Abstract
We treat several classes of hyperbolic stochastic partial differential equations in the framework of white noise analysis, combined with Wiener–Itô chaos expansions and Fourier integral operator methods. The input data, boundary conditions and coefficients of the operators are assumed to be generalized stochastic processes that have both temporal and spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate Sobolev–Kondratiev or weighted-Sobolev–Kondratiev spaces. Moreover, an explicit chaos form of the solutions is obtained, useful for calculating expectations, variances and higher order moments of the solution.
Highlights
Hyperbolic stochastic partial differential equations arise as models of various phenomena used in mathematical physics, economy, molecular biology and many other areas of science, where random fluctuations and uncertainties are incorporated into the equation by white noise or other singular generalized stochastic processes such as Poissonian processes or general Lévy processes
In this paper we will present techniques for solving singular hyperbolic stochastic partial differential equations resulting from the synergy of these, nowadays classical, two powerful methods: chaos expansions and Fourier integral operators
The main idea we present in this paper relies on the chaos expansion method: first, one uses the chaos expansion of all stochastic data in the equation to convert the SPDE into an infinite system of deterministic PDEs, the PDEs are recursively solved, and one must sum up these solutions to obtain the chaos expansion form of the solution of the initial SPDE
Summary
Hyperbolic stochastic partial differential equations arise as models of various phenomena used in mathematical physics, economy, molecular biology and many other areas of science, where random fluctuations and uncertainties are incorporated into the equation by white noise or other singular generalized stochastic processes such as Poissonian processes or general Lévy processes. In this paper we will consider two types of hyperbolic problems for suitable differential operators, acting by the Wick product instead of classical multiplication This is due to the fact that we allow random terms to be present both in the initial conditions and right-hand side of the equations, as well as in the coefficients of the involved operators. In this paper we will present techniques for solving singular hyperbolic stochastic partial differential equations resulting from the synergy of these, nowadays classical, two powerful methods: chaos expansions and Fourier integral operators. To perform our analysis in this case, we will again use chaos expansions, but this time in connection with the properties of a class of Fourier integral operators, defined through objects globally defined on Rd. Hyperbolic SPDEs via Wiener chaos expansion methods have been studied in [21], but our approach is more powerful and allows more singular input data in the model. We mention that random-field solutions of hyperbolic SPDEs via Fourier integral operator methods have been recently studied in [5,8], while function-valued solutions for associated semilinear hyperbolic SPDEs have been obtained in [7]
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