Abstract
A quenched large deviation principle for Brownian motion in a stationary potential is proved. As the proofs are based on a method developed by Sznitman [Comm. Pure Appl. Math. 47 (1994) 1655–1688] for Brownian motion among obstacles with compact support no regularity conditions on the potential is needed. In particular, the sufficient conditions are verified by potentials with polynomially decaying correlations such as the classical potentials studied by Pastur [Teoret. Mat. Fiz. 32 (1977) 88–95] and Fukushima [J. Stat. Phys. 133 (2008) 639–657] and the potentials recently introduced by Lacoin [Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 1010–1028; 1029–1048].
Highlights
Introduction and the main resultConsider a standard Brownian motion on Rd, (Zs; s ≥ 0), moving in a non-negative stationary ergodic potential
Where V0 is a real-valued non-negative random variable not identically zero on a probability space (Ω, F, P) and is a family of measurable maps on (Ω, F, P) which satisfies τx ◦ τy = τx+y for all x, y ∈ Rd, (x, ω) → τxω is measurable on the cartesian product Rd × Ω, P is invariant under τx for all x ∈ Rd and is ergodic, that is, if for some A ∈ F, τx(A) = A for all x ∈ Rd P(A) = 0 or 1
The goal of this paper is to extend the quenched LDP for the speed of the Brownian motion in a random potential to stationary random potentials without imposing a regularity condition
Summary
Consider a standard Brownian motion on Rd, (Zs; s ≥ 0), moving in a non-negative stationary ergodic potential. The expression of the rate function in terms of Lyapunov exponents allows to prove that the change in regime of the Brownian motion with constant drift observed by Sznitman ([36], Thm. 0.3) in a Poissonian potential associated to obstacles with compact support occurs for a large class of measurable potentials. This phase transition was further studied by Flury [14, 15] both in the discrete and the continuous settings.
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