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.

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 ON UNIQUENESS OF MILD SOLUTIONS FOR DISSIPATIVE STOCHASTIC EVOLUTION EQUATIONS

2010 ◽  
Vol 13 (03) ◽  
pp. 363-376 ◽  
Author(s):  
CARLO MARINELLI ◽  
MICHAEL RÖCKNER
Keyword(s):  
Evolution Equations ◽  
Broad Class ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Strong Solutions ◽  
Fundamental Issue ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Monotonicity Assumption ◽  
Semigroup Approach

In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.

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 Existence and approximate controllability of Sobolev type fractional stochastic evolution equations

2014 ◽  
Vol 62 (2) ◽  
pp. 205-215 ◽  
Author(s):  
N.I. Mahmudov
Keyword(s):  
Linear System ◽  
Hilbert Spaces ◽  
Approximate Controllability ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Sobolev Type ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Approximately Controllable

Abstract We study the existence of mild solutions and the approximate controllability concept for Sobolev type fractional semilinear stochastic evolution equations in Hilbert spaces. We prove existence of a mild solution and give sufficient conditions for the approximate controllability. In particular, we prove that the fractional linear stochastic system is approximately controllable in [0, b] if and only if the corresponding deterministic fractional linear system is approximately controllable in every [s, b], 0 ≤ s < b. An example is provided to illustrate the application of the obtained results.

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