1)$ et le paramètre de scaling $\gamma$ satisfont certaines conditions. Notre travail est partiellement motivé par l’étude des intersections ą courte portée de trajectoires, le polaron avec couplage fort et le modèle parabolique d’Anderson avec un potentiel donné par un bruit blanc fractionnaire en espace–temps.

In this paper, we investigate the long time asymptotics of the exponential moment for the following time–space Hamiltonian

$${\int }_0^t{\int }_0^t\frac{1}{{|r-s|}^{{\alpha }_0}}\gamma (B_r-B_s)\mathrm{d}s\mathrm{d}r,t\ge 0,$$

where $(B_s,s\ge 0)$ is a $d$-dimensional Brownian motion, the kernel $\gamma (·):{ℝ}^d\to [0,\infty )$ is a homogeneous function with singularity at zero; and ${\alpha }_0\in (0,1)$ together with the scaling parameter of $\gamma$ satisfies certain conditions. Our work is partially motivated by the studies of the short-range sample-path intersection, the strong coupling polaron, and the parabolic Anderson models with a time–space fractional white noise potential.

Mots clés : Time–space hamiltonian, brownian motion, Feynman–Kac large deviations

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