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Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process

Zdzisław Brzeźniak, Szymon Peszat (1999)

Studia Mathematica

Stochastic partial differential equations on ℝ d are considered. The noise is supposed to be a spatially homogeneous Wiener process. Using the theory of stochastic integration in Banach spaces we show the existence of a Markovian solution in a certain weighted L q -space. Then we obtain the existence of a space continuous solution by means of the Da Prato, Kwapień and Zabczyk factorization identity for stochastic convolutions.

Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

Abdellah Chkifa, Albert Cohen, Ronald DeVore, Christoph Schwab (2013)

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

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations...

Sparse finite element methods for operator equations with stochastic data

Tobias von Petersdorff, Christoph Schwab (2006)

Applications of Mathematics

Let A V → V ' be a strongly elliptic operator on a d -dimensional manifold D (polyhedra or boundaries of polyhedra are also allowed). An operator equation A u = f with stochastic data f is considered. The goal of the computation is the mean field and higher moments ℳ 1 u ∈ V , ℳ 2 u ∈ V ⊗ V , ... , ℳ k u ∈ V ⊗ ⋯ ⊗ V of the solution. We discretize the mean field problem using a FEM with hierarchical basis and N degrees of freedom. We present a Monte-Carlo algorithm and a deterministic algorithm for the approximation of the moment ℳ k u for k ≥ 1 . The key tool...

Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains

Petru A. Cioica, Stephan Dahlke, Stefan Kinzel, Felix Lindner, Thorsten Raasch, Klaus Ritter, René L. Schilling (2011)

Studia Mathematica

We use the scale of Besov spaces B τ , τ α ( ) , 1/τ = α/d + 1/p, α > 0, p fixed, to study the spatial regularity of solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains ⊂ ℝ. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

SPDEs with coloured noise: Analytic and stochastic approaches

Marco Ferrante, Marta Sanz-Solé (2006)

ESAIM: Probability and Statistics

We study strictly parabolic stochastic partial differential equations on ℝ d , d ≥ 1, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weak formulation of the equation are related, deriving...

SPDEs with pseudodifferential generators: the existence of a density

Samy Tindel (2000)

Applicationes Mathematicae

We consider the equation du(t,x)=Lu(t,x)+b(u(t,x))dtdx+σ(u(t,x))dW(t,x) where t belongs to a real interval [0,T], x belongs to an open (not necessarily bounded) domain 𝒪 , and L is a pseudodifferential operator. We show that under sufficient smoothness and nondegeneracy conditions on L, the law of the solution u(t,x) at a fixed point ( t , x ) ∈ [ 0 , T ] × 𝒪 is absolutely continuous with respect to the Lebesgue measure.

Spectral statistics for random Schrödinger operators in the localized regime

François Germinet, Frédéric Klopp (2014)

Journal of the European Mathematical Society

We study various statistics related to the eigenvalues and eigenfunctions of random Hamiltonians in the localized regime. Consider a random Hamiltonian at an energy E in the localized phase. Assume the density of states function is not too flat near E . Restrict it to some large cube Λ . Consider now I Λ , a small energy interval centered at E that asymptotically contains infintely many eigenvalues when the volume of the cube Λ grows to infinity. We prove that, with probability one in the large volume...

Speed of the Brownian loop on a manifold

Rémi Léandre (2006)

Banach Center Publications

We define the speed of the curved Brownian bridge as a white noise distribution operating on stochastic Chen integrals.

Splitting the conservation process into creation and annihilation parts

Nicolas Privault (1998)

Banach Center Publications

The aim of this paper is the study of a non-commutative decomposition of the conservation process in quantum stochastic calculus. The probabilistic interpretation of this decomposition uses time changes, in contrast to the spatial shifts used in the interpretation of the creation and annihilation operators on Fock space.

Stability of solutions of BSDEs with random terminal time

Sandrine Toldo (2006)

ESAIM: Probability and Statistics

In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time. More precisely, we are going to show that if (Wn) is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion W and if ( τ n ) is a sequence of stopping times that converges to a stopping time τ, then the solution of the BSDE driven by Wn with random terminal time τ n converges to the solution...

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