Nonlinear Stochastic Partial Differential Equations
01 Jan 2014-
TL;DR: In this article, a continuous real Gaussian random variable with mean 0 and covariance t s is defined, such that for any 0 s < t, the random variables, B(t) B(s) are independent.
Abstract: Brownian motion A real Brownian motion B 1⁄4 (B(t))t 0 on (O, F , P) is a continuous real stochastic process in [0, +1) such that (1) B(0)1⁄4 0 and for any 0 s< t, B(t) B(s) is a real Gaussian random variable with mean 0 and covariance t s, and (2) if 0< t1< < tn, the random variables, B(t1), B(t2) B(t1), . . ., B(tn) B(tn 1) are independent. Continuous stochastic process Let (O, F , P) be a probability space. A continuous stochastic process (with values in H) is a family of (H-valued) random variables (X(t) 1⁄4 X(t, o))t 0 (o2O) such that X( , o) is continuous for P – almost all o2O. Cylindrical Wiener process A cylindrical Wiener process in a Hilbert space H is a process of the form
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01 Feb 1988
TL;DR: In this paper, a family of probability spaces Ωℱ,Pγ, γ < 0 associated with the Euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow φt (time evolution) leavingPγ globally invariant.
Abstract: We construct a family of probability spacesΩℱ,Pγ), γ<0 associated with the Euler equation for a two dimensional inviscid incompressible fluid which carries a pointwise flow φt (time evolution) leavingPγ globally invariant. φt is obtained as the limit of Galerkin approximations associated with Euler equations.Pγ is also in invariant measure for a stochastic process associated with a Navier-Stokes equation with viscosity, γ, stochastically perturbed by a white noise force.
112 citations
TL;DR: Kim and Xu as mentioned in this paper proved the strongly convexity of a real Banach space to a zero of accretive operators in a uniformly smooth space, where the modifieations of the Mann iterations satisfied some conditions.
Abstract: Let X be a real Banach space, we will introduce a modifieations of the Mann iterations in a uniformly smooth Banach, Where satisfied some conditions, then we will prove the strongly space, ((1))(1)xuxJx {}, 1nnnnrn n to a zero of accretive operators. This theorem extend (Kim and Xu, Nonlinear convergence of the sequence {}x n Analysis) results. (Kim, 2005, pp.51-60)
10 citations
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..., Bensoussan et al. (1978), Freidlin (1996)....
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Book•
10 Sep 1993
TL;DR: In this article, the authors give bounds on the number of degrees of freedom and the dimension of attractors of some physical systems, including inertial manifolds and slow manifolds.
Abstract: Contents: General results and concepts on invariant sets and attractors.- Elements of functional analysis.- Attractors of the dissipative evolution equation of the first order in time: reaction-diffusion equations.- Fluid mechanics and pattern formation equations.- Attractors of dissipative wave equations.- Lyapunov exponents and dimensions of attractors.- Explicit bounds on the number of degrees of freedom and the dimension of attractors of some physical systems.- Non-well-posed problems, unstable manifolds. lyapunov functions, and lower bounds on dimensions.- The cone and squeezing properties.- Inertial manifolds.- New chapters: Inertial manifolds and slow manifolds the nonselfadjoint case.
5,038 citations
Book•
01 Jan 1978TL;DR: In this article, the authors give a systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate.
Abstract: This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.
4,869 citations
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01 Dec 1992TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
Abstract: Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.
4,042 citations