Rate of Convergence of Space Time Approximations for stochastic evolution equations

Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition assumed for the solution, the rate of convergence of various numerical approximations are estimated under strong monotonicity and Lipschitz conditions. The abstract setting involves general consistency conditions and is then applied to a class of quasilinear stochastic PDEs of parabolic type.Comment: 33 page

Rate of Convergence of Implicit Approximations for stochastic evolution equations

Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators are considered. Under some regularity condition assumed for the solution, the rate of convergence of implicit Euler approximations is estimated under strong monotonicity and Lipschitz conditions. The results are applied to a class of quasilinear stochastic PDEs of parabolic type.Comment: 25 page

Approximating rough stochastic PDEs

We study approximations to a class of vector-valued equations of Burgers type driven by a multiplicative space-time white noise. A solution theory for this class of equations has been developed recently in [Hairer, Weber, Probab. Theory Related Fields, to appear]. The key idea was to use the theory of controlled rough paths to give definitions of weak / mild solutions and to set up a Picard iteration argument. In this article the limiting behaviour of a rather large class of (spatial) approximations to these equations is studied. These approximations are shown to converge and convergence rates are given, but the limit may depend on the particular choice of approximation. This effect is a spatial analogue to the It\^o-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.Comment: 80 pages; Corrects a mistake in the proof of Lemma 3.

Time--space white noise eliminates global solutions in reaction diffusion equations

We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.Comment: New results included. To be published in Physica

SPDE in Hilbert Space with Locally Monotone Coefficients

In this paper we prove the existence and uniqueness of strong solutions for SPDE in Hilbert space with locally monotone coefficients, which is a generalization of the classical result of Krylov and Rozovskii for monotone coefficients. Our main result can be applied to different types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic $p$-Laplace equation and stochastic porous media equation with some non-monotone perturbations.Comment: 20 pages, add Remark 3.1 for stochastic Burgers equatio