Almost automorphic solutions to integral equations on the line

Given a∈L ¹(ℝ) and A the generator of an L ¹-integrable family of bounded and linear operators defined on a Banach space X, we prove the existence of almost automorphic solution to the semilinear integral equation u(t)=∫ _−∞ ^t a(t−s)[Au(s)+f(s,u(s))]ds for each f:ℝ×X→X almost automorphic in t, uniformly in x∈X, and satisfying diverse Lipschitz type conditions. In the scalar case, we prove that a∈L ¹(ℝ) positive, nonincreasing and log-convex is already sufficient.     

Multivalued nonlinear equations on the half line: A fixed point approach

New fixed point theorems of the authors are used to establish the existence of one (or more) C[0, ∞) solutions to the nonlinear integral inclusion y(t)∈ ∫₀^∞ K(t, s) F(s, y(s))ds fort ∈ [0,∞).     

Moduli of continuity for <Emphasis Type="Italic">l</Emphasis><Superscript>∞</Superscript>-valued Gaussian random fields

This paper establishes the general moduli of continuity for l ^∞-valued Gaussian random fields {X(t):= (X ₁(t),X ₂(t), h.), t ∈ [0, ∞) ^N } indexed by the N-dimensional parameter t:= (t ₁,…,t _N ), under the explicit condition yielding that the covariance function of distinct increments of X _k (t) for fixed k ≧ 1 is positive or nonpositive. Supported by KOSEF-R01-2008-000-11418-0.     

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     

The Quasi-Canonical Solution Operator to $${\bar \partial }$$ Restricted to the Fock-Space

We consider the solution operator S: ℱ_μ,(p,q) → L ²(μ)_{(p, q)} to the -operator restricted to forms with coefficients in ℱ_μ = {f: f is entire and ∫_ℂn |f(z)|² dμ(z) < ∞}. Here ℱ_μ,(p,q) denotes (p,q)-forms with coefficients in ℱ_μ, L ²(μ) is the corresponding L ²-space and μ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula S to . This solution operator will have the property Sv ⊥ ℱ_(p,q) ∀v ∈ ℱ_(p,q+1). As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators : ℱ_μ → L ²(μ).     

An Application of a Mountain Pass Theorem

We are concerned with the following Dirichlet problem: −Δu(x) = f(x, u), x∈Ω, u∈H ¹ ₀(Ω), (P) where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L ^∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0, 0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR) is no longer true, where F(x, s) = ∫ ^s ₀ f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming (AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞. Received June 24, 1998, Accepted January 14, 2000.     

Logarithmic Estimates for Submartingales and Their Differential Subordinates

In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then

Hölder continuity for spatial and path processes via spectral analysis

For ν(dθ), a σ-finite Borel measure on R ^d , we consider L ²(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ ^t ₀ e ^{−λ(θ)( t − s )} dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ ^t ₀ g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process _t (τ)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space

In this article we study the exponential behavior of the continuous stochastic Anderson model, i.e. the solution of the stochastic partial differential equation u(t,x)=1+∫₀^tκΔ_xu (s,x) ds+∫₀^t W(ds,x) u (s,x), when the spatial parameter x is continuous, specifically x∈R, and W is a Gaussian field on R₊×R that is Brownian in time, but whose spatial distribution is widely unrestricted. We give a partial existence result of the Lyapunov exponent defined as lim_t→∞t⁻¹ log u(t,x). Furthermore, we find upper and lower bounds for lim sup_t→∞t⁻¹ log u(t,x) and lim inf_t→∞t⁻¹ log u(t,x) respectively, as functions of the diffusion constant κ which depend on the regularity of W in x. Our bounds are sharper, work for a wider range of regularity scales, and are significantly easier to prove than all previously known results. When the uniform modulus of continuity of the process W is in the logarithmic scale, our bounds are optimal. This author's research partially supported by NSF grant no. : 0204999     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997     

Admissibility of the natural estimator of the mean of a Gaussian process

Let {X(t): t [a, b]} be a Gaussian process with mean μ L₂[a, b] and continuous covariance K(s, t). When estimating μ under the loss ∫_a^b ( (t)−μ(t))² dt the natural estimator X is admissible if K is unknown. If K is known, X is minimax with risk ∫_a^b K(t, t) dt and admissible if and only if the three by three matrix whose entries are K(t_i, t_j) has a determinant which vanishes identically in t_i [a, b], i = 1, 2, 3.     

Saturation classes for a sequence of convolution operators with limited oscillation

Saturation classes for the sequenceK _n (f, x) = ∫f(x −t)dμ _n (t) of linear operators whereK _n(f, x) is of the limited oscillation type, that is,μ _n (t) is monotonic fort ≠ [−Aσ _n ,Aσ _n ],σ _n =o(1),n → ∞ and ∫t ^2m dμ _n (t), are obtained. Examples of applications to some sequences of non-positive operators are given.     

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K⊂ℝ ^d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K ^Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX _i (t) = ∑ _j a(j−i) (X _j (t) −X _i (t))dt + σ (X _i (t))dB _i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a _S (i) = a(i) + a(−i) is recurrent, then each component X _i (∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a _S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a _S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000     

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.     

HOMOCLINIC ORBITS FOR SECOND ORDER HAMILTONIAN SYSTEM WITH QUADRATIC GROWTH

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A Note on Edgeworth Expansions for <Emphasis Type="Italic">U</Emphasis>-Statistics under Minimal Conditions

A one-term Edgeworth expansion for U-statistics with kernel h(x, y) was derived by Jing and Wang [3] under optimal moment conditions. In this note, we show that one of the optimal moment conditions E| h(X ₁, X ₂|^5/3 < ∞ can be weakened to lim _t→∞ t ^5/3 P(|h(X ₁, X ₂)| > t) → 0. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 453–440, July–September, 2005.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

Sharp forms of nevanlinna’s error terms

Let f(z) be a meromorphic function in the plane. If ψ(t)/t andp(t) are two positive, continuous and non-decreasing functions on [1,∞) with ∫ ₁ ^∞ dt/ψ(t) = ∞ and ∫ ₁ ^∞ dt/p(t) = ∞, then asr → ∞ outside a small exceptional set, provided that the divergence of the integral ∫ ₁ ^r dt/ψ(t) is slow enough. The same forms for the logarithmic derivative and for the ramification term are obtained. It is shown by example that the estimates are best possible. Author supported by Max-Planck-Gesellschaft Z.F.D.W and by NSFC.     

On a class of nonlinear integral equations

In this paper we consider abstract equations of the typeK _ν ν +ν =w ₀, in a closed convex subset of a separable Hilbert spaceH. For eachv in the closed convex subset,K _v :H →H is a bounded linear map. As an application of our abstract result we obtain an existence result for nonlinear integral equations of the typeν(s)+ν(s)∫ ₀ ¹ k(s,t)ν(t)dt =W ₀(s) in the spaceL ² [0,1].