Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus

A novel approach is developed to deal with the weak error estimates of temporal semi-discretisation of semi-linear stochastic partial differential equations (SPDEs) by the exponential Euler method. A weak error representation formula is first derived for the exponential integrator scheme in the context of truncated SPDEs. The formula enjoys the absence of the irregular term involved with the unbounded operator. As an application, the obtained formula is then applied to the exponential Euler scheme for SPDEs of parabolic type. Under certain mild assumptions on the nonlinearity, we provide an easy weak error analysis, which does not rely on the Malliavin calculus. AMS subject classification: 60H35, 60H15, 65C30.

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