Approximating rough stochastic PDEs
Provides a rigorous analysis of approximation-dependent limits for a class of rough stochastic PDEs, relevant to researchers in stochastic analysis and SPDEs.
The paper studies spatial approximations to a class of Burgers-type stochastic PDEs driven by multiplicative space-time white noise, proving convergence with rates but showing the limit depends on the approximation scheme, analogous to the Itô-Stratonovich correction.
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 Itô-Stratonovich correction in the theory of stochastic ordinary differential equations, where it is well known that different approximation schemes may converge to different solutions.