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Register a new userPrecise large deviations for random walk in random environment
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We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.
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Let $sigma(u)$, $uin mathbb{R}$ be an ergodic stationary Markov chain, taking a finite number of values $a_1,...,a_m$, and $b(u)=g(sigma(u))$, where $g$ is a bounded and measurable function. We consider the diffusion type process $$ dX^epsilon_t = b(X^epsilon_t/epsilon)dt + epsilon^kappasigmabig(X^epsilon_t/epsilonbig)dB_t, tle T $$ subject to $X^epsilon_0=x_0$, where $epsilon$ is a small positive parameter, $B_t$ is a Brownian motion, independent of $sigma$, and $kappa> 0$ is a fixed constant. We show that for $kappa<1/6$, the family ${X^epsilon_t}_{epsilonto 0}$ satisfies the Large Deviations Principle (LDP) of the Freidlin-Wentzell type with the constant drift $mathbf{b}$ and the diffusion $mathbf{a}$, given by $$ mathbf{b}=sumlimits_{i=1}^mdfrac{g(a_i)}{a^2_i}pi_iBig/ sumlimits_{i=1}^mdfrac{1}{a^2_i}pi_i, quad mathbf{a}=1Big/sumlimits_{i=1}^mdfrac{1}{a^2_i}pi_i, $$ where ${pi_1,...,pi_m}$ is the invariant distribution of the chain $sigma(u)$.
We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We describe precise asymptotics of upper large deviations, i.e. $mathbb{P}[Z_n > e^{rho n}]$. Moreover in the subcritical case, under the Cramer condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size.
We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.
We derive properties of the rate function in Varadhans (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations.
Let $xi(n, x)$ be the local time at $x$ for a recurrent one-dimensional random walk in random environment after $n$ steps, and consider the maximum $xi^*(n) = max_x xi(n,x)$. It is known that $limsup xi^*(n)/n$ is a positive constant a.s. We prove that $liminf_n (logloglog n)xi^*(n)/n$ is a positive constant a.s.; this answers a question of P. Revesz (1990). The proof is based on an analysis of the {em valleys /} in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time $n$ large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.
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