Discretisations of rough stochastic partial differential equations

Doctoral thesis English OPEN
Matetski, Kanstantsin
  • Subject: QA

This thesis consists of two parts, in both of which we consider approximations of rough stochastic PDEs and investigate convergence properties of the approximate solutions. In the first part we use the theory of (controlled) rough paths to define a solution for one-dimensional stochastic PDEs of Burgers type driven by an additive space-time white noise. We prove that natural numerical approximations of these equations converge to the solution of a corrected continuous equation and that their optimal convergence rate in the uniform topology (in probability) is arbitrarily close to 1/2 . In the second part of the thesis we develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical �43 model on the dyadic grid converges after renormalisation to its continuous counterpart. This result in particular implies that, as expected, the �43 measure is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition
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    C. REUTENAUER. Free Lie algebras, vol. 7 of London Mathematical Society Monographs. New Series. The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications.

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