scholarly journals Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps

2008 ◽  
Vol 118 (5) ◽  
pp. 864-895 ◽  
Author(s):  
Jiaowan Luo ◽  
Kai Liu
Keyword(s):  
Evolution Equations ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Infinite Dimensional ◽  
With Memory

P-th moment growth bounds of infinite-dimensional stochastic evolution equations

1998 ◽  
Vol 128 (1) ◽  
pp. 107-121 ◽  
Author(s):  
Kai Liu
Keyword(s):  
Evolution Equations ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Infinite Dimensional Space ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Infinite Dimensional ◽  
Sample Paths ◽  
Logarithmic Growth

The aim of this paper is to investigate the p-th moment growth bounds wilh a general rate function λ(t) of the strong solution for a class of stochastic differential equations in infinite dimensional space under various sufficient hypotheses. The results derived here extend the usual situations to some extent, containing for example the polynomial or iterated logarithmic growth cases studied by many authors. In particular, more generalised sufficient conditions, ensuring the p-th moment upper-bound of sample paths given by solutions of a class of nonlinear stochastic evolution equations, are captured. Applications to parabolic itô equations are also considered.

INTEGRATION BY PARTS AND SMOOTHNESS OF THE LAW FOR A CLASS OF STOCHASTIC EVOLUTION EQUATIONS

2004 ◽  
Vol 07 (01) ◽  
pp. 89-129 ◽  
Author(s):  
STEFANO BONACCORSI ◽  
MARCO FUHRMAN
Keyword(s):  
Additive Noise ◽  
Evolution Equations ◽  
Transition Probabilities ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Integration By Parts ◽  
Infinite Dimensional ◽  
Integration By Parts Formula ◽  
Derivatives Of ◽  
Logarithmic Derivatives

We consider a Markov process X in a Hilbert space H, solution of a semilinear stochastic evolution equation driven by an infinite-dimensional Wiener process, occurring in the equation as an additive noise. Using techniques of the Malliavin calculus, under suitable assumptions, we prove an integration by parts formula for the transition probabilities νt, t>0 (the laws of Xt). We deduce results on differentiability (i.e. existence of logarithmic derivatives) of νt along a set of directions h∈H which can be described in terms of the coefficients of the equation. The general results are then applied to various classes of non linear stochastic partial differential equations and systems.

scholarly journals Stability with a general rate function for a class of stochastic evolution equations in infinite dimensional spaces

1997 ◽  
Vol 63 (1) ◽  
pp. 128-144 ◽  
Author(s):  
Kai Liu
Keyword(s):  
Evolution Equations ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Polynomial Stability ◽  
Almost Sure Stability ◽  
Infinite Dimensional ◽  
Liapunov Functions ◽  
Special Cases

AbstractThe aim of this paper is to investigate the almost sure stability with a certain rate function λ(t) for a class of stochastic evolution equations in infinite dimensional spaces under various sufficient conditions. The results obtained here include exponential and polynomial stability as special cases. Much more refined sufficient conditions than the usual ones, for example, those described in [14], are obtained under our framework by the method of Liapunov functions. Two examples are given to illustrate our theory.

scholarly journals Random motion of strings and related stochastic evolution equations

1983 ◽  
Vol 89 ◽  
pp. 129-193 ◽  
Author(s):  
Tadahisa Funaki
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Basic Equation ◽  
Evolution Equations ◽  
Random Motion ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Infinite Dimensional ◽  
The Third ◽  
Elastic String

In this paper, we shall investigate the random motion of an elastic string by using the theory of infinite dimensional stochastic differential equations. The paper consists of three main parts and appendices. In the first part (§2), we shall derive a basic equation which describes the random motion of a string. Several properties of this equation will be investigated in § 3, 4 and 5. In the third part (§ 6), we shall deal with a stochastic differential equation on a Hilbert space as a generalization of the equation of the string.

Sign in / Sign up

Export Citation Format

Share Document