Almost sure exponential behavior of a directed polymer in a fractional Brownian environment

Abstract

This paper studies the asymptotic behavior of a one-dimensional directed polymer in a random medium. The latter is represented by a Gaussian field B H on R + × R with fractional Brownian behavior in time (Hurst parameter H ) and arbitrary function-valued behavior in space. The partition function of such a polymer isHere b is a continuous-time nearest neighbor random walk on Z with fixed intensity 2κ, defined on a complete probability space P b independent of B H . The spatial covariance structure of B H is assumed to be homogeneous and periodic with period 2π . For H < 1 2 , we prove existence and positivity of the Lyapunov exponent defined as the almost sure limit lim t→∞ t −1 log u(t). For H > 1 2 , we prove that the upper and lower almost sure limits lim sup t→∞ t −2H log u(t) and lim inf t→∞ (t −2H log t) log u(t) are non-trivial in the sense that they are bounded respectively above and below by finite, strictly positive constants. Thus, as H passes through 1 2 , the exponential behavior of u(t) changes abruptly. This can be considered as a phase transition phenomenon. Novel tools used in this paper include sub-Gaussian concentration theory via the Malliavin calculus, detailed analyses of the long-range memory of fractional Brownian motion, and an almost-superadditivity property.

Feynman–Kac formula for heat equation driven by fractional white noise

“…A process similar to (1.3) was studied by Viens and Zhang in [12], although it does not have a relation with a stochastic heat equation and, most likely, the asymptotic results obtained in [12] can be extended to the process (1.3).…”

Quenched asymptotics for Brownian motion in generalized Gaussian potential

“…Let r ≥ 2. By Proposition 1 in [13], also by Lemma 4.6 in [14], there is a nonnegative and continuous function Φ(x) on R d whose support is contained in the 1neighborhood of the grid 2rZ d , such that for any R > r and any generalized function ξ, 25) where Φ y (x) = Φ(x + y), and we use the notation Q R = (−R, R) d for any R > 0.…”

“…under the condition lim |x|→∞ γ(x) = 0. See [14] for the asymptotics of the second order, and [4,6,16,24] and [25] for a variety of versions in literature.…”

“…In general, going from the time-independent potential to the timedependent potential is a big step. We specially mention the work [25] by Viens and Zhang for their effort beyond the sub-additivity treatment. It is our hope that some ideas developed in the current paper may play a role in the future investigation of this direction.…”

Stochastic elastic equation driven by multiplicative multi-parameter fractional noise

“…For more contents for asymptotic behavior of a stochastic partial differential equation, see [6,33] and the references therein. In [33], the authors studied the asymptotic behavior of a 1-dimensional directed polymer in a random medium. The random was represented by a Gaussian field with fractional Brownian behavior in time (Hurst parameter H) and arbitrary function-valued behavior in space.…”