On stationarity of stochastic retarded linear equations with unbounded drift operators

This article continues the study of Liu [Statist. Probab. Lett. 78(2008): 1775–1783; Stoch. Anal. Appl. 29(2011): 799–823] for stationary solutions of stochastic linear retarded functional differential equations with the emphasis on delays which appear in those terms including spatial partial derivatives. As a consequence, the associated stochastic equations have unbounded operators acting on the point or distributed delayed terms, while the operator acting on the instantaneous term generates a strongly continuous semigroup. We present conditions on the delay systems to obtain a unique stationary solution by combining spectrum analysis of unbounded operators and stochastic calculus. A few instructive cases are analyzed in detail to clarify the underlying complexity in the study of systems with unbounded delayed operators.     

Nonexistence of Global Solutions to Nonlinear Stochastic Wave Equations in Mean L p -Norm

This article is concerned with explosive solutions of the initial-boundary problem for a class of nonlinear stochastic wave equations in a domain 𝒟 ? ? ^d . Under appropriate conditions on the initial data, the nonlinear term and the noise intensity, it is proved in Theorem 3.4 that there cannot exist a global solution and the local solution will blow up at a finite time in the mean L ^p  ? norm for p ≥ 1. An example is given to show the application of this theorem.     

Orthogonalpolynome in x und q ?x als L?sungen von reellen q -Operatorgleichungen zweiter Ordnung

 The explicit form and the three term recurrance relation for all the polynomial solutions in x and q ^−x from q-operator equations of second order are given. The known 17 q-orthogonal polynomial systems are special cases of 7 comprehensive q-systems.     

Orthogonalpolynome in x und q −x als Lösungen von reellen q-Operatorgleichungen zweiter Ordnung

 The explicit form and the three term recurrance relation for all the polynomial solutions in x and q ^−x from q-operator equations of second order are given. The known 17 q-orthogonal polynomial systems are special cases of 7 comprehensive q-systems. (Eingegangen 27. M?rz 2000; in revidierter Fassung 2. November 2000)     

Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+|x| ^p ) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O(1+|x| ^p ) at infinity. This latter case encompasses some equations related to backward stochastic differential equations.     

An Impact of Stochastic Dynamic Boundary Conditions on the Evolution of the Cahn-Hilliard System

Abstract

Nonlinear systems are often subject to random influences. Sometimes the noise enters the system through physical boundaries and this leads to stochastic dynamic boundary conditions. A dynamic, as opposed to static, boundary condition involves the time derivative as well as spatial derivatives for the system state variables on the boundary. Although stochastic static (Neumann or Dirichet type) boundary conditions have been applied for stochastic partial differential equations, not much is known about the dynamical impact of stochastic dynamic boundary conditions. The purpose of this article is to study possible impacts of stochastic dynamic boundary conditions on the long term dynamics of the Cahn-Hilliard equation arising in the materials science. We show that the dimension estimation of the random attractor increases as the coefficient for the dynamic term in the stochastic dynamic boundary condition decreases. However, the dimension of the random attractor is not affected by the corresponding stochastic static boundary condition.     

A class on non-local linear operators for vorticity waves

A steady longitudinal current in the nearshore can, in some conditions, support oscillations known as vorticity waves or shear waves. In this article, we consider a family of nonlinear evolution equations derived by Shrira and Voronovitch to describe the dynamics of vorticity waves near the coastal line and make the study of the dispersion and smoothing properties of the associated nonlocal free problems. More precisely, after establishing long and short time uniform estimates for a certain class of oscillatory integrals, we derive “L ^p ?L ^q ” and Strichartz-type estimates for the solutions of the linearized equations.     

A Support Theorem for a Generalized Burgers SPDE

When the initial condition u ₀ to a parabolic Burgers SPDE (containing a quadratic term) belongs to L ^q [0,1],2q, the trajectories of the solution u(t,x) a.s. belong to the space C([0,T],L ^q [0,1]). We characterize the support of the law of u in this space; the proof is based on an approximation of u by a sequence of stochastic processes obtained by replacing the Brownian sheet by linear adapted interpolations.     

Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity

This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the L^p−L^q time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials.     

Lp – Lq time decay estimate to the solution of the Cauchy problem for the system of equations describing nonlocal model of the thermoviscoelastic body

The aim of this paper is to present a new system of equations describing nonlocal model of thermoviscoelastic theory. We used the Papkin and Gurtin approach based on the constitutive relations for stress tensor σ(x), internal energy e(x) and heat flux q(x), with integral terms. Using the modified Cagniard-de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three-dimensional space. Basing on this matrix we represent in the explicit formula the solution of the Cauchy problem to this system of equations. Next, applying the method of Sobolev spaces, we proved the L^p–L^q time decay estimate to the solution of the Cauchy problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Measure diffusions and related explosion problems

For random measure-valued stochastic partial differential equations for biological processes, growth represented by scalar partial differential equations at each point of the support and spread being a diffusion on R ^d, solutions are constructed by smearing the growth processes at each spatial point and composing the resulting generator with the generator for the spread. If these solutions are unique the equation is called solvable. We find conditions for the noise term of a solvable equations to have trivial effect and we identify some non-solvable equations, for example the diffusion-free bilinear equation. The search led to an investigation of explosion and the effect of point barriers for scalar stochastic differential equations with linear drift; this is used to explain the clustering effect in the usual superprocess.     

Galois Theory of Difference Equations with Periodic Parameters

We develop a Galois theory for systems of linear difference equations with periodic parameters, for which we also introduce linear difference algebraic groups. We apply this to constructively test if solutions of linear q-difference equations, with q ∈ ?* and q not a root of unity, satisfy any polynomial ζ-difference equations with ζ ^t  = 1, t ≥ 1.     

Projection of diffusions on submanifolds: Application to mean force computation

We consider the problem of sampling a Boltzmann‐Gibbs probability distribution when this distribution is restricted (in some suitable sense) on a submanifold Σ of ?ⁿ implicitly defined by N constraints q₁(x) = ? = q_N(x) = 0 (N < n). This problem arises, for example, in systems subject to hard constraints or in the context of free energy calculations. We prove that the constrained stochastic differential equations (i.e., diffusions) proposed in [7, 13] are ergodic with respect to this restricted distribution. We also construct numerical schemes for the integration of the constrained diffusions. Finally, we show how these schemes can be used to compute the gradient of the free energy associated with the constraints. © 2007 Wiley Periodicals, Inc.     

Lp-Lq time decay estimate to the solution of the Cauchy problem of the system of equations for nonlocal model of hyperbolic thermoelasticity theory

The aim of this paper is to present a new system of equations describing nonlocal model of hyperbolic thermoelasticity theory. We used the Papkin and Gurtin approach based on the constitutive relations for internal energy e(x), and heat flux q(x), with integral terms. Such system of equations describes the propagation of thermal perturbation with finite velocity. Using the modified Cagniard–de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three–dimensional space. Basing on the constructed matrix of fundamental solutions in the explicit formula we represent the solution of the Cauchy problem to this system of equations in the form of some kind of convolutions. Next, applying the method of Sobolev spaces, we obtain the L^p−L^q time decay estimate to the solution of the Cauchy problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some Norm Inequalities for Gaussian Wick Products

We provide several inequalities for the ? ^q (𝒫)-norm of the Wick product of random variables. These estimates are based on a Jensen's type inequality for the Wick multiplication, which we derive via a positivity argument. As an application we study a certain type of anticipating stochastic differential equation whose solution is shown to be an element of ? ^q (𝒫) for some q ≥ 1.     

Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:

We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the L^p–L^q estimates of fundamental solutions for linear damped wave equations [K. Nishihara, L^p–L^q estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The L^p–L^q estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and L^p–L^q estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, L^p–L^q estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].     

Sur les fonctions q-Bessel de Jackson

(About Jackson q-Bessel functions)Laplace transform allows to resolve differential equations in the neighborhood of an irregular singular point. The purpose of the article is to study how to apply a basic Borel–Laplace transformation to q-difference equations satisfied by the q-Bessel functions of F.H. Jackson. Connection matrices are obtained between solutions at the origin and solutions at infinity.     

Stabilization by noise for a class of stochastic reaction-diffusion equations

We prove uniqueness, ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction-diffusion equations with multiplicative noise, in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable. To this purpose, we show that the L¹-norm of the difference of two solutions starting from any two different initial data converges ℙ-a.s. to zero, as time goes to infinity.This paper was written while the author was visiting the Scuola Normale Superiore, Pisa     

An infinite series of regular edge‐ but not vertex‐transitive graphs

Let n be an integer and q be a prime power. Then for any 3 ≤ n ≤ q?1, or n=2 and q odd, we construct a connected q‐regular edge‐but not vertex‐transitive graph of order 2qⁿ⁺¹. This graph is defined via a system of equations over the finite field of q elements. For n=2 and q=3, our graph is isomorphic to the Gray graph. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 249–258, 2002     

A viability theorem of stochastic semilinear evolution equations

A viability theorem of stochastic semilinear evolution equations is discussed under a dissipative condition in terms of uniqueness functions and a stochastic subtangential condition. Our strategy is to interpret a stochastic viability problem into a characterization problem of evolution operators associated with stochastic semilinear evolution equations. The main theorem is a generalization of the results due to Aubin and Da Prato in the case of stochastic differential equations in ℝ ^d .