Remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay

In this paper, we obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space BC((-∞,0];R^d) which denotes the family of bounded continuous R^d-value functions defined on (-∞,0] with norm under non-Lipschitz condition with Lipschitz condition being considered as a special case and a weakened linear growth condition. The solution is constructed by the successive approximation.     

Linear average and stochastic n-widths of Besov embeddings on Lipschitz domains

In this paper, we determine the asymptotic degree of the linear average and stochastic n-widths of the compact embeddings where is a Besov space defined on the bounded Lipschitz domain .     

Almost sure and moment Lyapunov exponents for a stochastic beam equation

The transverse vibrations of an Euler-Bernoulli beam with axial tension P and axial white noise forcing are given by

     

Dynamics of a stochastic Lotka-Volterra model perturbed by white noise

This paper continues the study of Mao et al. investigating two aspects of the equation

     

Stochastic delay Lotka-Volterra model

We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka-Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model into the Itô form , and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A.     

Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the -valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the -valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.     

Strong solutions to stochastic wave equations with values in Riemannian manifolds

Let M be a d-dimensional compact Riemannian manifold. We prove existence of a unique global strong solution of the stochastic wave equation , where Y is a C¹-smooth transformation and W is a spatially homogeneous Wiener process on whose spectral measure has finite moments up to order 2.     

Stochastic differential delay equations of population dynamics

In this paper we stochastically perturb the delay Lotka-Volterra model

     

The solution manifold and C-smoothness for differential equations with state-dependent delay

Let h>0, open, and continuously differentiable. If f satisfies two mild additional smoothness conditions then the set

     

Barycentric selectors and a Steiner-type point of a convex body in a Banach space

Let F be a mapping from a metric space into the family of all m-dimensional affine subsets of a Banach space X. We present a Helly-type criterion for the existence of a Lipschitz selection f of the set-valued mapping F, i.e., a Lipschitz continuous mapping satisfying . The proof of the main result is based on an inductive geometrical construction which reduces the problem to the existence of a Lipschitz (with respect to the Hausdorff distance) selector S_X^(m) defined on the family of all convex compacts in X of dimension at most m. If X is a Hilbert space, then the classical Steiner point of a convex body provides such a selector, but in the non-Hilbert case there is no known way of constructing such a point. We prove the existence of a Lipschitz continuous selector for an arbitrary Banach space X. The proof is based on a new result about Lipschitz properties of the center of mass of a convex set.     

Existence and uniqueness of solutions of semilinear stochastic infinite-dimensional differential systems with H-regular noise

Existence and uniqueness of approximate strong solutions of stochastic infinite-dimensional systems

     

Existence of positive solutions of halflinear delay differential equation

In this paper we consider the halflinear delay differential equation