Stratonovich Solution for the Wave Equation

Raluca M. Balan ()
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Raluca M. Balan: University of Ottawa

Journal of Theoretical Probability, 2022, vol. 35, issue 4, 2643-2689

Abstract: Abstract In this article, we construct a Stratonovich solution for the stochastic wave equation in spatial dimension $$d \le 2$$ d ≤ 2 , with time-independent noise and linear term $$\sigma (u)=u$$ σ ( u ) = u multiplying the noise. The noise is spatially homogeneous and its spectral measure satisfies an integrability condition which is stronger than Dalang’s condition. We give a probabilistic representation for this solution, similar to the Feynman–Kac-type formula given in Dalang et al. (Trans Am Math Soc 360:4681–4703, 2008) for the solution of the stochastic wave equation with spatially homogeneous Gaussian noise, that is white in time. We also give the chaos expansion of the Stratonovich solution and we compare it with the chaos expansion of the Skorohod solution from Balan et al. (Exact asymptotics of the stochastic wave equation with time independent noise, 2020. arXiv:2007.10203 ).

Keywords: Stochastic wave equation; Stratonovich solution; Malliavin calculus; Primary 60H15; Secondary 60H07 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01144-y

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