Large Deviations for the solution of a Kac-type kinetic equation

The aim of this paper is to study large deviations for the self-similar solution of a Kac-type kinetic equation. Under the assumption that the initial condition belongs to the domain of normal attraction of a stable law of index $\alpha<2$ and under suitable assumptions on the collisional kernel, precise asymptotic behavior of the large deviations probability is given.


Introduction
This paper deals with the probability of large deviations for the solutions of a class of one dimensional Boltzmann-like equations. Specifically, given an initial probability distributionρ 0 on B(R), the Borel σ-field of R, we consider a time-dependent probability measure ρ t solution of the homogeneous kinetic equation (1) ∂ t ρ t + ρ t = Q + (ρ t , ρ t ) ρ 0 =ρ 0 .
Following [3,11], we assume that Q + is the smoothing transformation defined by where ρ is the law of X 1 , X 2 , (L, R) is a given random vector of R 2 , and (L, R), X 1 and X 2 are stochastically independent. The first model of type (1)- (2) has been introduced by Kac [22], with collisional parameters L = sinθ and R = cosθ for a random angleθ uniformly distributed on [0, 2π). In the original Kac equation ρ t represents the probability distribution of the velocity of a particle in a homogeneous gas. In addition to the Kac equation, also some one dimensional dissipative Maxwell models for colliding molecules, see e.g. [8,25,27], can be seen as special cases of (1)- (2). Moreover, equations (1)- (2) have been used to describe socio-economical dynamics see, e.g., [5,7,15,24,26,29] and the references therein. In this last case particles are replaced by agents in a market and velocities by some quantities of interest (money, wealth, information,...). Finally, it is worth recalling that, using results in [10,11], it can be shown that the isotropic solution of the multidimensional inelastic Boltzmann equation [9] can be expressed in terms of the solution of equation (1) for a suitable choice of (L, R).
As for the speed of convergence to equilibrium, explicit rates with respect to suitable probability metrics have been derived in various papers. For the Kac equation see [13,14,17], for the inelastic Kac equation see [4], for the solutions of the general model (1)-(2) see [2,3,6].
Many of the above mentioned results are based on a probabilistic representation of the solution ρ t . In point of fact, as we will briefly explain in Section 2.2, it can be proved that the unique solution ρ t of (1)- (2) is the law of the stochastic process where ν t is a Yule process, [β jn ] jn are suitable random weights and X j are independent identically distributed (i.i.d., for short) random variables with lawρ 0 .
The aim of this paper is to study large deviations for the (eventually rescaled) solution ρ t when the initial conditionρ 0 belongs to the domain of normal attraction of an α-stable law. More precisely, we will study the large deviation probability for e −tµ(α) V t when, for a suitable µ(α), e −tµ(α) V t converges in law to a scale mixture of α-stable distributions. In the following we shall assume that α < 2, the study of the case α = 2 is postponed to future work since it requires completely different techniques.
In view of the probabilistic representation (3) it is not surprising that the study of the large deviation probabilities for ρ t is strictly related to large deviations for sums of i.i.d. random variables.
Let us briefly recall these classical results. If α ∈ (0, 1) ∪ (1, 2) and if (X n ) n≥1 is a sequence of i.i.d. random variables in the domain of normal attraction of an α-stable law, centered if α > 1, then, n −1/α n i=1 X i converges in law to an α-stable random variable. Moreover, if x n → +∞, then where c 0 is a positive constant determined by the law of X 1 . See [18,19,20]. For more information on large deviations for sums of i.i.d. random variables see, for example, [12,28] and the references therein.
Our main result, which is stated in Theorem 3.1, is reminiscent of (4). It can be summarized by saying that if the initial distributionρ 0 belongs to the domain of normal attraction of an α-stable law with α < 2 and the collision coefficients (L, R) satisfy some additional assumptions, then c 0 x α t as x t goes to +∞. As in the i.i.d. case, this result can be interpreted by saying that the main part of probability of large deviations is generated by one large summand comparable with the whole sum process V t . The paper is organized as follows. Section 2.1 is devoted to a brief review of some known results on the self-similar asymptotics for the solutions of (1). Section 2.2 contains the detailed description of the probabilistic representation (3). In Section 2.3 we provide some results on the process H t = max j=1,...,νt |β jνt X j |. In particular we show that the law of H t satisfies a kinetic equation of type (1) for a suitable collisional kernel. Section 3 contains the large deviation results for ρ t . Section 4 deals with the study of large deviation probabilities for weighted sums of i.i.d. random variables. The proofs of the results stated in Section 2 and 3 are collected in Section 5.

Self-similar asymptotics for the solutions
In the following, all the random elements are defined on a given probability space (Ω, F , P ) and E denotes the expectation with respect to P .
Throughout the paper we assume that L and R are non-negative random variables such that P {L > 0} + P {R > 0} > 1.
As for the initial probability distributionρ 0 is concerned, we will assume that it belongs to the domain of normal attraction of an α-stable law. It is well-known that, provided α = 2, a probability measureρ 0 belongs to the domain of normal attraction of an α-stable law if and only if its distribution function F 0 (x) :=ρ 0 {(−∞, x]} satisfies (5) lim Typically, one also requires that c + 0 + c − 0 > 0. See for example Chapter 2 of [21]. be the so called spectral function of Q + , see [2] and [11].
2.1. Convergence to self-similar solutions. In the study of the asymptotic behavior of the solutions of (1), a fundamental role is played by the fixed point equation for distributions where Z, Z 1 , Z 2 are i.i.d. positive random variables, Θ is a random variable with uniform distribution on (0, 1), (Z, Z 1 , Z 2 ), Θ and (L, R) are stochastically independent. As already recalled in the introduction, the unique solution ρ t to (1)-(2) is the law of the stochastic process V t defined in (3). Further details on this probabilistic representation will be given in Section 2.2. The next results, concerning the convergence of a suitable rescaling of V t to the so-called self-similar solutions of (1), are proved in [2]. Theorem 2.1 (CLT when α = 1, [2]). Let α ∈ (0, 1) ∪ (1, 2) and let condition (5) be satisfied for If µ(δ) < µ(α) < +∞ for some δ > α, then e −µ(α)t V t converges in distribution, as t → +∞, to a random variable V ∞ with the following characteristic function: Further information on the mixing random variable Z ∞ (α) are given in Proposition 5.2. See also [2].

Remark 1.
In order to study the large deviations for ρ t , in what follows we will need to assume that c + 0 + c − 0 > 0, even if both Theorem 2.1 and Theorem 2.2 hold also for c + 0 + c − 0 = 0. In this last case, Theorem 2.1 is valid with λ = η = 0 and hence V ∞ = 0 with probability one, while Theorem 2.2 is valid with V ∞ = γ 0 Z ∞ (1).

Remark 2.
Let us consider a random vector (L, R) such that µ(α) = 0, that is E[L α + R α ] = 1. As a consequence of the previous results, if E[L δ + R δ ] < 1 for some δ > α, then V t converges in distribution to V ∞ . In this case Z ∞ (α) satisfies the fixed point equation and it is easy to see that the law ρ ∞ of V ∞ is a steady state for equation (1), i.e. ρ ∞ = Q + (ρ ∞ , ρ ∞ ). This case has been extensively studied in [3].
we start by providing some results on this last process. First of all, it is worth noticing that the law of H t satisfies an homogeneous kinetic equation of the form (1) with Q + replaced by the kernel where, as usual, X 1 , X 2 , (L, R) are independent and X i has law ρ for i = 1, 2.
Following the same line of reasoning of [2,3] we prove the next result on the asymptotic behavior of e −µ(α)t H t . Theorem 2.4. Let α ∈ (0, 1) ∪ (1, 2) and the hypotheses of Theorem 2.1 be in force, or let α = 1 and the hypotheses of Theorem 2.2 hold. Assume also that c 0 = c + 0 + c − 0 > 0. Then e −µ(α)t H t converges in distribution, as t → +∞, to a random variable H ∞ with the following probability distribution function: It is useful to note that Theorem 2.4 states that the law of H ∞ is a scale mixture of Fréchet distributions.

Main results: large deviations for ρ t
As a consequence of Theorems 2.1-2.2, one has that, if x t → +∞ as t → +∞, then The main result of this paper concerns the study of the speed of convergence of such a probability to zero under suitable conditions on the function µ(s). In order to state the results, we need some more notation.  (15), then (16)-(17) hold true.
where the law of V ∞ is a steady state for equation (1).
As pointed out in the Introduction, the results stated in the previous theorem are related to large deviations for sums of i.i.d. random variables: Let α ∈ (0, 1) ∪ (1, 2) and let (X n ) n≥1 be a sequence of i.i.d. random variables in the domain of normal attraction of an α-stable law, centered for α > 1, then, [19] and [20]. It follows from (5) since each stable random variable belongs to its own domain of normal attraction. Consequently (20) lim At this stage, it should be clear that equations (16)- (17)- (18) provide analogous results for our processes.

Large deviation for sum of weighted i.i.d. random variables
The present section deals with the study of the probability of large deviations for weighted sums of i.i.d. random variables. This study is a generalization of the large deviation estimates presented in [19,20] and, besides the interest it could hold in itself, it is the first step in the proof of Theorem 3.1.
Taking the expectation on both side of the last inequality we get Now, recalling that (M n (α)) n≥1 is a martingale, we obtain where h(t) is defined in (15) and C µ is a suitable constant.
Proof. As above the symbol C designates a constant, not necessarily the same at each occurrence. We shall repeatedly use the following two simple facts: for any γ > −1 and any t > 0 and, for every t ≥ 1, Relation (49) follows by a simple Taylor expansion of log(1 − x), while (48) follows from (45) and from the inequality .

Proofs of the main theorems.
Proof of Theorem 2.2. The proof follows the same steps of the one of Theorem 2.2 in [2], using in place of Lemma 5.1 in [2] the following simple result: Let (X n ) n≥1 be a sequence of iid random variables with common distribution function F 0 . Assume that (a jn ) j≥1,n≥1 is an array of positive weights such that lim n→+∞ n j=1 a jn = a ∞ and lim n→+∞ max 1≤j≤n a jn = 0.
If F 0 satisfy (5) with α = 1, c + 0 = c − 0 > 0 and (9) holds, then n j=1 a jn X j converges in law to a Cauchy random variable of scale parameter πa ∞ c 0 and position parameter a ∞ γ 0 . To prove this claim, according to the classical general central limit theorem for array of independent random variables, it is enough to prove that x dQ j,n (x) 2 (ǫ > 0), x dQ j,n (x) , See, e.g., Theorem 30 and Proposition 11 in [16]. Conditions (51) and (52) can be proved exactly as the analogous conditions of Lemma 5 in [3]. As for condition (53) note that Using the assumptions on F 0 and on (a jn ) jn it follows immediately that This gives (53).
Sketch of the proof of Theorem 2.3. Using the results in [23] one proves that is a solution of an homogeneous kinetic equation of the form (1) with Q + replaced byQ + . At this stage, following the same arguments used to prove Proposition 1 in [3], one proves that q t is the law of H t .
Proof of Theorem 2.4. Let x > 0 and let B * the σ-field generated by the array of weights [β jn ] j,n and by the Yule process [ν t ] t≥0 . Then where x α e −S(α)t β α jν t .
Proof of Theorem. 3.1. Recalling that B denotes the σ-field generated by the array of random variables [β jn ] jn , using (24) one can write one gets that for every t > 0 and by (41) M νt (α)e −S(α)t → Z ∞ (α) in L 1 . Furthermore |∆ t | ≤ 1 and ∆ t → 1 in probability by Lemma 5.3. Finally by (44) and by (22), one gets in probability. Combining these facts one obtains that in probability for t → +∞ and, by the generalized dominated convergence theorem, that ∆ t D t → Z ∞ (α) in L 1 . Hence, in view of (42) one obtains As far as the term B On the other hand, applying (25), one gets t .
As in the previous part U Remark 4 and the assumptions on x t , according to the value of S(α) and S(2α), one can choose 1/2 < γ < 1 in order that U (2) t → 0 for t → +∞. Hence, lim sup t→+∞ x α t P {|e −S(α)t V t | > x t } ≤ c 0 /(1 − ǫ) α for every ǫ > 0 which implies In view of (56) and (58) we obtain In order to complete the proof of (16) it is sufficient to show that As already noted, by convexity of S and the condition µ(δ) < µ(α), it follows that µ(s) < µ(α) if α < s < δ. Hence, without loss of generality, we can assume that α < δ < 2α. Let Z ∞ (α) be as in Theorems 2.1 and 2.2. Then where S α is a stable r.v. with index α, Z ∞ (α) and S α being independent. If F ∞ and G α denote the distribution functions of V ∞ and S α , respectively, then since E(Z ∞ (α)) = 1 by (42). From the properties of the tails of stable distributions one can write that for x > 0, since α < δ < 2α. See, e.g., [21]. Hence To prove (17), use (26) to writẽ