2011
DOI: 10.1080/07362994.2011.548998
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The Stochastic Dirichlet Problem Driven by the Ornstein–Uhlenbeck Operator: Approach by the Fredholm Alternative for Chaos Expansions

Abstract: In this article, we prove the Fredholm alternative theorem for mappings defined on spaces of generalized stochastic processes given by their Wiener-Itô chaos expansion form. We apply the result to solve the stochastic Dirichlet problem with a perturbation term driven by the Ornstein-Uhlenbeck operator.

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Cited by 15 publications
(20 citation statements)
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“…We generalize the definition of the Wick product of random variables to the set of generalized stochastic processes in the way as it is done in [10,21,22,27].…”
Section: Generalized Processesmentioning
confidence: 99%
See 1 more Smart Citation
“…We generalize the definition of the Wick product of random variables to the set of generalized stochastic processes in the way as it is done in [10,21,22,27].…”
Section: Generalized Processesmentioning
confidence: 99%
“…In [1,3,14,15,19,24] the Malliavin derivative and the Skorokhod integral are defined on a subspace of ðLÞ 2 so that the resulting process after application of these operators always remains in ðLÞ 2 . In [9,10,12] we allowed values in ðSÞ 21 and thus obtained a larger domain for all operators.…”
Section: Operators Of the Malliavin Calculusmentioning
confidence: 99%
“…We generalize the definition of the Wick product of random variables to the set of generalized stochastic processes in the way as it is done in [20,40] and [41]. From now on we assume that X is closed under multiplication, i.e.…”
Section: Multiplication Of Stochastic Processesmentioning
confidence: 99%
“…We will recall of these classical results and denote the corresponding domains with a "zero" in order to retain a nice symmetry between test and generalized processes. In [19,20,23,24] we allowed values in (S) −1 and thus obtained larger domains for all operators. These domains will be denoted by a "minus" sign to reflect the fact that they correspond to generalized processes.…”
Section: Operators Of the Malliavin Calculusmentioning
confidence: 99%
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