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Maximal inequalities and space-time regularity of stochastic convolutions

Szymon Peszat, Jan Seidler (1998)

Mathematica Bohemica

Space-time regularity of stochastic convolution integrals J = 0 S(-r)Z(r)W(r) driven by a cylindrical Wiener process W in an L 2 -space on a bounded domain is investigated. The semigroup S is supposed to be given by the Green function of a 2 m -th order parabolic boundary value problem, and Z is a multiplication operator. Under fairly general assumptions, J is proved to be Holder continuous in time and space. The method yields maximal inequalities for stochastic convolutions in the space of continuous...

Maximum principle for forward-backward doubly stochastic control systems and applications

Liangquan Zhang, Yufeng Shi (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example...

Maximum principle for forward-backward doubly stochastic control systems and applications*

Liangquan Zhang, Yufeng Shi (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an...

Multiscale Finite Element approach for “weakly” random problems and related issues

Claude Le Bris, Frédéric Legoll, Florian Thomines (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We address multiscale elliptic problems with random coefficients that are a perturbation of multiscale deterministic problems. Our approach consists in taking benefit of the perturbative context to suitably modify the classical Finite Element basis into a deterministic multiscale Finite Element basis. The latter essentially shares the same approximation properties as a multiscale Finite Element basis directly generated on the random problem. The specific reference method that we use is the Multiscale...

Multiscale stochastic homogenization of convection-diffusion equations

Nils Svanstedt (2008)

Applications of Mathematics

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form u ε ω / t + 1 / ϵ 3 𝒞 T 3 ( x / ε 3 ) ω 3 · u ε ω - div α T 1 ( x / ε 1 ) ω 1 , T 2 ( x / ε 2 ) ω 2 , t u ε ω = f . It is shown, under certain structure assumptions on the random vector field 𝒞 ( ω 3 ) and the random map α ( ω 1 , ω 2 , t ) , that the sequence { u ϵ ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem u / t - div ( ( t ) u ) = f .

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