Random walks in continuous time are largely employed in several fields of both theoretical and applied interest. In this paper we consider a class of continuous-time Markov chains on integers, called the basic model, which can have transitions to adjacent states only, and with alternating transition rates to their adjacent states; namely we assume to have the same transition rates for the odd states, and the same transition rates for the even states. We also consider some independent random time-changes of the basic model.

Markov chains with alternating rates are useful in the study of chain molecular diffusions. We recall the paper [31], where a molecule is modeled as a freely-joined chain of two regularly alternating kinds of atoms, which have alternating jump rates. Another reference is [6] where a simple birth-death process with alternating rates has been studied as a model for an infinitely long chain of atoms joined by links which are subject to random alternating shocks. Recent results on the transient probabilities of such model, also in the presence of suitable reflecting or absorbing states, are provided in [32, 33] and [34].

In this paper we also consider independent random time-changes of the basic model which provide more flexible versions of the chemical models in the references cited above. More precisely we consider the inverse stable subordinator or, alternatively, the (possibly tempered) stable subordinator. In the first case the particle is subject to a sort of trapping and delaying effect; on the contrary, in the second case, we allow positive jumps in the random time-changed argument, which produces a possible rushing effect.

We start with a more rigorous presentation of the basic model in terms of the generator. In general we consider a continuous-time Markov chain {X(t):t≥0} on Z (where Z is the set of integers), and we consider the state probabilities (1) pk,n(t):=P(X(t)=n|X(0)=k), which satisfy the condition pk,n(0)=1{k=n} ; the generator G=(gk,n)k,n∈Z of {X(t):t≥0} is defined by gk,n:=limt→0pk,n(t)−pk,n(0)t. Then, for some α1,α2,β1,β2>0 0$]]>, we assume to have (see Figure 1) gk,n:=α1ifn=k+1andkis evenβ1ifn=k+1andkis oddα2ifn=k−1andkis evenβ2ifn=k−1andkis odd0otherwise(fork≠n); therefore gn,n=−(α1+α2)ifnis even−(β1+β2)ifnis odd.

We remark that this is a generalization of the model in [10]; in fact we recover that model by setting α1=λη+μ(1−η)β1=μη+λ(1−η)α2=λθ+μ(1−θ)β2=μθ+λ(1−θ) for λ,μ>0 0$]]> and η,θ∈[0,1] ; moreover the case (θ,η)=(1,1) was studied in [8], whereas the case (θ,η)=(0,1) identifies the model investigated in [6] and [34].

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 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.

Moreover we consider some random time-changes of the basic model {X(t):t≥0} , with independent processes. This is motivated by the great interest that the theory of random time-changes (and subordination) is being receiving starting from [5] (see also [30]). In particular this theory allows to construct non-standard models which are useful for possible applications in different fields; indeed, in many circumstances, the process is more realistically assumed to evolve according to a random (so-called operational) time, instead of the usual deterministic one. For instance, in applications to finance, the particle jumps usually represent price changes separated by a random waiting time between trades; then a time-changed version captures the role of information flow and activity time in modeling price changes (see e.g. [17]). Similarly, in applications to hydrology, the velocity irregularities caused by a heterogeneous porous media can be described by heavy tailed particle jumps, whereas suitable assumptions concerning the distribution of the waiting times allow to model particle sticking or trapping (see e.g. [4]).

A wide class of random time-changes concerns subordinators, namely nondecreasing Lévy processes (see, for example, [29, 19, 22, 24] and [9]); recent works with different kind of random time-changes are [11, 3] and [12]. The random time-changes of {X(t):t≥0} studied in this paper are related to fractional differential equations and stable processes. More precisely we consider: 1.

the inverse of the stable subordinator {Tν(t):t≥0} ;

We also try to extend the large deviation results for {X(t):t≥0} to the cases with a random time-change considered in this paper. It is useful to remark that all the large deviation principles in this paper are proved by applications of the Gärtner Ellis Theorem; moreover these large deviation principles yield the convergence (at least in probability) to the values at which the large deviation rate functions uniquely vanish. Thus, motivated by potential applications, when dealing with large deviation principles with the same speed function, we compare the rate functions to establish if we have a faster or slower convergence (if they are comparable). In conclusion the evaluation of the rate function can be an important task, in particular when they are given in terms of a variational formula (as happens with the application of the Gärtner Ellis Theorem).

The applications of the Gärtner Ellis Theorem are based on suitable limits of moment generating functions. So, in view of the applications of this theorem, we study the probability generating functions of the random variables of the processes; in particular the formulas obtained for {X(Tν(t)):t≥0} have some analogies with many results in the literature for other time-fractional processes (for instance the probability generating functions are expressed in terms of the Mittag-Leffler function), with both continuous and discrete state space (see, for example, [22, 14, 2] and [16]). For {X(Tν(t)):t≥0} we can consider large deviations only (the difficulties to obtain a moderate deviation result are briefly discussed); moreover we compute (and plot) different large deviation rate functions for various choices of ν∈(0,1) and we conclude that, the smaller is ν, the faster is the convergence of Xν(t)t to zero (as t→∞ ). For {X(S˜ν,μ(t)):t≥0} we can obtain large and moderate deviations for the tempered case μ>0 0$]]> only; in fact in this case we can apply the Gärtner Ellis Theorem because we have light-tailed distributed random variables (namely the moment generating functions of the involved random variables are finite in a neighborhood of the origin).

There are some references in the literature with applications of the Gärtner Ellis Theorem to time-changed processes. However there are very few cases where the random time-change is given by the inverse of the stable subordinator; see e.g. [13] and [35] where the time-changed processes are fractional Brownian motions (see also [20] and [25] for other asymptotic results for time-changed Gaussian processes with inverse stable subordinators). We are not aware of any other references where the time-changed process takes values on Z .

We conclude with the outline of the paper. Section 2 is devoted to some preliminaries on large deviations. In Section 3 we present the results for the basic model, i.e. the (non-fractional) process {X(t):t≥0} . Then we present some results for the process {X(t):t≥0} with random time-changes: the case with the inverse of the stable subordinator is studied in Section 4, the case with the (possibly tempered) stable subodinator is studied in Section 5. We conclude with the short final Section 6 devoted to some conclusions. We also present a final appendix (Section A) with the state probabilities expressions.

Some results in this paper concerns the theory of large deviations; so, in this section, we recall some preliminaries (see e.g. [7], pages 4–5). A family of probability measures {πt:t>0} 0\}$]]> on a topological space Y satisfies the large deviation principle (LDP for short) with rate function I and speed function vt if: limt→+∞vt=+∞ , I:Y→[0,+∞] is lower semicontinuous, lim inft→+∞1vtlogπt(O)≥−infy∈OI(y) for all open sets O, and lim supt→+∞1vtlogπt(C)≤−infy∈CI(y) for all closed sets C. A rate function is said to be good if all its level sets {{y∈Y:I(y)≤η}:η≥0} are compact.

We also present moderate deviation results. This terminology is used when, for each family of positive numbers {at:t>0} 0\}$]]> such that at→0 and vtat→∞ , we have a family of laws of centered random variables (which depend on at ), which satisfies the LDP with speed function 1/at , and they are governed by the same quadratic rate function which uniquely vanishes at zero (for every choice of {at:t>0} 0\}$]]>). More precisely we have a rate function J(y)=y22σ2 , for some σ2>0 0$]]>. Typically moderate deviations fill the gap between a convergence to zero of centered random variables, and a convergence in distribution to a centered Normal distribution with variance σ2 .

The main large deviation tool used in this paper is the Gärtner Ellis Theorem (see e.g. Theorem 2.3.6 in [7]).

In this section we present the results for the basic model. Some of them will be used for the models with random time-changes in the next sections. We start with some non-asymptotic results, where t is fixed, which concern probability generating functions, means and variances. In the second part we present the asymptotic results, namely large and (moderate) deviation results as t→∞ .

In particular the probability generating functions {Fk(·,t):k∈Z,t≥0} are important in both parts; they are defined by Fk(z,t):=EzX(t)|X(0)=k= ∑n=−∞∞znpk,n(t)(fork∈Z), where {pk,n(t):k,n∈Z,t≥0} are the state probabilities in (1).

We also have to consider the function Λ:R→R defined by (2) Λ(γ):=h(eγ)eγ−α1+α2+β1+β22, where (3) h(z):=12h˜(z;α_,β_),where we meanh˜(z;α_,β_)=h˜(z;α1,α2,β1,β2)andh˜(z;α_,β_):=(α1+α2−(β1+β2))2z2+4(β1z2+β2)(α1z2+α2).

The function Λ is the analogue of the function Λ in equation (14) in [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 Λ′(0) and Λ″(0) and therefore we present the following lemma. Lemma 3.1.

Let Λ be the function in (2). Then we have Λ′(0)=2(α1β1−α2β2)α1+α2+β1+β2 and Λ″(0)=4(α1β1+α2β2)α1+α2+β1+β2−8(α1β1−α2β2)2(α1+α2+β1+β2)3. Moreover Λ″(0)>0 0$]]>; in fact Λ″(0)=4(α1+α2+β1+β2)3{(α1β1+α2β2)×[(α1+α2)2+(β1+β2)2+2α1β2+2α2β1]+8α1α2β1β2}.