Hairer, Martin M; Maas, JanIST Austria ; Weber, Hendrik
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 Probability Theory Related Fields by Hairer and Weber. 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 behavior 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.
Communications on Pure and Applied Mathematics
JM is supported by Rubicon grant 680-50-0901 of the Netherlands Organisation for Scientific Research (NWO). MH is supported by EPSRC grant EP/D071593/1 and by the Royal Society through a Wolfson Research Merit Award. Both MH and HW are supported by the Le
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Hairer M, Maas J, Weber H. Approximating Rough Stochastic PDEs. Communications on Pure and Applied Mathematics. 2014;67(5):776-870. doi:10.1002/cpa.21495
Hairer, M., Maas, J., & Weber, H. (2014). Approximating Rough Stochastic PDEs. Communications on Pure and Applied Mathematics. Wiley-Blackwell. https://doi.org/10.1002/cpa.21495
Hairer, Martin, Jan Maas, and Hendrik Weber. “Approximating Rough Stochastic PDEs.” Communications on Pure and Applied Mathematics. Wiley-Blackwell, 2014. https://doi.org/10.1002/cpa.21495.
M. Hairer, J. Maas, and H. Weber, “Approximating Rough Stochastic PDEs,” Communications on Pure and Applied Mathematics, vol. 67, no. 5. Wiley-Blackwell, pp. 776–870, 2014.
Hairer M, Maas J, Weber H. 2014. Approximating Rough Stochastic PDEs. Communications on Pure and Applied Mathematics. 67(5), 776–870.
Hairer, Martin, et al. “Approximating Rough Stochastic PDEs.” Communications on Pure and Applied Mathematics, vol. 67, no. 5, Wiley-Blackwell, 2014, pp. 776–870, doi:10.1002/cpa.21495.