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Taylor expansions of solutions of stochastic partial differential equations with additive noise. (English) Zbl 1220.35202

The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an Itó formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of an SODE using an iterated application of the Itó formula. Consequently, until recently, only low order numerical approximation results for such an SPDE have been available. Here, the fact that the solution of an SPDE driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the SPDE provides an alternative approach for deriving stochastic Taylor expansions for the solution of such an SPDE. Essentially, the exponential factor has a mollifying effect and ensures that all integrals take values in the Hilbert space under consideration. The iteration of such integrals allows us to derive stochastic Taylor expansions of arbitrarily high order, which are robust in the sense that they also hold for other types of driving noise processes such as fractional Brownian motion. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.

MSC:

35R60 PDEs with randomness, stochastic partial differential equations
35K90 Abstract parabolic equations
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems

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References:

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