On Some Degenerate Large Deviation Problems

This paper concerns the issue of obtaining the large deviation principle for solutions of stochastic equations with possibly degenerate coe–cients. Speciﬂcally, we explore the potential of the methodology that consists in establishing exponential tightness and identifying the action functional via a maxingale problem. In the author’s earlier work it has been demonstrated that certain convergence properties of the predictable characteristics of semimartingales ensure both that exponential tightness holds and that every large deviation accumulation point is a solution to a maxingale problem. The focus here is on the uniqueness for the maxingale problem. It is ﬂrst shown that under certain continuity hypotheses existence and uniqueness of a solution to a maxingale problem of diﬁusion type are equivalent to Luzin weak existence and uniqueness, respectively, for the associated idempotent Ito equation. Consequently, if the idempotent equation has a unique Luzin weak solution, then the action functional is speciﬂed uniquely, so the large deviation principle follows. Two kinds of application are considered. Firstly, we obtain results on the logarithmic asymptotics of moderate deviations for stochastic equations with possibly degenerate diﬁusion coe–cients which, as compared with earlier results, relax the growth conditions on the coe–cients, permit certain non-Lipshitz-continuous coe–cients, and allow the coe–cients to depend on the entire past of the process and to be discontinuous functions of time. The other application concerns multiple-server queues with impatient customers.


Introduction
There are certain difficulties in establishing the large deviation principle (LDP) for solutions of stochastic equations with degenerate coefficients. As a motivating example, let us consider the setting of a d-dimensional diffusion process with small noise where ↓ 0, the B = (B t , t ∈ R + ) are standard d-dimensional Wiener processes, functions b(x) and σ(x) with values in R d and R d×d , respectively, are bounded and continuous in x. If the diffusion matrix c(x) = σ(x)σ(x) T , where T denotes transposition, is uniformly elliptic and the functions b(x) and σ(x) are, in addition, uniformly continuous, a seminal result by Freidlin and Wentzell [18] states that the processes X = (X t , t ∈ R + ) obey the LDP in the metric space C of R dvalued continuous functions on R + with compact-open topology. (Strictly speaking, Freidlin and Wentzell's original result concerns the space of continuous functions defined on the [0, 1] interval, however, the extension to C is routine.) The action functional for absolutely continuous functions x = (x t , t ∈ R + ) with x 0 = x 0 is given by whereẋ denotes the derivative and u · v denotes the inner product of vectors u and v. For all other x, I(x) = ∞. The proof employs the change-of-measure technique. If one retains some form of the ellipticity condition, this LDP can be generalised to the settings of diffusions with jumps, which was already done in Freidlin and Wentzell [18], and of continuous in x unbounded coefficients, depending on , time and the past history, see Dupuis and Ellis [11], Feng [14], Feng and Kurtz [15], Friedman [19], Liptser and Pukhalskii [27], Mikami [29], Narita [31], Stroock [40], and Wentzell [43]. Certain cases of a discontinuous drift coefficient can also be analysed, see, e.g., Boué, Dupuis and Ellis [3], Korostelëv and Leonov [24,25]. Allowing c(x) to be degenerate complicates matters significantly. The large deviation lower bound can no longer be obtained by a straightforward application of a change of measure. It is possible to prove, however, that if one requires, in addition, that b(x) and c(x) 1/2 be Lipshitz continuous or locally Lipshitz continuous, then the above LDP holds without an ellipticity requirement on the diffusion matrix. The action functional is still given by (1.1), where the inverse of c(x) has to be replaced with the pseudo-inverse, provided x is absolutely continuous with x 0 = x 0 anḋ x t − b(x t ) belongs to the range of c(x t ) a.e., and I(x) is equal to infinity otherwise. This version of the Freidlin-Wentzell LDP has been established under various additional assumptions and various techniques have been used.
Cutland [6] invokes the apparatus of infinitesimal analysis and is able to tackle non-timehomogeneous diffusions with bounded Lipshitz-continuous-in-the-space-variable coefficients depending on the past. The other results are confined to the time-homogeneous Markov setting. Azencott [1] considers the case of locally Lipshitz continuous b and continuously differentiable σ. No growth conditions are assumed, so diffusions with explosion are also allowed. Baldi and Chaleyat-Maurel [2], Dembo and Zeitouni [10], and Feng [14] analyse the case of diffusion processes with bounded Lipshitz-continuous coefficients. The first two articles use discretization considerations, while Feng [14] applies methods of nonlinear semigroup convergence. Azencott [1], Baldi and Chaleyat-Maurel [2], and Feng [14] also allow the coefficients to depend on . (Baldi and Chaleyat-Maurel [2] actually state the LDP for locally Lipshitz-continuous coefficients with no growth conditions, requiring instead that the equationġ t = b(g t ) + σ(g t )ḟ t , g 0 = x 0 , have a solution for every function f with a locally square-integrable derivative. However, the authors do not provide enough substantiation for the extension from the case of bounded Lipshitz continuous coefficients to the case of locally Lipshitz continuous coefficients.) In recent work Feng and Kurtz [15] improve on the results of Feng [14] and derive the LDP for diffusions with jumps having Lipshitz-continuous coefficients satisfying the linear growth condition. Along with discretization, a tool often used for tackling the case of degenerate diffusions is the perturbation approach dating back to Varadhan [42]. For the setting of (1.1), the approach consists in introducing a small nonsingular diffusion term and taking a limit in the lower bound for the perturbed equation as the perturbation's magnitude tends to zero. Dupuis and Ellis [11] combine the perturbation approach with discretization methods and representation formulas. Their results cover the case of bounded Lipshitz continuous coefficients. Liptser [26] applies the perturbation approach to study diffusions with averaging, which have Lipshitz continuous coefficients. De Acosta [9], who considers diffusions with jumps and bounded Lipshitz continuous coefficients, invokes both the perturbation approach and discretization arguments as well as a Banach-space version of the method of stochastic exponentials, cf., Puhalskii [33,35,36]. An interesting example of an equation with a non-Lipshitz-continuous degenerate diffusion coefficient is considered by Dawson and Feng [7,8], Feng [16], and Feng and Xiong [17], who are motivated by the study of Fleming-Viot processes. The authors obtain the LDP for a particular form of equation (1.1) with the (i, j) entry of c(x) given by x i (δ ij − x j ), where x = (x 1 , . . . , x d ) T , the entries being in [0, 1], and δ ij is Kronecker's delta. To sum up, we are aware of no results for equation (1.1) that would simultaneously allow the coefficients to be degenerate, to grow at linear rate, to be time-dependent, and to be discontinuous in the time variable. Nor are there, to our knowledge, any results available that combine the hypotheses that the coefficients depend on the past history, are possibly degenerate and unbounded.
In Puhalskii [32,35,36], we introduced and developed an approach to establishing the LDP for semimartingales, which, in analogy with the martingale problem method in the weak convergence theory, reduces the task to establishing exponential tightness and identifying the action functional via a maxingale problem. The establishing of exponential tightness is often routine. In particular, certain convergence conditions on the predictable characteristics ensure both that exponential tightness holds and that every large deviation accumulation point is a solution to the maxingale problem. The difficult part, as in the weak convergence theory, is proving uniqueness of a solution to the maxingale problem. The uniqueness is known to hold if certain nondegeneracy conditions are met, Puhalskii [36,Section 2.8]. The motivation for this article has been to incorporate degenerate maxingale problems of diffusion type.
Attaining this objective puts us in a position to derive results on the LDP for a range of large deviation settings that lead to a maxingale problem of diffusion type. In particular, the gaps mentioned above can be filled. For equation (1.1) we are able to tackle the case of possibly degenerate coefficients, which are locally Lipshitz-continuous in the space variable, satisfy the lineargrowth condition and depend on the entire past of the process. Besides, the time-non-homogeneous Markov setting of the coefficients measurably depending on time is incorporated. In addition, the coefficients may depend on and grow slightly faster than linearly, say, at rate |x| log(1 + |x|). For time-homogeneous non-degenerate diffusions the latter possibility has been known since the work of Narita [31]. If the drift is directed towards the origin, then no restrictions are needed on the growth rate of the drift coefficient, which is consistent with what is known for stochastic differential equations, see, e.g., Ethier and Kurtz [12]. Slightly non-Lipshitz behaviour of the coefficients is also permitted, e.g., in the one-dimensional setting one can take σ(x) = x log|x|. We are not able, though, to obtain by our methods the results of Dawson and Feng [7,8], Feng [16], and Feng and Xiong [17].
In fact, our results concern a more general setting than that of (1.1) by allowing "compensated" Poisson jumps. The jump sizes are assumed to be large as compared with time periods between the jumps, so in the end the jump part also contributes to "the limiting c" in (1.2). It is thus a form of moderate deviation asymptotics. Moreover, we also obtain results on another type of moderate deviation asymptotics, where "the limiting b" is a linear function. For nondegenerate c both types of asymptotics have been analysed by Wentzell [43]. No results for possibly degenerate c are known to us. (The results of de Acosta [9], Dupuis and Ellis [11], and Feng and Kurtz [15] concern large deviation asymptotics when the time spans between the jumps are of the same order of magnitude as the jump sizes.) Finally, we derive moderate deviation asymptotics in the form of the LDP for the queueing model of a multiple-server system with impatient customers. Before explaining how the outlined results are obtained, we recall the main steps of the maxingale problem approach by continuing with equation (1.1). This should help the interested reader to follow the developments below. We still assume that b(x) and c(x) are continuous and bounded functions. Let E denote expectation. Given λ ∈ R d , the stochastic processes t 0 λ · c(X s )λ ds , t ∈ R + are martingales relative to the associated filtrations, so on taking λ/ as λ we have that for an arbitrary bounded function f (x) on C that is measurable with respect to the σ-algebra on C generated by the projections x → x r , where r ≤ s ≤ t, the following equality holds Boundedness of b(x) and c(x) implies the exponential uniform integrability condition that for where 1(Γ) denotes the indicator function of an event Γ. The boundedness also makes it straightforward to see that the net of laws of the X is exponentially tight, so there exists a subnet obeying the LDP with some (tight) action functional. Let us assume, in addition, that the function f is continuous. By a version of Varadhan's lemma (see, e.g., Dembo and Zeitouni [10]) on raising both sides of (1.3) to the power of and letting ↓ 0 we conclude that if I(x) is the action functional in the LDP for some subnet, then If we treat the set function Π(Γ) = sup x∈Γ exp(−I(x)), Γ ⊂ C, as a maxitive, or idempotent, analogue of probability, then equality (1.4) represents a maxitive analogue of one of characterisations of martingales, so we call the function exp λ·x t − t 0 λ·b(x s ) ds−1/2 t 0 λ·c(x s )λ ds , t ∈ R + , x ∈ C an exponential maxingale. The problem of identifying I from (1.4) is accordingly referred to as a maxingale problem. If I is specified uniquely by (1.4), then the LDP holds with action functional I. In fact, one can show that (1.4) is equivalent to an idempotent analogue of the definition of a martingale: for arbitraryx ∈ C sup x∈C: xq=xq, q≤s We note similarity with the dynamic programming equation. It was shown in Puhalskii [36] that in (1.5) it is possible to replace constants λ with functions λ(s, x) from a certain class. If the diffusion matrix c(x) is nondegenerate, this class is large enough to identify I. In the present article we build on that result but adopt a "probabilistic" rather than an analytical approach to the uniqueness problem.
Let us recall that the key element in obtaining results on weak convergence to stochastic processes with possibly degenerate coefficients is the fact that solutions to the limit martingale problems are weak solutions to the associated stochastic equations, see, e.g., Ikeda and Watanabe [22], Jacod and Shiryaev [23], Liptser and Shiryaev [28], Stroock and Varadhan [41]. In a similar vein, one can associate idempotent differential equations with maxingale problems and show that weak solutions to these equations are solutions to the maxingale problems. For instance, Theorem 2.8.9 in Puhalskii [36] establishes conditions under which a Luzin weak solution to the idempotent Ito equation where W = (W t , t ∈ R + ) is an idempotent Wiener process, is a solution to problem (1.4). (The terminology of idempotent probability theory is recalled in Section 2.) As for the converse, it has been known to hold for nondegenerate maxingale problems, Puhalskii [36, Theorems 2.6.30 and 2.8.21], and for a certain degenerate maxingale problem associated with idempotent Poisson processes, Puhalskii [38]. In this paper, the result that solutions to maxingale problems are weak solutions to the corresponding idempotent equations is extended to maxingale problems of the diffusion type (1.4) with possibly degenerate matrices c. Thus, in complete analogy with the situation in weak convergence theory, the issue of uniqueness for maxingale problems of diffusion type is equivalent to that of Luzin weak uniqueness for idempotent Ito differential equations. Interestingly, the proof draws on the perturbation idea, which is so useful in the stochastic setting, however, the perturbation is introduced in the idempotent equation (1.6) rather than in a stochastic process (of course, there are no stochastic processes to speak of !). With the equivalence between maxingale problems of diffusion type and idempotent Ito differential equations in hand, we again follow in the footsteps of the stochastic setting to deduce that pathwise uniqueness for idempotent Ito equations implies uniqueness for the associated maxingale problems. Equation (1.6) is in fact an ordinary differential equation of a certain kind. The uniqueness issue for ordinary differential equations is well understood, e.g., Lipshitz continuity of the coefficients suffices. This enables us to obtain results on uniqueness for degenerate maxingale problems of diffusion type. In particular, as in the theory of ordinary differential equations, it is possible to relax the requirements of Lipshitz continuity and of linear growth.
The paper is structured as follows. Section 2 contains a background on idempotent probability. In Section 3 existence and uniqueness issues for diffusion maxingale problems are analysed. Section 4 is concerned with large deviation applications.

Technical preliminaries
This section reviews necessary results from idempotent probability theory. More details can be found in Puhalskii [36]. We let denote the operator norm of a matrix, let ∨ denote the maximum, and let ∧ denote the minimum.
A property P(υ), υ ∈ Υ, pertaining to the elements of Υ is said to hold Π-a.e. if Π(P(υ) does not hold) = 0. A τ -algebra A on Υ is defined as a collection of subsets of Υ that are unions of the elements of some partitioning of Υ into disjoint sets. We call the elements of the partitioning the atoms of A and denote as [υ] the atom containing υ. τ -algebras are closed under set-differences, arbitrary unions and intersections of sets. The power set of Υ is referred to as the discrete τ -algebra. A τ -algebra A is called complete (or Π-complete, or complete with respect to Π if the idempotent probability needs to be specified) if each one-point set {υ} with Π(υ) = 0 is an atom of A; the completion (or the Π-completion, or the completion with respect to Π) of a τ -algebra A is defined as the τ -algebra obtained by taking as atoms the elements of Υ of idempotent probability 0 and the maximal subsets of the atoms of A that do not contain elements of idempotent probability 0; the completion of a τ -algebra is a complete τ -algebra. If Υ is also a set equipped with idempotent probability Π and τ -algebra A , then the product idempotent A function f from a set Υ equipped with idempotent probability Π to a set Υ is called an idempotent variable. If Υ and Υ are equipped with τ -algebras A and A , respectively, the idempotent variable f is said to be A/A -measurable, or simply measurable if the τ -algebras are understood, We say that f is A-measurable if it is measurable for the discrete τ -algebra on Υ , i.e., {υ ∈ Υ : f (υ) = υ } ∈ A for all υ ∈ Υ . The τ -algebra on Υ generated by f is defined by the atoms {υ ∈ Υ : f (υ) = υ }, υ ∈ Υ .
As in probability theory, we routinely omit the argument υ in the notation for an idempotent variable. The idempotent distribution of an idempotent variable f is defined as the set function is a deviability on Υ, and f is a mapping from Υ to a metric space Υ , which is continuous when restricted to the sets {υ ∈ Υ : Subsets A and A of Υ are said to be independent if Π(A ∩ A ) = Π(A)Π(A ); τ -algebras A and A are said to be independent if sets A and A are independent for arbitrary A ∈ A and A ∈ A ; Υ -valued idempotent variables f and f are said to be independent if An idempotent variable f and a τ -algebra A are said to be independent (or f to be independent of A) if the τ -algebra generated by f and A are independent. If f is R + -valued, the idempotent expectation of f is defined by Sf = sup υ∈Υ f (υ)Π(υ), it is also denoted as S Π f if the reference idempotent probability needs to be indicated. The following analogue of the Markov inequality holds: where f (υ) is an R + -valued idempotent variable constant on the atoms of A. Conditional idempotent expectation is thus specified uniquely Π-a.e. It has many of the properties of conditional expectation, in particular, , and if f and A are independent then S(f |A) = Sf Π-a.e. A flow of τ -algebras, or a τ -flow, on Υ is defined as a collection A = (A t , t ∈ R + ) of τ -algebras on Υ such that A s ⊂ A t for s ≤ t; the latter condition is equivalent to the atoms of A s being unions of the atoms of A t . A τ -flow is called complete if it consists of complete τ -algebras, the completion of a τ -flow is obtained by completing its τ -algebras. An idempotent variable κ : Υ → R + is called an idempotent A-stopping time, or a stopping time relative to A, if {υ : κ(υ) = t} ∈ A t for t ∈ R + . Given a τ -flow A and an idempotent A-stopping time κ, we define A κ as the τ -algebra with atoms The τ -flow generated by X is defined as consisting of the τ -algebras generated by the X t . It is the minimal τ -flow, to which X is adapted. If the idempotent process X is A-adapted and R-valued with unbounded above continuous paths, then κ = inf{t ∈ R + : X t ≥ a}, where a ∈ R, is an idempotent A-stopping time.
An A-adapted R + -valued idempotent process M = (M t , t ∈ R + ) is said to be an A-exponential maxingale, or an exponential maxingale relative to A, if the M t are maximable and S(M t |A s ) = M s Π-a.e. for s ≤ t. If, in addition, the collection M t , t ∈ R + , is uniformly maximable, then M is said to be a uniformly maximable exponential maxingale. An A-adapted R + -valued idempotent process M = (M t , t ∈ R + ) is called an A-local exponential maxingale, or a local exponential maxingale relative to A, if there exists a sequence τ n of idempotent A-stopping times such that τ n ↑ ∞ as n → ∞ and the stopped idempotent processes (M t∧τn , t ∈ R + ) are uniformly maximable A-exponential maxingales; the sequence τ n is referred to as a localising sequence for M .
Proof. Let τ n and τ n be localising sequences for M and M , respectively. By Puhalskii [36, Lemma is called a local maxingale with quadratic characteristic M relative to A if the idempotent processes (exp(λ · M t − λ · M t λ/2), t ∈ R + ) are A-local exponential maxingales for all λ ∈ R d . The next lemma presents a version of Lenglart-Rebolledo's and Doob's inequalities, see, e.g., Liptser and Shiryaev [28].
Lemma 2.2. If M is a local maxingale with quadratic characteristic M relative to A and M 0 = 0, then for arbitrary a > 0, b > 0, and t > 0 If the M s do not depend on υ ∈ Υ, then Let us recall that C denotes the metric space of continuous R d -valued functions on R + with compact-open topology. The canonical τ -flow on C is defined as the τ -flow C = (C t , t ∈ R + ) with the C t having as atoms the sets Wiener idempotent probability (or Wiener deviability) is a deviability on C defined by A standard idempotent Wiener process is defined as an idempotent process with idempotent distribution Π W . Thus, it has Π-a.e. absolutely continuous paths. The definition implies that the canonical idempotent process on C is idempotent Wiener under Π W . The component processes of an R d -valued standard idempotent Wiener process are independent one-dimensional standard idempotent Wiener processes.
Wiener process on (Υ, Π), then the idempotent process exp(λ · W t − |λ| 2 t/2), t ∈ R + is an exponential maxingale relative to the flow where the A W t are the τ -algebras generated by W s , s ≤ t. An R d -valued continuous-path idempotent process W is said to be standard idempotent Wiener relative to a τ -flow A if W 0 = 0 and the idempotent process exp(λ · W t − |λ| 2 t/2), t ∈ R + is an A-exponential maxingale for arbitrary λ ∈ R d . If an idempotent process is idempotent Wiener relative to a τ -flow, then it is idempotent Wiener.
Let W be an R d -valued standard idempotent Wiener process and (σ t (υ), t ∈ R + ) be an R d×dvalued idempotent process, which is Lebesgue measurable in t for all υ and is such that σ s (υ)σ s (υ) T is locally integrable for all υ. We define the idempotent Ito integral (σ W ) t by The integral is thus specified uniquely Π-a.e. The idempotent process σ W = (σ W ) t , t ∈ R + has Π-a.e. continuous paths. If (W t , t ∈ R + ) and (σ t , t ∈ R + ) are adapted to a complete τ -flow A, then σ W is A-adapted. The next lemma is a version of Theorem 2.5.19 in Puhalskii [36]. Let tr denote the trace of a matrix.
Lemma 2.3. Let (σ t (x), t ∈ R + , x ∈ C) be an R k×m -valued function, which is continuous in x, Lebesgue measurable in t, and is adapted to the canonical τ -flow on C. Let (X t , t ∈ R + ) be a continuous-path R d -valued idempotent process, which is adapted to a complete τ -flow A on Υ and whose idempotent distribution is a deviability on C. Let (W t , t ∈ R + ) be an R m -valued standard idempotent Wiener process relative to A. If, in addition, t 0 tr σ s (x)σ s (x) T ds is a well-defined continuous function of x ∈ C for every t ∈ R + , then σ W is a local maxingale relative to A with quadratic characteristic t 0 σ s (X)σ s (X) T ds, t ∈ R + ). Proof. Continuity of σ t (x) and x for all A. By Dini's theorem, they converge to 0 as A → ∞ uniformly on compacts, which checks the hypotheses of Theorem 2.5.19 in Puhalskii [36].
Below, we use t 0 σ sẆs ds to denote (σ W ) t . Let σ t (x), x ∈ C, t ∈ R + , and b t (x), x ∈ C, t ∈ R + , be functions with values in R d×d and R d , respectively, which are Lebesgue-measurable in t and C t -measurable in x. Let also the functions b t (x) and σ t (x)σ t (x) T be locally integrable in t. Let W be an R d -valued standard idempotent Wiener process on an idempotent probability space (Υ, Π) relative to a complete τ -flow A and let C W t for t ∈ R + denote the completion of C t with respect to the Wiener deviability on C. We say that, given x 0 ∈ R d , an R d -valued idempotent process X on (Υ, Π) is a strong solution to the idempotent Ito equation if equality (2.1) holds Π-a.e. and there exists a function J : C → C, which is C W t /C t -measurable for every t ∈ R + , such that X = J(W ) Π-a.e. As a consequence, X is A-adapted. A strong solution is called Luzin if the function J is continuous when restricted to the sets {x ∈ C : Π W (x) ≥ a} for a ∈ (0, 1]. We say that there exists a unique strong solution (respectively, Luzin strong solution) if any strong solution (respectively, Luzin strong solution) can be written as X = J(W ) Π-a.e. for the same function J.
We say that idempotent probability Π x 0 on C is a weak solution of (2.1) if there exists an idempotent probability space (Υ, Π) with a complete τ -flow A, on which are defined a continuouspath R d -valued A-adapted idempotent process X = (X t , t ∈ R + ) and a continuous-path R d -valued A-adapted idempotent process W = (W t , t ∈ R + ) such that W is standard idempotent Wiener relative to A, X has idempotent distribution Π x 0 , and (2.1) holds for all t ∈ R + Π-a.e. on Υ. If the joint idempotent distribution of (X, W ) (and, hence, Π x 0 ) is, in addition, a deviability, then we call Π x 0 a Luzin weak solution. If a strong (respectively, Luzin strong) solution exists, then its idempotent distribution is a weak (respectively, Luzin weak) solution. Occasionally, we call the idempotent process X itself a weak solution (respectively, Luzin weak solution) so that, with a slight abuse of terminology, one can say that a (Luzin) strong solution is a (Luzin) weak solution. It is worth mentioning that weak solutions can always be realised on the canonical space C × C, with X = (X t , t ∈ R + ) and W = (W t , t ∈ R + ) being the projection processes defined by X t (x, x ) = x t and W t (x, x ) = x t for (x, x ) ∈ C × C. Then the joint idempotent distribution of (X, W ) coincides with Π and Pathwise uniqueness is said to hold for (2.1) if for every two weak solutions (X, W ) and (X , W ), defined on a common idempotent probability space (Υ, Π) with the same τ -flow A, we have X = X Π-a.e. provided W = W Π-a.e. Pathwise uniqueness holds if for every w ∈ C such that Π W (w) > 0 the initial value problemẋ t = b t (x) + σ t (x)ẇ t , x 0 = x 0 , has at most one solution x. If pathwise uniqueness holds, then (2.1) has at most one weak solution so that if, in addition, there exists a (Luzin) weak solution, then it is unique, and there exists a unique (Luzin) strong solution.
As a consequence, we have the following result on strong existence and uniqueness. Let the functions b t (x) and σ t (x) be, in addition, locally Lipshitz continuous in x, i.e., for every a > 0 there exists an R + -valued locally integrable in t function l a t , t ∈ R + , such that |b t (x) − b t (y)| ≤ l a t sup s≤t |x s − y s | and σ t (x) − σ t (y) 2 ≤ l a t sup s≤t |x s − y s | 2 if sup s≤t |x s | ≤ a and sup s≤t |y s | ≤ a. Let them also satisfy the linear-growth condition that there exists an R + -valued locally integrable function l t , t ∈ R + , such that |b t (x)| ≤ l t (1 + sup s≤t |x s |) and σ t (x) 2 ≤ l t (1 + sup s≤t |x s | 2 ) for x ∈ C and t ∈ R + . Then (2.1) has a unique weak solution, which is also a Luzin strong solution.
Let Υ be a metric space. A net Π φ , φ ∈ Φ, where Φ is a directed set, of idempotent probabilities on Υ is said to converge weakly to idempotent probability Π on Υ if lim φ∈Φ S Π φ f = S Π f for every R + -valued bounded and continuous function f on Υ; equivalently, lim sup φ∈Φ Π φ (F ) ≤ Π(F ) for all closed sets F ⊂ Υ and lim inf φ∈Φ Π φ (G) ≥ Π(G) for all open sets G ⊂ Υ. A net of idempotent variables with values in a common metric space is said to converge in idempotent distribution if their idempotent distributions weakly converge. One has the continuous mapping principle for convergence in idempotent distribution: if a net X φ , φ ∈ Φ, of idempotent variables with values in Υ converges in idempotent distribution to an idempotent variable X with values in Υ and f is a continuous function from Υ to a metric space Υ , then the net f (X φ ), φ ∈ Φ, converges in idempotent distribution to f (X). A net Π φ , φ ∈ Φ, of deviabilities on Υ is said to be tight if inf K∈K lim sup φ∈Φ Π φ (Υ \ K) = 0, where K denotes the collection of compact subsets of Υ. A tight net of deviabilities is weakly compact, i.e., it contains a subnet that converges weakly to a deviability (if Π φ is a sequence, then it contains a weakly convergent subsequence).

Maxingale problems of diffusion type and idempotent Ito equations
The purpose of this section is to establish equivalence between solutions of maxingale problems of diffusion type and Luzin weak solutions to the associated idempotent Ito equations. We also present results on existence and uniqueness for idempotent Ito equations.
As above, C denotes the metric space of continuous R d -valued functions x = (x t , t ∈ R + ) endowed with compact-open topology, C s , for s ∈ R + , denotes the τ -algebra on C generated by the mappings x → x t , t ∈ [0, s], and the canonical τ -flow on C is defined by C = (C t , t ∈ R + ). We identify the product spaceĈ = C × C with the space of continuous R d × R d -valued functions (x, x ) = (x t , x t ), t ∈ R + with compact-open topology; the canonical τ -flow onĈ is denoted bŷ C = (Ĉ t , t ∈ R + );ĈΠ = ĈΠ t , t ∈ R + denotes the completion of the τ -flowĈ with respect to a deviabilityΠ onĈ. Besides, "a.e." refers to Lebesgue measure unless specified otherwise.
Let functions b s (x) and c s (x), where s ∈ R + and x ∈ C, with values in R d and the set of positive semi-definite d × d-matrices, respectively, be C s -measurable and continuous in x ∈ C for s ∈ R + , Lebesgue measurable in s for x ∈ C, and such that the integrals t 0 |b s (x)| ds and t 0 tr c s (x) ds are finite and continuous in x ∈ C for all t ∈ R + . We define the function G Let us introduce the idempotent Ito equation  We preview the proof with a lemma in the theme of the continuous mapping principle.
Lemma 3.1. Let idempotent variables X and X φ , φ ∈ Φ, where Φ is a directed set, with values in a metric space E be defined on idempotent probability spaces (Υ, Π) and respectively. Let f be a function from E to a metric space E . If the X φ converge in idempotent distribution to X as φ ∈ Φ and there exists a sequence of closed sets F m ⊂ E, m ∈ N, such that lim m→∞ lim sup φ∈Φ Π(X φ ∈ F m ) = 0 and f is continuous when restricted to each F m , then the f (X φ ) converge in idempotent distribution to f (X).
Proof. Let f m denote a continuous function that coincides with f on F m , which exists by the Tietze-Urysohn extension theorem. Then the f m (X φ ) converge in idempotent distribution to f m (X) as φ ∈ Φ by the continuous mapping principle, The claim follows.
We prove necessity. By hypotheses, under Π x 0 the idempotent process exp(λ · ( , t ∈ R + is a local exponential maxingale relative to C. Let X and W denote the coordinate processes onĈ, i.e., X t (x, x ) = x t and W t (x, x ) = x t , and let Π = Π x 0 × Π W . By the maxingale property of an idempotent Wiener process and Lemma 2.1, the idempotent process exp(λ · ( λ · (c s (X) + I)λ/2 ds, t ∈ R + , where I is the d × d identity matrix and > 0, is a local exponential maxingale on (Ĉ,Π) relative to the canonical τ -flow. Hence, by Theorem 2.5.22 in Puhalskii [36], on (Ĉ,Π) there exists a standard idempotent Wiener process W relative toĈΠ such that Let Π be the deviability onĈ that is the idempotent distribution of (X, W ) on (Ĉ,Π). Since the idempotent distributions of X and W are Π x 0 and Π W , respectively, the net Π is tight as → 0. Let a subnet Π of Π weakly converge to a deviabilityΠ onĈ. SinceΠ has Π x 0 and Π W as the coordinate projections, we can assume that the (X, W ) converge in idempotent distribution to the canonical idempotent process (X,W ) on (Ĉ,Π) and thatW is a standard idempotent Wiener process. Let us show thatW is idempotent Wiener relative toĈΠ. It suffices to prove thatW t −W s is independent ofĈ s on (Ĉ,Π) for s ≤ t. Since W is standard idempotent Wiener relative toĈΠ and X isĈΠ-adapted, W t − W s is independent of (X r , W r ) on (Ĉ,Π) for r ≤ s. By convergence in idempotent distribution of the (X, W ) to (X,W ),W t −W s is independent of (X r ,W r ) on (Ĉ,Π) for r ≤ s. The required property follows sinceĈ s is generated by (X r ,W r ), r ≤ s.
In addition, on noting that under the hypotheses the mapping x → (c s (x), s ∈ [0, t]) is continuous in x ∈ C with respect to the norm t 0 c s (x) ds, we have that the function (x, y) → t 0 c s (x) 1/2ẏ s ds, t ∈ R + fromĈ to C, which is defined arbitrarily when the integral is not well defined, is continuous when restricted to the sets F m = {(x, y) : Equality (3.4) follows. Since X has idempotent distribution Π x 0 andW is a standard idempotent Wiener process relative toĈΠ underΠ, Π x 0 is a Luzin weak solution of (3.4).
As a consequence of the theorem, uniqueness holds for (x 0 , G) if and only if equation (3.2) has a unique Luzin weak solution. We now provide sufficient conditions for the equation to have a Luzin weak solution, which extend the results of Puhalskii [36,37]. Let us assume that for every a > 0 there exists a locally integrable in s nonnegative function r a s , s ∈ R + , such that |b t (x)|+ c t (x) ≤ r a t for all t ∈ R + and x ∈ C satisfying the condition sup s≤t |x s | ≤ a. Let there exist a differentiable real-valued function V (x), x ∈ R d , such that |V (x)| → ∞ as |x| → ∞, an increasing real-valued positive function k(q), q ∈ R + , such that k(q) → ∞ as q → ∞ and ∞ 1 1/k(q) dq = ∞, and a locally integrable R + -valued function l s , s ∈ R + , for which where ∇ denotes gradient. Let c s (x) ⊕ denote the pseudo-inverse of c s (x) and let idempotent probability Π * x 0 on C be defined by if x is absolutely continuous, x 0 = x 0 , andẋ s − b s (x) is in the range of c s (x) a.e., and Π * x 0 (x) = 0 otherwise. The next lemma implies, in particular, that Π * x 0 is a deviability. Proof. Let σ t (x) = c t (x) 1/2 . We first prove that the ordinary differential equatioṅ where w t is an absolutely continuous function such that ∞ 0 |ẇ t | 2 dt < ∞, satisfies the extension condition, Puhalskii [36], i.e., given arbitrary T > 0, every absolutely continuous functionx t that satisfies (3.8) on [0, T ] can be extended to a solution on [T, ∞). Let, given N ∈ N and . By hypotheses, the functions b N t (x) and σ N t (x) are continuous in x, and the functions |b N t (x)| and σ N t (x) 2 are bounded by the locally integrable function r N t , so the equationṡ have solutions x N = (x N t , t ∈ R + ) (the latter follows, e.g., by the method of successive approximations). We have by (3.6a) and (3.6b) for t ≥ T a.e.
which implies that the y N t are bounded in N for a given t. Therefore, for suitable constants Let idempotent probabilityΠ onĈ be defined byΠ(x, w) = Π W (w) if (x, w) satisfies (3.8) andΠ(x, w) = 0 otherwise. Then the first component process onĈ has idempotent distribution Π * x 0 underΠ, the second component process has idempotent distribution Π W , and (3.2) holdŝ Π-a.e. In addition, by the argument of the proof of part 1 of Lemma 2.6.17 in Puhalskii [36], the second component process is standard idempotent Wiener relative toĈΠ. Thus, Π * x 0 is a weak solution to (3.2). It is a Luzin weak solution ifΠ is a deviability onĈ. Upper semi-continuity of Π(x, w) is established similarly to how it is done in the proof of Theorem 2.6.24 in Puhalskii [36]. To show that the sets {(x, w) :Π(x, w) ≥ a} are, moreover, compact for a ∈ (0, 1], let us consider sequences w (n) ∈ C, n ∈ N, and x (n) ∈ C, n ∈ N, such thaṫ
As mentioned in Section 2, pathwise uniqueness holds for (3.2) under Lipshitz continuity conditions, so there exists a unique weak solution, which is in fact a Luzin strong solution. The following uniqueness result relaxes these requirements.
Corollary 3.1. Let, in addition to the hypotheses of Lemma 3.2, for every a > 0 there exist a differentiable real-valued function V a (x), x ∈ R d , such that V a (x) = 0 if x = 0, an increasing R +valued function k a (q), q ∈ R + , such that k a (0) = 0, k a (q) > 0 for q > 0 and 0 dq/k a (q) = ∞ for > 0, and a locally integrable R + -valued function l a t , t ∈ R + , for which the inequalities (3.2) has a unique strong solution, which is a Luzin strong solution. This solution has idempotent distribution Π * x 0 , which is a unique weak solution and is a unique Luzin weak solution of (3.2).
The proof is omitted since it is much the same as that of the next corollary, which permits non-Lipshitz behaviour for the Markov setting. The result is analogous to the one from the theory of ordinary differential equations, see, e.g., Coddington and Levinson [5], Hartman [21]. are Lebesgue-measurable in s ∈ R + , continuous in x ∈ R d , and such that the functions ess sup x∈R d : |x|≤A |b s (x)| and ess sup x∈R d : |x|≤A ĉ s (x) are locally integrable in s for A ∈ R + . Let the following growth conditions be satisfied: where the functions V (x), k(q), and l t are as in the hypotheses of Lemma 3.2. Let for every a > 0 there exist functions V a (x), k a (q), and l a t as in the statement of Corollary 3.1 such that the inequalities ∇V a (x − y) · (b t (x) −b t (y)) ≤ l a t k a |V a (x − y)| and ∇V a (x − y) · ĉ t (x) 1/2 −ĉ t (y) 1/2 2 ∇V a (x − y) ≤ l a t k a (|V a (x − y)|) 2 hold when |x| ≤ a and |y| ≤ a. Then (3.2) has a unique strong solution, which is a Luzin strong solution. This solution has idempotent distribution Π * x 0 , which is a unique weak solution and is a unique Luzin weak solution of (3.2).
Proof. By Lemma 3.2 under the hypotheses there exists a Luzin weak solution, so it suffices to prove pathwise uniqueness for (3.2). Let X and Y be two solutions. Given arbitrary a > 0 and T > 0, we define τ a = inf{t ∈ R + : |X t | ∨ |Y t | ≥ a} ∧ T . Then a.e. for t ≤ τ a Let ϕ t = |V a (X t − Y t )| and ψ t = 1 + l a t + l a t |Ẇ t |. Then ψ t is positive, locally integrable, and ϕ t ≤ t 0 ψ s k a (ϕ s ) ds. Denoting as y t the right-hand side of the latter inequality, we have that t 0 dy s /k a ( + y s ) ≤ t 0 ψ s ds for arbitrary > 0. Hence, the integral +yt dq/k a (q) = t 0 dy s /k a ( + y s ) is bounded as → 0, so y t = 0. We have thus proved that X t = Y t for t ≤ τ a . Letting a → ∞, we conclude that X t = Y t for t ≤ T .
Remark 3.1. The most common choice of functions V , V a , k, and k a is V (x) = V a (x) = |x| 2 and k(q) = k a (q) = q. One then obtains extensions of linear-growth and Lipshitz continuity conditions. Remark 3.2. As an example of non-Lipshitz-continuous coefficients with super-linear growth at infinity, one can takeĉ the v i are elements of R d , and Λ 1 and Λ 2 are d × d matrices.

Applications to large deviations of stochastic equations
We now consider applications to large deviation convergence of solutions of stochastic equations to solutions of idempotent Ito equations with possibly degenerate coefficients. In the standard large deviation theory classification, this is the setting of moderate deviations. To recall the terminology, we say that a net P φ , φ ∈ Φ, Φ being a directed set, of probability measures on the Borel σ-algebra of a metric space Υ large deviation (LD) converges as As a matter of fact, the net P φ , φ ∈ Φ, LD converges at rate k φ to Π if and only if it obeys the LDP with action functional I = − log Π for scale k φ , so LD convergence is another way of formulating LDP results, which we find more natural for our approach. Next, a net of random variables X φ , φ ∈ Φ, with values in Υ is said to LD converge in distribution (denoted as X φ ld − → X) at rate k φ to a Luzin idempotent variable X with values in Υ if the net of laws of the X φ LD converges at rate k φ to the idempotent distribution of X. We also say that the net X φ , φ ∈ Φ, converges to a point υ ∈ Υ superexponentially in probability at rate k φ and write X φ where ρ is the metric on Υ. The theorems on LD convergence presented in this section refer to the Skorohod space D of right-continuous with left-hand limits R d -valued functions x = (x t , t ∈ R + ) endowed with the Skorohod-Prohorov-Lindvall metric, see, e.g., Jacod and Shiryaev [23] for more detail. We denote as x s− the left-hand limit of x at s.
The first application expands on Example 5.4.6 in Puhalskii [36]. Let B(E) denote the Borel σ-algebra of a metric space E and P(D) denote the predictable σ-algebra on R + × D. Let (U, U) be a measurable space and m(dϑ) be a non-negative σ-finite measure on (U, U). We assume as given functions b , γ s (x), σ , γ s (x), and f , γ s (x, ϑ), where > 0, γ > 0, s ∈ R + , x ∈ D, and ϑ ∈ U , which are P(D)/B(R d ), P(D)/B(R d × R d ), and P(D) ⊗ U/B(R d )-measurable, respectively. Let there exist weak solutions to the stochastic equations , and |f ,γ s (x, ϑ)| ≤ h ,γ s (ϑ)(1 + sup t≤s |x t |), where l ,γ s and h ,γ s (ϑ) are R + -valued and increasing in s, h ,γ s (ϑ) is U/B(R + )-measurable for every s ∈ R + , and U h ,γ s (ϑ) 2 m(dϑ) < ∞, Gihman and Skorohod [20]. Another set of conditions is given by assuming that b ,γ , and |f ,γ s (x, ϑ)| ≤ĥ ,γ s (ϑ)(1 + |x|), wherel ,γ s andĥ ,γ s (ϑ) are R + -valued and increasing in s,ĥ ,γ s (ϑ) is U/B(R + )-measurable for every s ∈ R + , and Uĥ ,γ s (ϑ) 2 m(dϑ) < ∞, Gihman and Skorohod [20], Ikeda and Watanabe [22], and Jacod and Shiryaev [23]. To ensure uniqueness one may require, in addition, that either the coefficients be Lipshitz-continuous or that they be bounded and a non-degeneracy condition hold, see Gihman and Skorohod [20], Ikeda and Watanabe [22], and Jacod and Shiryaev [23] for more detail. We also remark that for diffusions without jumps the linear-growth and (or) Lipshitz continuity conditions can be relaxed in analogy with the hypotheses of Lemma 3.2, Corollary 3.1, and Corollary 3.2, cf., Narita [30], Ethier and Kurtz [12], Ikeda and Watanabe [22], Fang and Zhang [13].) Let us denote x * t = sup s≤t |x s | and introduce the following moment conditions on the jumps of the X ,γ : It is assumed that the functions b s (x) and c s (x) as defined at the beginning of Section 3 are extended for x ∈ D such that they are locally integrable in s, are adapted to the canonical τ -flow on D, and are continuous in x at x ∈ C. (The canonical τ -flow on D is defined similarly to the one on C as consisting of τ -algebras generated by projection mappings.) Let us denote c ,γ If, in addition, either condition (P ) holds and log(γ −1 ) → ∞, or condition ( SE) holds and 2−β /γ β → ∞, then X ,γ ld → X at rate 1/ .
Proof. The proof consists in checking the hypotheses of Theorem 5.2.12 in Puhalskii [36] and proceeds similarly to the proof of Lemma 8 in Puhalskii [37]. There are several groups of conditions to be checked. The first group concerns the limit idempotent equation. For the setting at hand these are the continuity and local majoration conditions on B t (x) = t 0 b s (x) ds and C t (x) = t 0 c s (x) ds, see p.375 of Puhalskii [36]. These conditions readily follow from the hypotheses. There is also condition (N E), which requires certain regularity properties of deviability Π * x 0 from (3.7): not only does the function Π * x 0 (x) have to be upper compact in x but also the sets ∪ s∈[0,t] {x * s : Π * x 0 ,s (x) ≥ a} must be bounded for a ∈ (0, 1] and t ∈ R + , where in analogy with the definition of Π * is in the range of c r (x) a.e. on [0, s], and Π * x 0 ,s (x) = 0 otherwise. The stated property follows, as in the proof of Lemma 8 in Puhalskii [37], by the fact that Π * x 0 is a deviability and the equationẋ t = b t (x) satisfies the extension condition The second group of conditions concerns the predictable measure of jumps of X ,γ , which has the form These are conditions (A) loc + (a) loc on p.377 of Puhalskii [36]. They require, roughly speaking, that ν ,γ in the limit reside on sets Γ that are "close to the origin" and follow by the moment conditions (P ) or (SE) as in the proof of Theorem 5.4.4 in Puhalskii [36]. There is also condition (0) requiring that the initial value X ,γ 0 converge superexponentially in probability to x 0 , which holds since X ,γ 0 = x 0 . The final group of conditions concerns convergence of the predictable characteristics of X ,γ to the characteristics of X. Under the moment conditions (P ) or (SE) the measure ν ,γ satisfies condition (L 2 ) loc on p.410 of Puhalskii [36]. Therefore, as in Theorem 5.3.5 and in view of the assertion of Lemma 5.3.2 in Puhalskii [36], it suffices to check conditions (sup B ) loc and (C 0 ) loc on p.410 of Puhalskii [36]. These conditions feature the first and modified second characteristics of X ,γ without truncation, which are given by the respective equalities It is required that B ,γ t∧τ ,γ N −B t∧τ ,γ N (X ,γ ) converge to 0 superexponentially in probability uniformly over t from bounded domains and −1C ,γ t∧τ ,γ N − C t∧τ ,γ N (X ,γ ) converge to 0 superexponentially in probability for all t, where τ ,γ N = inf{t ∈ R + : |X ,γ t | + t ≥ N } and N ∈ N is arbitrary. These convergences follow by the convergence conditions in the hypotheses.
Thus, according to Theorem 5.2.12 in Puhalskii [36] the sequence of laws of the X ,γ is exponentially tight and every limit deviability solves the maxingale problem (x 0 , G) with G from (3.1). According to Theorem 3.1 such a deviability is a weak solution (3.2), so is specified uniquely. Theorem 4.1 implies, in particular, that, if, in addition, the equation x t = x 0 + t 0 b s (x) ds has a unique solution, then under the hypotheses the X ,γ tend in probability to this solution. We now consider deviations of order o(1) from such a solution. The limit proves to be an idempotent Ornstein-Uhlenbeck process.
The probability objects associated with the nth system are defined on a complete probability space (Ω n , F n , P n ). Let A n = (A n t , t ∈ R + ) denote the exogenous arrival process in the nth queue. Let L n,k = (L n,k t , t ∈ R + ), k ∈ N, and B n,k = (B n,k t , t ∈ R + ), k ∈ N, be Poisson processes of respective rate θ n t and µ n t at time t, which are responsible for reneging and service, respectively, in the sense made precise by equation (4.2) below. The processes A n , L n,k , and B n,k are considered as random elements of space D and are assumed to be mutually independent. They are also independent of the initial queue length.