Continuity results and estimates for the Lyapunov exponent of Brownian motion in stationary potential

Rueß, Johannes

doi:10.1214/15-bjps288

Cited by 4 publications

(9 citation statements)

References 41 publications

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“…Recently, Ruess [32] considered Brownian motion in a stationary potential. Inspired by Schröder [34], he showed the existence of Lyapunov exponents for a large class of potentials and he expressed them in terms of a variational formula.…”

Section: Resultsmentioning

confidence: 99%

“…Therefore the shape theorem 2.2 and the LDP given in Theorem 1.1 hold for this family of potentials. The regularity of the potential, as defined in [32], is not needed for these results. However, for regularized versions of the potentials, [32] gave a variational expression for quenched free energy.…”

Section: 2mentioning

confidence: 99%

“…It does not seem possible to extend his results by approximating a measurable potential by regular potentials. Especially since Ruess [33] gave an example where the Lyapunov exponents are not continuous with respect to the potential.…”

Section: Introductionmentioning

confidence: 99%

“…is a positive constant c 10 such that W (x) ≥ c 10 (|x| −γ ∧ 1) for all x ∈ R d thenCov(V (0), V (x)) > C B(0) |x − u| −γ du > C|x| −γ .Ruess' potential. Ruess[32] gave an example of a two-dimensional Brownian motion in a stationary potential constructed from a planar Poissonian tessellation. A line in the plane is parametrized by its distance to the origin, denoted by ρ, and the angle θ ∈[0, π[ formed by the line and the horizontal axis.…”

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confidence: 99%

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Large deviations for Brownian motion in a random potential

Boivin

Lê

2020

ESAIM: PS

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A quenched large deviation principle for Brownian motion in a non-negative, stationary potential is proved. A sufficient moment condition on the potential is given but unlike the results of Armstrong and Tran (2014) no regularity is assumed. The proof is based on a method developed by Sznitman (1994) for Brownian motion among Poissonian potential. In particular, the LDP holds for potentials with polynomially decaying correlations such as the classical potentials studied by L. Pastur (1977) and R. Fukushima (2008) and the potentials recently introduced by H. Lacoin (2012).Subject Classification: 82B41, 60K37

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Section: Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Large deviations for Brownian motion in a random potential

Boivin

Lê

2020

ESAIM: PS

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“…In the case of the standard FPP model, a related variational formula in terms of cocycles was derived in [26] and the (level 1) entropy formula was recently proved in [3], where the formula was utilized to answer some questions about asymptotic properties of geodesics. The entropy variational formula was also proved for the Green's function decay rate for the Schrödinger operator with a periodic potential in the aforementioned [42] and this was extended to the case of a more general random ergodic potential in [40]. In [41], this variational expression was used to prove regularity properties of the Lyapunov exponent, as a function of the potential and of the law of the environment.…”

Section: Previous Workmentioning

confidence: 98%

A shape theorem and a variational formula for the quenched Lyapunov exponent of random walk in a random potential

Janjigian¹,

Sergazy²,

Rassoul-Agha³

2020

Preprint

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We prove a shape theorem and derive a variational formula for the limiting quenched Lyapunov exponent and the Green's function of random walk in a random potential on a square lattice of arbitrary dimension and with an arbitrary finite set of steps. The potential is a function of a stationary environment and the step of the walk. This potential is subject to a moment assumption whose strictness is tied to the mixing of the environment. Our setting includes directed and undirected polymers, random walk in static and dynamic random environment, and, when the temperature is taken to zero, our results also give a shape theorem and a variational formula for the time constant of both site and edge directed last-passage percolation and standard first-passage percolation.

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