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      How to determine the law of the noise driving a SPDE

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          Abstract

          We consider a stochastic partial differential equation (SPDE) on a lattice \partial_t X=(\Delta-m^2)X-\lambda X^p+\eta where \(\eta\) is a space-time L\'evy noise. A perturbative (in the sense of formal power series) strong solution is given by a tree expansion, whereas the correlation functions of the solution are given by a perturbative expansion with coefficients that are represented as sums over a certain class of graphs, called Parisi-Wu graphs. The perturbative expansion of the truncated (connected) correlation functions is obtained via a Linked Cluster Theorem as a sums over connected graphs only. The moments of the stationary solution can be calculated as well. In all these solutions the cumulants of the single site distribution of the noise enter as multiplicative constants. To determine them, e.g. by comparison with a empirical correlation function, one can fit these constants (e.g. by the methods of least squares) and thereby one (approximately) determines law of the noise.

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          Feynman graph representation of the perturbation series for general functional measures

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            Author and article information

            Journal
            29 October 2006
            Article
            math/0610906
            3657363e-b0fa-4ea6-9d31-16769e038bfc
            History
            Custom metadata
            60H15; 60H35
            25 pages, 2 figures
            math.PR

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