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International Journal for Uncertainty Quantification

Impact factor: 1.000

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2011003485
pages 195-213

DIFFERENTIAL CONSTRAINTS FOR THE PROBABILITY DENSITY FUNCTION OF STOCHASTIC SOLUTIONS TO THE WAVE EQUATION

Daniele Venturi
Division of Applied Mathematics, Brown University, Providence, Rhode Island 09212, USA
George Em Karniadakis
Division of Applied Mathematics Brown University Providence, RI 02912, USA

ABSTRACT

By using functional integral methods we determine new types of differential constraints satisfied by the joint probability density function of stochastic solutions to the wave equation subject to uncertain boundary and initial conditions. These differential constraints involve unusual limit partial differential operators and, in general, they can be grouped into two main classes: the first one depends on the specific field equation under consideration (i.e., on the stochastic wave equation), the second class includes a set of intrinsic relations determined by the structure of the joint probability density function of the wave and its derivatives. Preliminary results we have obtained for stochastic dynamical systems and first-order nonlinear stochastic particle differential equations (PDEs) suggest that the set of differential constraints is complete and, therefore, it allows determining uniquely the probability density function of the solution to the stochastic problem. The proposed new approach can be extended to arbitrary nonlinear stochastic PDEs and it could be an effective way to overcome the curse of dimensionality for random boundary and initial conditions. An application of the theory developed is presented and discussed for a simple random wave in one spatial dimension.