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     

On isometries in GMV-algebras

Let A = (A,⊕,⁻,^∼, 0, 1) be a GMV-algebra and ρ: A × A → A the distance function on A defined by ρ(x, y) = (x∨y)−(x∧y) for each x, y ∈ A.     

On polynomial symbols for subdivision schemes

Given a dilation matrix A :ℤ^d→ℤ^d, and G a complete set of coset representatives of 2π(A ^−Tℤ^d/ℤ^d), we consider polynomial solutions M to the equation ∑ _g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order. Research partially supported by the Danish Technical Science Foundation, Grant No. 9701481, and by the Danish SNF-PDE network.     

On the partially microlocal hypoellipticity for a class of totally characteristic operators with characteristics of constant multiplicity

In this paper, I study the microlocal hypoellipticity for a class of totally characteristic operators (1.1). My main result is as follows: Under the conditions (I), (II), if the indicial operator of (1.1) is microlocally hypoelliptic in the complement ofWF _x(Pu(t,·)) for anyu(t,x)∈C _b ^∞ ([0,T], ℰ),t∈[0,T], λ∈ℤ, then the operator (1.1) is microlocally hypoelliptic in the variablex. Supported by the Natural Science Foundation and Young Men's Science Foundation of Academia Sinica     

Generalization of the perron effect whereby the characteristic exponents of all solutions of two differential systems change their sign from negative to positive

We obtain a generalization of the complete Perron effect whereby the characteristic exponents of all solutions change their sign from negative for the linear approximation system to positive for a nonlinear system with perturbations of higher-order smallness [Differ. Uravn., 2010, vol. 46, no. 10, pp. 1388–1402]. Namely, for arbitrary parameters λ ₁ ≤ λ ₂ < 0 and m > 1 and for arbitrary intervals [b _i , d _i ) ⊂ [λ _i ,+∞), i = 1, 2, with boundaries d ₁ ≤ b ₂, we prove the existence of (i) a two-dimensional linear differential system with bounded coefficient matrix A(t) infinitely differentiable on the half-line t ≥ 1 and with characteristic exponents λ ₁(A) = λ ₁ ≤ λ ₂(A) = λ ₂ < 0; (ii) a perturbation f(t, y) of smallness order m > 1 infinitely differentiable with respect to time t > 1 and continuously differentiable with respect to y ₁ and y ₂, y = (y ₁, y ₂) ∈ R ² such that all nontrivial solutions y(t, c), c ∈ R ², of the nonlinear system .y = A(t)y + f(t, y), y ∈ R ², t ≥ 1, are infinitely extendible to the right and have characteristic exponents λ[y] ∈ [b ₁, d ₁) for c ₂ = 0 and λ[y] ∈ [b ₂, d ₂) for c ₂ ≠ 0.     

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     

Some simple Haar-type wavelets in higher dimensions

An orthonormal wavelet system in ℝ^d, d ∈ ℕ, is a countable collection of functions {ψ _j,k ^ℓ }, j ∈ ℤ, k ∈ ℤ^d, ℓ = 1,..., L, of the form that is an orthonormal basis for L² (ℝ^d), where a ∈ GL_d (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L = d = 1, ψ¹(x) = ψ(x) = κ[0,1/2)^(x) − κ_[l/2,1) ^(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate products Φ(x₁, x₂, ..., x_d) = φ₁ (x₁)φ₂(x₂) ... φ_d(x_d) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems. For example, if a = ( _1-1 ^{1 1} ) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed in [7] and involve dilations by matrices that are products of the form a^jb, j ∈ ℤ, where a ∈ GL_d(ℝ) has some “expanding” property and b belongs to a group of matrices in GL_d(ℝ) having |det b| = 1.     

Aging in two-dimensional Bouchaud's model

Let E _x be a collection of i.i.d. exponential random variables. Symmetric Bouchaud's model on ℤ² is a Markov chain X(t) whose transition rates are given by w _xy = ν exp (−βE _x ) if x, y are neighbours in ℤ². We study the behaviour of two correlation functions: ℙ[X(t _w +t) = X(t _w )] and ℙ[X(t') = X(t _w ) ∀ t'∈ [t _w , t _w + t]]. We prove the (sub)aging behaviour of these functions when β > 1.     

Two-Point Boundary Value Problems for Duffing Equations across Resonance

In this paper, we consider the equation y″+f(x,y)=0 with a nonresonance condition of the form A≤f _y (x,y)≤B, where (k−1)²<A≤k ²<⋅⋅⋅<m ²≤B<(m+1)², k,m∈ℤ⁺. With optimal control theory and the Schauder fixed-point theorem, by introducing a new cost functional, we obtain a new existence and uniqueness result for the above equation with two-point boundary-value conditions. This work was supported by NSFC Grant 10501017 and 985 Project of Jilin University.     

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

On a class of stochastic partial differential equations related to turbulent transport

Summary. We consider the Cauchy problem for the mass density ρ of particles which diffuse in an incompressible fluid. The dynamical behaviour of ρ is modeled by a linear, uniformly parabolic differential equation containing a stochastic vector field. This vector field is interpreted as the velocity field of the fluid in a state of turbulence. Combining a contraction method with techniques from white noise analysis we prove an existence and uniqueness result for the solution ρ∈C ^1,2([0,T]×ℝ ^d ,(S)^*), which is a generalized random field. For a subclass of Cauchy problems we show that ρ actually is a classical random field, i.e. ρ(t,x) is an L ²-random variable for all time and space parameters (t,x)∈[0,T]×ℝ ^d . Received: 27 March 1995 / In revised form: 15 May 1997     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

Derivations with power central values on multilinear polynomials in prime rings

Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x ₁, ..., x _n ) a nonzero multilinear polynomial over C. Suppose that x ^s d(x)x ^t ∈ Z(R) for all x ∈ {d(f(x ₁, ..., x _n ))|x ₁, ..., x _n ∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ C = eRC for some idempotent e in the socle of RC and f(x ₁, ..., x _n ) ^N is central-valued in eRCe, where N = s + t + 1.      

Mixing properties of the generalized T, T-1-process

Consider a general random walk on ℤ^d together with an i.i.d. random coloring of ℤ^d. TheT, T ^-1-process is the one where time is indexed by ℤ, and at each unit of time we see the step taken by the walk together with the color of the newly arrived at location. S. Kalikow proved that ifd = 1 and the random walk is simple, then this process is not Bernoulli. We generalize his result by proving that it is not Bernoulli ind = 2, Bernoulli but not Weak Bernoulli ind = 3 and 4, and Weak Bernoulli ind ≥ 5. These properties are related to the intersection behavior of the past and the future of simple random walk. We obtain similar results for general random walks on ℤ^d, leading to an almost complete classification. For example, ind = 1, if a step of sizex has probability proportional to l/|x|^α (x ⊋ 0), then theT, T ^-1-process is not Bernoulli when α ≥2, Bernoulli but not Weak Bernoulli when 3/2 ≤α < 2, and Weak Bernoulli when 1 < α < 3/2. Research partially carried out while a guest of the Department of Mathematics, Chalmers University of Technology, Sweden in January 1996. Research supported by grants from the Swedish Natural Science Research Council and from the Royal Swedish Academy of Sciences.     

An Extremal Problem on Degree Sequences of Graphs

 Let G=(I _n ,E) be the graph of the n-dimensional cube. Namely, I _n ={0,1} ⁿ and [x,y]∈E whenever ||x−y||₁=1. For A⊆I _n and x∈A define h _A (x) =#{y∈I _n A|[x,y]∈E}, i.e., the number of vertices adjacent to x outside of A. Talagrand, following Margulis, proves that for every set A⊆I _n of size 2 ⁿ⁻¹ we have for a universal constant K independent of n. We prove a related lower bound for graphs: Let G=(V,E) be a graph with . Then , where d(x) is the degree of x. Equality occurs for the clique on k vertices. Received: January 7, 2000 RID="*" ID="*" Supported in part by BSF and by the Israeli academy of sciences     

On generalized derivations satisfying certain identities

Let R be a prime ring with char R ≠ 2 and let d be a generalized derivation on R. We study the generalized derivation d satisfying any of the following identities:

Regev’s and Amitsur’s conjectures for codimensions of generalized polynomial identities

Let A be a finite-dimensional associative algebra over a field of characteristic 0. Then there exist C ∈ ℚ₊ and t ∈ ℤ₊ such that gc _n (A) ∼ Cn ^t d ⁿ as n → ∞, where d = PIexp(A). In particular, Amitsur’s and Regev’s conjectures hold for the codimensions gc _n (A) of generalized polynomial identities.     

On the intersection of infinite geometric and arithmetic progressions

We prove that the intersection G ∩ A of an infinite geometric progression G = u, uq, uq ², uq ³, ..., where u > 0 and q > 1 are real numbers, and an infinite arithmetic progression A contains at most 3 elements except for two kinds of ratios q. The first exception occurs for q = r ^1/d , where r > 1 is a rational number and d ∈ ℕ. Then this intersection can be of any cardinality s ∈ ℕ or infinite. The other (possible) exception may occur for q = β ^1/d , where β > 1 is a real cubic algebraic number with two nonreal conjugates of moduli distinct from β and d ∈ ℕ. In this (cubic) case, we prove that the intersection G ∩ A contains at most 6 elements.