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__Weak convergence in second moments for linear SPDEs__

R. Kruse, A. Andersson and S. Larsson

**Abstract**

In numerical analysis of stochastic differential equations (SDEs) one usually differentiates between the so called strong and weak convergence. While the first notion ensures a good pathwise approximation of the SDE, a weakly convergent scheme only gives a good approximation of the law of the exact solution. Strong convergence implies weak convergence and, by a rule of thumb, the order of weak convergence is up to twice the order of strong convergence.
In this talk we confirm this rule for an Euler Galerkin finite element approximation of a linear stochastic PDE and a certain class of test functions, which ensures the so called convergence in second moments. The main feature of our analysis is that we avoid the classical ansatz relying on the associated Kolmogorov's backward equation. Instead we follow a more direct approach with Gronwall's lemma.
If time permits we indicate several possible ways to generalize this approach to semilinear SPDEs and a richer class of test functions.