Almost-sure central limit theorem for a Markov model of random walk in dynamical random environment

Summary We consider a model of random walk on ℤ^ν, ν≥2, in a dynamical random environment described by a field ξ={ξ _t (x): (t,x)∈ℤ^ν+1}. The random walk transition probabilities are taken as P(X _{t +1}= y|X _t = x,ξ _t =η) =P ₀( y−x)+ c(y−x;η(x)). We assume that the variables {ξ _t (x):(t,x) ∈ℤ^ν+1} are i.i.d., that both P ₀(u) and c(u;s) are finite range in u, and that the random term c(u;·) is small and with zero average. We prove that the C.L.T. holds almost-surely, with the same parameters as for P ₀, for all ν≥2. For ν≥3 there is a finite random (i.e., dependent on ξ) correction to the average of X _t , and there is a corresponding random correction of order to the C.L.T.. For ν≥5 there is a finite random correction to the covariance matrix of X _t and a corresponding correction of order to the C.L.T.. Proofs are based on some new L ^p estimates for a class of functionals of the field. Received: 4 January 1996/In revised form: 26 May 1997     

Complete asymptotic decomposition of the sojourn probability of a diffusion process in thin domains with moving boundaries

We investigate a diffusion process ξ(t) with absorption defined in a thin domainD _ε ={(x,t)∶εG ₁ (t)<x<εG ₂ (t), t≥0}. We obtain the complete decomposition of the sojourn probability of ξ(t) inD _ε with respect to ε→0. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 9, pp. 1155–1164, September, 1999.     

Stochastic Equations with Time-Dependent Drift Driven by Levy Processes

The stochastic equation dX _t =dS _t +a(t,X _t )dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X ₀=x ₀∈ℝ when (ℛe  ψ(ξ))⁻¹=o(|ξ|⁻¹) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 1974) in the case of a(t,x)=a(x). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.     

Large deviations for sums of independent random variables with dominatedly varying tails

In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0are proved, where {N(t); t≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.     

Modular curves and bases for the spaces of cuspidal modular forms

Let Γ⊂SL ₂(ℝ) be a Fuchsian group of the first kind. For a character χ of Γ→ℂ^× of finite order, we define the usual space S _m (Γ,χ) of cuspidal modular forms of weight m≥0. For each ξ in the upper half–plane and m≥3, we construct cuspidal modular forms Δ _k,m,ξ,χ ∈S _m (Γ,χ) (k≥0) which represent the linear functionals f?\fracd^kfdz^k|_z=xf\mapsto\frac{d^{k}f}{dz^{k}}|_{z=\xi} in terms of the Petersson inner product. We write their Fourier expansion and use it to write an expression for the Ramanujan Δ-function. Also, with the aid of the geometry of the Riemann surface attached to Γ, for each non-elliptic point ξ and integer m≥3, we construct a basis of S _m (Γ,χ) out of the modular forms Δ _k,m,ξ ,χ (k≥0). For Γ=Γ ₀(N), we use this to write a matrix realization of the usual Hecke operators T _p for S _m (N,χ).     

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξn) on R , and reproduce independently new particles according to a probability law p(ξn) on N. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean EξZ(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.     

Sojourn time of almost semicontinuous integral-valued processes in a fixed state

Let ξ(t) be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps

Correlation structure of intermittency in the parabolic Anderson model

Consider the Cauchy problem ∂u(x, t)/∂t = ℋu(x, t) (x∈ℤ^d, t≥ 0) with initial condition u(x, 0) ≡ 1 and with ℋ the Anderson Hamiltonian ℋ = κΔ + ξ. Here Δ is the discrete Laplacian, κ∈ (0, ∞) is a diffusion constant, and ξ = {ξ(x): x∈ℤ ^d } is an i.i.d.random field taking values in ℝ. G?rtner and Molchanov (1990) have shown that if the law of ξ(0) is nondegenerate, then the solution u is asymptotically intermittent. In the present paper we study the structure of the intermittent peaks for the special case where the law of ξ(0) is (in the vicinity of) the double exponential Prob(ξ(0) > s) = exp[−e ^{s /θ}] (s∈ℝ). Here θ∈ (0, ∞) is a parameter that can be thought of as measuring the degree of disorder in the ξ-field. Our main result is that, for fixed x, y∈ℤ ^d and t→∈, the correlation coefficient of u(x, t) and u(y, t) converges to ∥w _ρ∥⁻² _ℓ2Σ_{z ∈ℤd} w _ρ(x+z)w _ρ(y+z). In this expression, ρ = θ/κ while w _ρ:ℤ^d→ℝ⁺ is given by w _ρ = (v _ρ)^{⊗ d} with v _ρ: ℤ→ℝ⁺ the unique centered ground state (i.e., the solution in ℓ²(ℤ) with minimal l ²-norm) of the 1-dimensional nonlinear equation Δv + 2ρv log v = 0. The uniqueness of the ground state is actually proved only for large ρ, but is conjectured to hold for any ρ∈ (0, ∞). empty It turns out that if the right tail of the law of ξ(0) is thicker (or thinner) than the double exponential, then the correlation coefficient of u(x, t) and u(y, t) converges to δ _{x, y} (resp.the constant function 1). Thus, the double exponential family is the critical class exhibiting a nondegenerate correlation structure. Received: 5 March 1997 / Revised version: 21 September 1998     

Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Limiting process for integral functionals of a wiener process on a cylinder

We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge to the processw ₁(τ(t)), τ(t) = β₁ t + (β₂ − β₁)mes {s:w ₂(s)≥0,s<t}, wherew ₁(t andw ₂(t) are independent one-dimensional Wiener processes, β₁ and β₂ are nonrandom values, and β₂≥β₁≥0. Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 765–768, June, 1994.     

Asymptotic normality and efficiency of a weighted correlogram

For a process X(t)=Σ _j=1 ^M g _j (t)ξ _j (), where g_j(t) are nonrandom given functions, is a stationary vector-valued Gaussian process, Eξ_k(t) = 0, and Eξ_k(0) Eξ_l(τ) = r _kl(τ), we construct an estimate for the functions r _kl(τ) on the basis of observations X(t), t ∈ [0, T]. We establish conditions for the asymptotic normality of as T → ∞. We consider the problem of the optimal choice of parameters of the estimate depending on observations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 937–947, July, 1998.     

Transient phenomena for random walks in the absence of the expected value of jumps

Let ξ, ξ₁, ξ₂, ... be independent identically distributed random variables, and S _n :=Σ _j=1 ⁿ ,ξ _j , $ \bar S $ \bar S := sup _n≥0 S _n . If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $ \bar S $ \bar S as a → 0 and $ \bar S $ \bar S tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:

STABILITY AND ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

Itiswellknownthattheexistenceofalmostperiodicsolutionsiscloselyrelatedtothestabilityofsolutions.Forfunctionaldifferentialequationswithinfinitedelay,Y.Hin.[5'6]studiedtheproblemsontheexistenceofalmostperiodicsolutionsandthestability.However,therearefewpapersll2]dealingwithneutralfunctionaldifferentialequationswithinfinitedelay.Inthepresentpaper,forneutralfunctionaldifferentialequationswithinfinitedelay,weprovetheinherencetheoremfortheuniformlystableoperatorD(t),definethestabilitywithrespecttot…     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573     

Moderate Deviations for Random Sums of Heavy-Tailed Random Variables

Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E（X1） and let {N（t）; t ≥0} be a random process taking non-negative integer values with finite mean λ（t） = E（N（t）） and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P（（X1 ＋… ＋XN（t）） -λ（t）μ 〉 x） uniformly for x ∈[γb（t）, ∞） are obtained, where γ〉 0 and b（t） can be taken to be a positive function with limt→∞ b（t）/λ（t） = 0.     

A fragmentation process connected to Brownian motion

Let (B _s , s≥ 0) be a standard Brownian motion and T ₁ its first passage time at level 1. For every t≥ 0, we consider ladder time set ℒ ^(t) of the Brownian motion with drift t, B ^(t) _s = B _s + ts, and the decreasing sequence F(t) = (F ₁(t), F ₂(t), …) of lengths of the intervals of the random partition of [0, T ₁] induced by ℒ ^(t) . The main result of this work is that (F(t), t≥ 0) is a fragmentation process, in the sense that for 0 ≤t < t′, F(t′) is obtained from F(t) by breaking randomly into pieces each component of F(t) according to a law that only depends on the length of this component, and independently of the others. We identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by Aldous and Pitman [3]. Received: 19 February 1999 / Revised version: 17 September 1999 /?Published online: 31 May 2000     

Probabilities of high excursions of Gaussian fields

Let {ξ(t), t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E ξ(t) ≡ 0, D ξ(t) ≡ 1, and continuous trajectories defined on the m-dimensional interval T ì \mathbbR^m T \subset {\mathbb{R}^m} . The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{−v(t) < ξ(t) < u(t), t ∈ T}, when, for all t ∈ T, u(t), v(t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions of empirical processes and fields, Sov. Math., Dokl., 45(1):226–228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e^−Q  + Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R(t) = cov(ξ′(t), ξ′(t)).     

The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity