Anderson polymer in a fractional Brownian environment: asymptotic behavior of the partition function

Abstract

We consider the Anderson polymer partition function $$u(t):={\mathbb{E}}^X\left[e^{{\int }_0^t\mathrm{d}{B}_s^{X(s)}}\right],$$ where $\{B_t^x;t\ge 0{\}}_{x\in {\mathbb{Z}}^d}$ is a family of independent fractional Brownian motions all with Hurst parameter $H\in (0,1)$, and $\{X(t){\}}_{t\in {\mathbb{R}}^{\ge 0}}$ is a continuous-time simple symmetric random walk on ${\mathbb{Z}}^d$ with jump rate $\kappa$ and started from the origin. ${\mathbb{E}}^X$ is the expectation with respect to this random walk. We prove that when $H\le 1/2$, the function $u(t)$ almost surely grows asymptotically like $e^{lt}$, where $l>0$ is a deterministic number. More precisely, we show that as $t$ approaches $+\infty$, the expression $\{\frac{1}{t}\log u(t){\}}_{t\in {\mathbb{R}}^{>0}}$ converges both almost surely and in the ${\mathcal{L}}^1$ sense to some deterministic number $l>0$. For $H>1/2$, we first show that ${\lim }_{t\to \infty }\frac{1}{t}\log u(t)$ exists both almost surely and in the ${\mathcal{L}}^1$ sense, and equals a strictly positive deterministic number (possibly $+\infty$); hence almost surely $u(t)$ grows asymptotically at least like $e^{at}$ for some deterministic constant $a>0$. On the other hand, we also show that almost surely and in the ${\mathcal{L}}^1$ sense, ${\mathrm{limsup}}_{t\to \infty }\frac{1}{t\sqrt{\log t}}\log u(t)$ is a deterministic finite real number (possibly zero), hence proving that almost surely $u(t)$ grows asymptotically at most like $e^{bt\sqrt{\log t}}$ for some deterministic positive constant $b$. Finally, for $H>1/2$ when ${\mathbb{Z}}^d$ is replaced by a circle endowed with a Hölder continuous covariance function, we show that ${\mathrm{limsup}}_{t\to \infty }\frac{1}{t}\log u(t)$ is a finite deterministic positive number, hence proving that almost surely $u(t)$ grows asymptotically at most like $e^{ct}$ for some deterministic positive constant $c$.

References in corpus (2)

• Moments and Lyapunov exponents for the parabolic Anderson model
• Feynman-Kac representation for the parabolic Anderson model driven by fractional noise