scholarly journals Invariant manifolds for weak solutions to stochastic equations

2000 ◽  
Vol 118 (3) ◽  
pp. 323-341 ◽  
Author(s):  
Damir Filipović
Keyword(s):  
Weak Solutions ◽  
Invariant Manifolds ◽  
Stochastic Equations

On weak solutions of stochastic equations in Hilbert spaces

1994 ◽  
Vol 46 (1-2) ◽  
pp. 41-51 ◽  
Author(s):  
Dariusz Gątarek ◽  
Beniamin Gołdys
Keyword(s):  
Hilbert Spaces ◽  
Weak Solutions ◽  
Stochastic Equations

scholarly journals Attractors for the Stochastic 3D Navier–Stokes Equations

2003 ◽  
Vol 03 (03) ◽  
pp. 279-297 ◽  
Author(s):  
Pedro Marín-Rubio ◽  
James C. Robinson
Keyword(s):  
White Noise ◽  
Global Attractor ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Stochastic Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Additive White Noise ◽  
Solutions Of Equations

In a 1997 paper, Ball defined a generalised semiflow as a means to consider the solutions of equations without (or not known to possess) the property of uniqueness. In particular he used this to show that the 3D Navier–Stokes equations have a global attractor provided that all weak solutions are continuous from (0, ∞) into L2. In this paper we adapt his framework to treat stochastic equations: we introduce a notion of a stochastic generalised semiflow, and then show a similar result to Ball's concerning the attractor of the stochastic 3D Navier–Stokes equations with additive white noise.

On Multidimensional SDEs Without Drift and with A Time-Dependent Diffusion Matrix

2000 ◽  
Vol 7 (4) ◽  
pp. 643-664 ◽  
Author(s):  
H. J. Engelbert ◽  
V. P. Kurenok
Keyword(s):  
Stochastic Process ◽  
Weak Solutions ◽  
Sufficient Conditions ◽  
Stochastic Equations ◽  
Diffusion Matrix ◽  
Homogeneous Case ◽  
Initial State ◽  
Existence Of Weak Solutions ◽  
One Dimensional ◽  
Arbitrary Initial State

Abstract We study multidimensional stochastic equations where x o is an arbitrary initial state, W is a d-dimensional Wiener process and is a measurable diffusion coefficient. We give sufficient conditions for the existence of weak solutions. Our main result generalizes some results obtained by A. Rozkosz and L. Słomiński [Stochastics Stochasties Rep. 42: 199–208, 1993] and T. Senf [Stochastics Stochastics Rep. 43: 199–220, 1993] for the existence of weak solutions of one-dimensional stochastic equations and also some results by A. Rozkosz and L. Słomiński [Stochastic Process. Appl. 37: 187–197, 1991], [Stochastic Process. Appl. 68: 285–302, 1997] for multidimensional equations. Finally, we also discuss the homogeneous case.

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