Asymptotic Results for Random Walks in Continuous Time with Alternating Rates

Abstract

We investigate some large deviation problems for a random walk in continuous time {N (t); t ≥ 0} with spatially inhomogeneous rates of alternating type. We first deal with the large deviation principle for the convergence of N (t)/t to a suitable constant. Then, the case of moderate deviations is also discussed. Motivated by possible applications in chemical physics context, we finally obtain an asymptotic lower bound for level crossing probabilities both in the case of finite and infinite horizon.

How animals move along? Exactly solvable model of superdiffusive spread resulting from animal’s decision making

“…(2-4) and (9) shows three different time scales: (i) a short transient for small values of ωt, defined through the complete expressions of the moments in Eq. (24), where the initial ballistic behavior starts to slow down to a diffusion type of spreading; (ii) a long transient behavior, where the exponential contributions are neglected and the system may be approximated by the characteristic function (40), under which the pdf shows fatter than normal tails (almost exponential); (iii) large time asymptotics, defined by ωt 1, where the system converges to a diffusive type of spreading, with the diffusion coefficient given by…”

Random time-changes and asymptotic results for a class of continuous-time Markov chains on integers with alternating rates

“…The function is the analogue of the function in equation 14in [10], and plays a crucial role in the proofs of the large (and moderate) deviation results. However we refer to this function also for the non-asymptotic results in order to have simpler expressions; in particular we refer to the derivatives (0) and (0) and therefore we present the following lemma.…”

“…In this section we present Propositions 3.3 and 3.4, which are the generalization of Propositions 3.1 and 3.2 in [10]. In both cases we apply the Gärtner Ellis Theorem, and we use the probability generating function in Proposition 3.1.…”

“…In particular we extend the results in [10] by giving explicit expressions of the probability generating function, mean and variance of X(t) (for each fixed t > 0), and we study the asymptotic behavior (as t → ∞) in the fashion of large deviations. Here we also give explicit expressions of the state probabilities.…”

“…Then, for some α 1 , α 2 , β 1 , β 2 > 0, we assume to have (see We remark that this is a generalization of the model in [10]; in fact we recover that model by setting ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩…”