Expansion Theorems for Generalized Random Processes, Wick Products and Applications to Stochastic Differential Equations

Pilipović, Stevan; Seleši, Dora

doi:10.1142/s0219025707002634

Cited by 36 publications

(42 citation statements)

References 7 publications

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“…The most developed alternative approach is based on Hida's white noise analysis [7]. The white noise approach exploits the built-in set of stochastic spaces, such as Hida or Kondratiev spaces [11,12], or even larger exponential spaces [16]. The traditional approach [17,20,21], etc., has to select a stochastic space and then to study the largest possible class of equations admitting a solution in that space.…”

Section: Assumption a The Expectation Of The Highest Order (Differenmentioning

confidence: 99%

Elliptic equations of higher stochastic order

Lototsky¹,

Rozovskiĭ²,

Wan³

2010

ESAIM: M2AN

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Abstract. This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hp finite element approximation. The results of related numerical experiments are presented.Mathematics Subject Classification. 35R60, 65L60, 60H15, 60H35.

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Section: Assumption a The Expectation Of The Highest Order (Differenmentioning

confidence: 99%

Elliptic equations of higher stochastic order

Lototsky¹,

Rozovskiĭ²,

Wan³

2010

ESAIM: M2AN

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“…Especially, WS * is the dual of W 0 S = W 1,2 0 (I ) ⊗ (S) 1 . We state our main result from [11], which was used also in [12]: Theorem 1 Following conditions are equivalent:…”

Section: Preliminariesmentioning

confidence: 99%

“…. n are generalized random processes defined as in [11] and L is a linear elliptic operator of the form…”

Section: Introductionmentioning

confidence: 99%

On the Generalized Stochastic Dirichlet Problem—Part I: The Stochastic Weak Maximum Principle

Pilipović

Seleši

2009

Potential Anal

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We treat the stochastic Dirichlet problem L♦u = h + ∇ f in the framework of white noise analysis combined with Sobolev space methods. The input data and the boundary condition are generalized stochastic processes regarded as linear continuous mappings from the Sobolev space W 1,2 0 into the Kondratiev space (S) −1 . The operator L is assumed to be strictly elliptic in divergence form L♦u = ∇(A♦∇u + b ♦u) + c♦∇u + d♦u. Its coefficients: the elements of the matrix A and of the vectors b , c and d are assumed to be generalized random processes, and the product of two generalized processes, denoted by ♦, is interpreted as the Wick product. In this paper we prove the weak maximum principle for the operator L, which will imply the uniqueness of the solution to L♦u = h + ∇ f .

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“…Especially, one does not need to take care that the PDE-solution belongs to the domain of the inverse Hermite transform. We will also use the series expansion method as we did in [12,13] (also known as the propagator method in [10]) to obtain the coefficients in the chaos expansion of the solution.…”

Section: Introductionmentioning

confidence: 99%

On the Generalized Stochastic Dirichlet Problem—Part II: Solvability, Stability and the Colombeau Case

Pilipović

Seleši

2009

Potential Anal

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In this paper we consider the stochastic Dirichlet problem L♦u = h + ∇ f in the framework of white noise analysis combined with Sobolev space and Colombeau algebra methods. The operator L is assumed to be strictly elliptic inIts coefficients: the elements of the matrix A and of the vectors b , c and d are assumed to be generalized random processes, and the product of two generalized processes is interpreted as the Wick product. Generalized random processes are considered as linear bounded mappings from the Sobolev space W 1,2 0 into the Kondratiev space (S) −1 . In this paper we prove existence and uniqueness of the problem of this form in the case when the operator L generates a coercive bilinear form, and then extend this result to the general case. We also consider the case when the coefficients of L, the input data and the boundary condition are Colombeau-type generalized stochastic processes.

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Expansion Theorems for Generalized Random Processes, Wick Products and Applications to Stochastic Differential Equations

Cited by 36 publications

References 7 publications

Elliptic equations of higher stochastic order

Elliptic equations of higher stochastic order

On the Generalized Stochastic Dirichlet Problem—Part I: The Stochastic Weak Maximum Principle

On the Generalized Stochastic Dirichlet Problem—Part II: Solvability, Stability and the Colombeau Case

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