Stochastic partial differential equations arise as mathematical models of complex multiscale systems under random influences. Invariant manifolds often provide geometric structure for understanding stochastic dynamics. In this paper, a random invariant manifold reduction principle is proved for a class of stochastic partial differential equations. The dynamical behavior is shown to be described...

One general method of Lyapunov functionals construction which was used earlier both for stochastic differential equations with aftereffect and for stochastic difference equations with discrete time here is applied for stochastic difference equations with continuous time. © 2004 IMACS. Published by Elsevier B.V. All rights reserved.

Stochastic differential equations (SDE's) play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability. A general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here. High-order numerical methods are developed for the integration of stochastic differential equations with st...

In this paper stochastic Volterra equations admitting exponentially bounded resolvents are studied. After obtaining convergence of resolvents, some properties for stochastic convolutions are studied. Our main results provide sufficient conditions for strong solutions to stochastic Volterra equations.

Invariant sets of dynamic systems play an important role in many situations when the dynamic behavior is constrained in some way. Knowing that a set in the state space of a system is invariant means that we have bounds on the behavior. We can verify that pre-specified bounds which originate from, for example, safety restrictions, physical constraints, or state-feedback magnitude bounds are not ...

Using the formalism of stochastic embedding developed by [J. Cresson, D. Darses, J. Math. Phys. 48, 072703 (2007)], we study how dynamics classical Newton equation for a force deriving from potential is deformed under assumption that this can admit processes as solutions. We focus on two denitions Newton's called dierential and variational. rst prove virial theorem which natural generalization ...

When solving a partial dierential equation on a MIMD distributed memory computer there arises the problem of distributing the mesh so that the communication requirement of the numerical solver is minimised subject to a load balancing constraint. In this paper we introduce a new algorithm for this partitioning problem for unstructured meshes and study its properties, along with those of other co...