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Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE

Adam Andersson (Institutionen för matematiska vetenskaper, matematik) ; Raphael Kruse ; Stig Larsson (Institutionen för matematiska vetenskaper, matematik)
Stochastic Partial Differential Equations: Analysis and Computations (2194-0401). Vol. 4 (2016), 1, p. 113-149.
[Artikel, refereegranskad vetenskaplig]

We introduce a new family of refined Sobolev–Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.

Nyckelord: SPDE, Finite element method, Backward Euler, Weak convergence, Convergence of moments, Malliavin calculus, Duality, Spatio-temporal discretization



Denna post skapades 2015-10-29. Senast ändrad 2017-11-24.
CPL Pubid: 225023

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)

Ämnesområden

Numerisk analys
Matematisk statistik

Chalmers infrastruktur