Zdzisław Brzeźniak

On stochastic convolution in banach spaces and applications

Stochastic evolution equations are studied in M-type 2 Banach spaces framework. Using factorization method and Burkholder inequality we prove regularity properties of stochastic convolution processes. We prove also existence of local and global solutions with close to optimal regularity. We show that solution with cylindrical Wiener process can be approximated by solutions with finite dimensional Wiener processes. Application to reaction diffusion equations are presented

Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation

Using the theory of stochastic integration for processes with values in a UMD Banach space developed recently by the authors, an Ito formula is proved which is applied to prove the existence of strong solutions for a class of stochastic evolution equations in UMD Banach spaces. The abstract results are applied to prove regularity in space and time of the solutions of the Zakai equation.

Regularity of Ornstein–Uhlenbeck Processes Driven by a Lévy White Noise

The paper is concerned with spatial and time regularity of solutions to linear stochastic evolution equation perturbed by Lévy white noise “obtained by subordination of a Gaussian white noise”. Sufficient conditions for spatial continuity are derived. It is also shown that solutions do not have in general cádlág modifications. General results are applied to equations with fractional Laplacian. Applications to Burgers stochastic equations are considered as well.

Weak Solutions to Stochastic Wave Equations with Values in Riemannian Manifolds

Let M be a compact Riemannian manifold. We prove existence of a global weak solution of the stochastic wave equation D t u t = D x u x + (X u + λ0(u)u t + λ1(u)u x )Ẇ where X is a continuous vector field on M, λ0 and λ1 are continuous vector bundles homomorphisms from TM to TM, and W is a spatially homogeneous Wiener process on ℝ with finite spectral measure. We use recently introduced general method of constructing weak solutions of SPDEs that does not rely on any martingale representation theorem.

Approximations of the Wong–Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces

Introduction 1 Stochastic Integration in M-type 2 Banach Spaces 2 Approximations of SDEs with Lipschitz and bounded coefficients 3 Approximation of SDEs whose coefficients are locally Lipschitz 4 Applications to diffusion processes on loop spaces 4.1 Diffusion processes on loop manifolds 4.2 An approximation result for solutions to SDEs on M 5 Applications to stochastic flows Appendix References

Horizontal Lift of an Infinite Dimensional Diffusion

We prove existence of the horizontal lift to a line bundle of certain diffusion processes on some infinite-dimensional manifolds. We provide three classes of finite-dimensional manifolds for which the corresponding loop spaces have a line bundle and thus provide three classes of loop manifolds on which certain diffusion processes admit a horizontal lift. Applications to Quantum Field Theory are indicated.

Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces

Let M be a compact Riemannian homogeneous space (e.g., a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation Dt∂tu=∑dk=1Dxk∂xku+fu(Du)+gu(Du)W˙ in any dimension d≥1, where f and g are continuous multilinear maps, and W is a spatially homogeneous Wiener process on Rd with finite spectral measure. A nonstandard method of constructing weak solutions of SPDEs, that does not rely on martingale representation theorem, is employed.

Computational Studies for the Stochastic Landau--Lifshitz--Gilbert Equation

The stochastic Landau--Lifshitz--Gilbert equation describes the thermally induced dynamics of magnetic moments in ferromagnetic materials. Solutions of this highly nonlinear stochastic PDE are unit vector fields and satisfy an energy estimate. These are crucial properties to construct a convergent discretization in space and time. We propose a convergent finite element approximation of the problem based on the midpoint rule. The numerical scheme preserves the underlying properties of the continuous problem. Further, we construct a robust and efficient Newton-multigrid solver for the solution of the nonlinear systems associated with the discretized problems at each time level. Computational studies show the optimal convergence behavior of the scheme in the case of smooth solutions. Long-time dynamics for finite ensembles of spins evidence the ergodicity of an invariant measure of the continuum model. Numerical experiments in two dimensions demonstrate pathwise finite time blow-up behavior of the solution w...

Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity

The strong existence and the pathwise uniqueness of solutions with