Quenched asymptotics for Brownian motion in generalized Gaussian potential

Abstract

In this paper, we study the long-term asymptotics for the quenched moment \[\mathbb{E}_x\exp \biggl\{\int_0^tV(B_s)\,ds\biggr\}\] consisting of a $d$-dimensional Brownian motion $\{B_s;s\ge 0\}$ and a generalized Gaussian field $V$. The major progress made in this paper includes: Solution to an open problem posted by Carmona and Molchanov [Probab. Theory Related Fields 102 (1995) 433-453], the quenched laws for Brownian motions in Newtonian-type potentials and in the potentials driven by white noise or by fractional white noise.

Published in at http://dx.doi.org/10.1214/12-AOP830 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models

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• Longtime asymptotics of the two-dimensional parabolic Anderson model with white-noise potential
• Moment estimates for some renormalized parabolic Anderson models
• Phase Transitions in Asymptotically Singular Anderson Hamiltonian and Parabolic Model
• Spectral rigidity of random Schrödinger operators via Feynman-Kac formulas
• Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise
• Spatial asymptotic of the stochastic heat equation with compactly supported initial data
• A $K$-rough path above the space-time fractional Brownian motion