Random Operators and Stochastic Equations 

About: Random Operators and Stochastic Equations is an academic journal published by De Gruyter. The journal publishes majorly in the area(s): Stochastic differential equation & Stochastic partial differential equation. It has an ISSN identifier of 0926-6364. Over the lifetime, 653 publications have been published receiving 3791 citations.

Some skew-symmetric models

Abstract: The skew-normal distributions have been introduced by many authors, e.g. Azzalini (1985), Arnold et al. (1993), Aigner et al. (1977), Andel et al. (1984). This class of distributions includes the normal distribution and possesses several properties which coincide or are close to the properties of the normal family. However, this class has a skewness parameter which makes it possible to have a reasonable model for a skewed population distribution thus providing a more flexible model which represents the data as adequately as possible. Besides being useful in modeling, they are helpful in studying the robustness, and in Bayesian analysis as priors. The construction of such models is based on the following lemma (see Azzalini, 1985).

Localization for random perturbations of periodic Schrödinger operators

Abstract: We prove localization for Anderson-type random perturbations of periodic Schr dinger operators on R near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schr dinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.

Existence and uniqueness of solutions of stochastic functional differential equations

Abstract: Using a variant of the Euler-Murayama scheme for stochastic functional dierential equations with bounded memory driven by Brownian motion we show that only weak one-sided local Lipschitz (or ’monotonicity’) conditions are sucient for local existence and uniqueness of strong solutions. In case of explosion the method yields the maximal solution up to the explosion time. We also provide a weak growth condition which prevents explosions to occur. In an appendix we formulate and prove four lemmas which may be of independent interest: three of them can be viewed as rather general stochas

Asymptotic distribution of smoothed eigenvalue density. II. Wigner random matrices

Abstract: We study spectral properties of the Wigner ensemble of Ν Χ Ν random symmetric matrices W* '. We consider the smoothed eigenvalue density function determined by the resolvent ^ψ(^) z)-l wjtn z = \\ + yv~, α > 0. We prove that if the random matrix entries have the eighth finite moments, then the mentioned function, when centered and properly normalized, converges as Ν — > 00 for all α < 1/8 to a Gaussian random variable. We show that corresponding correlation function does not depend on the details of the probability distribution of the random matrix entries and coincides with that derived for the Gaussian Orthogonal Ensemble. This supports the universality conjecture for local spectral properties of large random matrices. 1. MAIN RESULTS AND DISCUSSION We Study the asymptotic behaviour of the eigenvalue distribution of random symmetric matrices when their dimension tends to infinity. We use the technique suggested in [3] to investigate the distribution of the smoothed eigenvalue density of Gaussian Orthogonal Ensemble of random matrices. In this paper we are concerned with the case when the matrix entries are independent arbitrarily distributed random variables. Let us consider the ensemble of random N x N real symmetric matrices W^ with entries ti;(x>y)> (1.1) w(x,y) = w(y,x), x, Let {w(x, τ/), x < t/, x, y 6 N} be independent random variables defined on the same probability space. We assume also that Ew(x,y)=0, £[«,(*,»)]' = { 2^ |^ί^ (1-2) where Ε denotes the mathematical expectation. The ensemble (1. !)-(!. 2) is known as the Wigner ensemble of random matrices. Let us consider the resolvent G(z) = (W^ z)\", Im z ^ 0 and denote by its normalized trace: 150 Asymptotic distribution of smoothed eigenvalue ·£*>-* ' where λ^ = λ ·̂ \\ j = 1, . . . , Ν are eigenvalues of W^\\ It is clear that the imaginary part of g(\\z) can be regarded as a smoothing of the formal density of eigenvalue distribution of Ξ y&(A-T)p< N >(r)dT, (1.3) with _/>ε(λ)αλ = 1 and where S(r) is the Dirac ί-function. The smaller ε is, the more detailed information about the eigenvalue distribution one obtains from the analysis of (1.3). This rs because the function (1.3), broadly speaking, is related to those eigenvalues of W W that fall into the ε-vicinity of λ. Since the total number of eigenvalues is TV, it is natural to consider (1.3) when ε goes to zero simultaneously with N. We will call the random variable π~ξα(\\) = π\" W>(A + V-), α > Ο the \"smoothed eigenvalue density function\". In this paper we continue the study of ξα started in [3, 11]. In [11] it was shown that if supEMi,y)] 0 / OO /»2V φ (χ _ r)pW(r)dr = (2TM) / φε(\\ τ) χΑυ r dr (1.7) -οο V-2v with probability 1 provided (1.4) holds (see [8] and [16] for weaker conditions). A. Beutet de Monvel and A. Khorunzhy 151 Let us note that (1.7) means weak convergence of the measures p^(r) to the measure with the support (-2v,2v) and the density π\"^), r G (-2v,2v). The proof of (1.5) presented in [11] also implies convergence in average Ihn Ε0<\">(ζΛ,) = /ι(λ) + ίξ(λ), (1.8) Ν-)· οο where z^ — Ayv + 'iN~ and λ^ν -> λ G (-2v,2v) as N ->· 00 (see also [1] where (1.8) is proved for ZN = X + iLW\"/ for £>, TV ->> oo). In this paper we study the centered random variable and prove a central limit theorem refining (1.5). THEOREM 1.1. Let the random variables ( w ( x , y ) , z,y G IE} satisfy the conditions E KZ, t/)]* = v2k < oo, fc = 2, 3, 4 (1.9) and Q, fc = l,2,3. (1.10) If λ G (-2υ,2ν) and α G (0, 1/8), then the random variables 9 (X + tfT), ^(λ) Ξ ΛΓ-^ ρ (λ + ίΝ~) converge in distribution, as Ν -> oo, to Gaussian random variables θ and η that are independent and have variance 1/4. Let us note that the Gaussian type of the distribution of the random variable ίε) Eg^(X + ie)], ε > 0 was proved for the first time for Wigner random matrices in [8] (see also [13] for more details). A similar result is obtained for Gaussian random matrices with correlated entries in [2]. Gaussian fluctuations of certain spectral statistics have been also observed in other ensembles of random matrices with statistically dependent entries [9, 10, 19]. The following statement concerns the asymptotic behaviour of the correlation function of the smoothed eigenvalue density. THEOREM 1.2. (i) If \\i and \\2 are fixed numbers in the interval (-2υ,2ι>), then, under the conditions of Theorem 1.1, E g (Xl + i^TV-) g (A2 + ΊδιΝ-*) = O(N~) for 61,62 = ±1. (ii) Let λ G (-2v,2v) and α G (0,1/9), and (1.9) and (1.10) hold. Then, E g (λ + nN~ + iN~) 9 (λ -f riN-P ίΛΤ) 152 Asymptotic distribution of smoothed eigenvalue for all β G (Ο, α) andri,r2 G (Ο,οο). (iii) If, in addition to (1.9) and (1.10), E [w(x, t/)] = 0and sup E [w(x, τ/)] < oo, (1.12) *,y then (1.11) holds for all α G (0, 1/8). Remark 1.1. We assume odd moments of iy(x, y) equal to zero to make the proofs of Theorems 1.1 and 1.2 shorter. Our computations show that one can avoid this restriction by using a truncation technique common in probability theory and in spectral theory of random matrices as well. We plan to study this problem in a separate publication. Remark 1.2. We are interested in minimal conditions on the probability distribution of w(x,y). As it follows from our computations and is illustrated particularly by item (iii) of Theorem 1.2 , the greater the values of α one wants to consider, the more moments of w ( x , y ) are required to be finite. It is somewhat surprising for us that to come closer to the real axis, one needs to impose conditions that are not related to the smoothness of the probability distribution of eigenvalues, i.e. smoothness of the probability distribution of w(x,yYs. It would be interesting to develop another approach that could be more adapted to the study of the smoothed eigenvalue density function. Remark 1.3. If w(x,y), χ < y are independent Gaussian random variables satisfying (1.2), then the ensemble {W^} in (1.1) defines the Gaussian Orthogonal Ensemble of random matrices (GOE). In [3] we proved that Theorems 1.1 and 1.2 are valid for the GOE with α G (0,1). Remark 1.4. Theorems 1.1 and 1.2 remain valid when the second equality of (1.2) is replaced by the simpler condition E[w(x,y)] = v, z,y GIF. To discuss our results, let us note that the properties of the random variables θ ο and η, as well as the correlation function of 9 (z), cf. (1.12) do not depend on the particular values of the moments v^ of tu(x,y) , cf. (1.9). Thus, one can conclude that the behaviour of the eigenvalue statistics related to the number of eigenvalues n that is much greater than 1 and much less than N in the limit N -> oo, does not depend on the details of the probability distribution of the Wigner random matrices. The latter statement strongly resembles the universality conjecture for large random matrices originally formulated for eigenvalue statistics with n ~ 1 as N — > oo (see e.g. [15, 17]). The spectral statistics related to a finite number of eigenvalues is referred to as the local one. Our considerations concern a case that can be regarded as an intermediate between the local and the global (n ~ N). It should be noted that the intermediate regime we study differs from these two regimes. Indeed, it is known [5, 15] that in GOE the variance of the number of eigenvalues lying inside the intervals of length LN' is of order (logL)\". It is shown in [5] that the distribution of the centered number of eigenvalues lying in such intervals converges to the Gaussian one. However, the logarithmic behaviour of the variance cannot be derived directly from (1.11), even if one could prove that (1.11) holds for z = \\ + ieyv, On the other hand, the global regime corresponds, in our notations, to α = 0 or, equivalently, ε > 0 in (1.3). In this case it is proved in [12] that the leading term Γ(λι, λ2;ε) of the correlation function = Ε Ρ (λι + ie) 9 (λ2 ίε) ̂ Τ(λ1,λ2;ε)[1 + 0(1)], ε > Ο A. outet de Monvel and A. Khorunzhy 153 depends on the cumulant «4 = E [w(x,y)} 3z;. See also [6], where Τ(λι,λ2;ε) is explicitly obtained, using diagrammatic and renormalization-group techniques. Both expressions in [12] and [6] coincide up to the sign in front of the term with «4. This dependence is shown to vanish in the asymptotics of infinitely small ε and λι λ2 -> 0. In this limit, the mentioned leading term takes the form (compare with (1.11)) However, the latter statement is not equivalent to our results because we consider but not its leading term. It should be pointed out that the universality of spectral statistics considered in the regime 1 0, N(\\\\ A2) ->· s, one obtains from (1.13) the expression -π~$~ that coincides with the Wigner-Dyson exact large-5 asymptotics of the eigenvalue density of the GOE (see e.g. [15]). Turning back to random matrices whose entries are arbitrarily distributed independent random variables, let us mention the recent paper [18]. It is proved that the centered moments Tr tf/ ETr v7 (1.14) converge to Gaussian random variables as Ν -> oo, k = N, χ € (0, 1/2). The covariance function of these random variables is shown to be independent from the particular values of the moments E [w(z,t/)]. This can be regarded as one more evidence for the universal behaviour of eigenvalue distribution. Our results are complemented to results of [18] in the sense that we study the spectral properties inside of the interval (-2υ,2ν) comprising the limiting spectrum of W^\\ while the asymptotic behaviour of the moments (1.14) with increasing numbers 2k is related to the eigenvalues lying outside of (-2ν,2υ). We organize this paper as follows. In Section 2 we study the variance of g^(z). To do this, we combine the general technique developed in [3] for the GOE with the cumulant expansion approach suggested in [14, 13] to study random matrices with arbitrarily distributed entries. In this section we also prove some estimates. In Sections 3 and 4 we prove Theorems 1.1 and 1.2 , respectively. In Section 5 we summarize our results. 2. GENERAL TECHNIQUE AND MAIN ESTIMATES The m