arXiv subjects
Publications and source records attributed to Z. Brzezniak.
The paper is concerned with the properties of solutions to linear evolution equation perturbed by cylindrical Lévy processes. It turns out that solutions, under rather weak requirements, do not have càdlàg modification. Some natural open questions are also stated.
The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative space-dependent noise is considered for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere $\mathbb S^2$. The regularity of weak solutions is also discussed. Some of the regularity results are new even for the deterministic Landau-Lifshitz-Gilbert equation.
In this article we show that three dimensional vector advection equation is self dual in certain sense defined below. As a consequence, we infer classical result of Serrin of existence of strong solution of Navier-Stokes equation. Also we deduce Feynman-Kac type formula for solution of the vector advection equation and show that the formula is not unique i.e. there exist flows which differ from standard flow along which vorticity is conserved.
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.
The Navier--Stokes equation in the bidimensional torus is considered, with initial velocity and forcing term in suitable Besov spaces. Results of local existence and uniqueness are proven; under further restriction on the indexes defining the Besov speces involved, we prove global existence.
We consider second order differential operators with coefficients which are Gaussian random fields. When the covariance becomes singular at short distances then the propagators of the Schrödinger equation as well as of the wave equation behave in an anomalous way. In particular, the Feynman propagator for the wave equation is less singular than the one with deterministic coefficients. We suggest some applications to quantum gravity.