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.

[1] X. Chen. Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs 157. Amer. Math. Soc., Providence, RI, 2010. | MR | Zbl

[2] X. Chen. Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models. Ann. Probab. 40 (2012) 1436–1482. | DOI | MR | Zbl

[3] X. Chen. Quenched asymptotics for Brownian motion in generalized Gaussian potential. Ann. Probab. 42 (2014) 576–622. | DOI | MR | Zbl

[4] X. Chen and J. Rosen. Large deviations and renormalization for Riesz potentials of stable intersection measures. Stochastic Proc. Appl 120 (2010) 1837–1878. | DOI | MR | Zbl

[5] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Wiener integrals for large time. In Functional Integration and Its Applications (Proc. Internat. Conf. London, 1974) 15–33. Clarendon Press, Oxford, 1975. | MR | Zbl

[6] M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. Pure Appl. Math. 29 (1976) 389–461. | DOI | MR | Zbl

[7] M. D. Donsker and S. R. S. Varadhan. Asymptotic for the Wiener sausage. Comm. Pure Appl. Math. 28 (1975) 525–565. | DOI | MR | Zbl

[8] M. D. Donsker and S. R. S. Varadhan. Asymptotics for the polaron. Comm. Pure Appl. Math. 36 (1983) 505–528. | DOI | MR | Zbl

[9] N. Dunford and J. Schwartz. Linear Operators 1. Wiley, New York, 1988. | Zbl

[10] J. Gärtner and W. König. The parabolic Anderson model. In Interacting Stochastic Systems 153–179. Springer, Berlin, 2005. | MR | Zbl

[11] R. Van Der Hofstad and W. König. A survey of one-dimensional random polymers. Stat. Phys. 103 (5/6) (2001) 915–944. | MR | Zbl

[12] Y. Hu and D. Nualart. Stochastic heat equation driven by fractional noise. Probab. Theory Related Fields 143 (2009) 285–328. | DOI | MR | Zbl

[13] Y. Hu, D. Nualart and J. Song. Feynman–Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39 (2011) 291–326. | MR | Zbl

[14] R. Léandre. Minoration en temps petit de la densité d’une diffusion dégénérée. J. Funct. Anal. 74 (1987) 399–414. | DOI | MR | Zbl

[15] E. H. Lieb and L. E. Thomas. Exact ground state energy of the strong coupling polaron. Comm. Math. Phys. 183 (1997) 511–519. | DOI | MR | Zbl

[16] U. Mansmann. The free energy of the Dirac polaron, an explicit solution. Stochastics Stochastics Rep. 34 (1991) 93–125. | DOI | MR | Zbl

[17] J. Song. Asymptotic behavior of the solution of heat equation driven by fractional white noise. Statist. Probab. Lett. 82 (2012) 614–620. | DOI | MR | Zbl

[18] K. Yosida. Functional Analysis. Springer, Berlin, 1966. | MR | Zbl