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A note on stochastic wave equations. (English) Zbl 0978.60066
Lumer, G√ľnter (ed.) et al., Evolution equations and their applications in physical and life sciences. Proceeding of the Bad Herrenalb (Karlsruhe) conference, Germany, 1999. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 215, 501-511 (2001).
Let \(T = \mathbb R^{d}/2\pi\mathbb Z^{d}\) be a \(d\)-dimensional torus, and \(W_\Gamma\) a spatially homogeneous Wiener process taking values in the space \(\mathcal D^\prime(T)\) of distributions on \(T\). That is, \[ \mathbf E(W_\Gamma(t),\varphi)(W_\Gamma(s),\psi) = (t\land s) (\Gamma,\varphi *\psi_{s}) \] for all \(s,t\geq 0\) and for all test functions \(\varphi,\psi\in\mathcal D(T)\), where \(\psi_{s}(\theta) := \psi(-\theta)\) and \(\Gamma\in\mathcal D^\prime(T)\) is a positive definite distribution. Consider a stochastic wave equation \[ {\partial^2 u\over \partial t^2}(t,\theta) = \Delta u(t,\theta) + {\partial W_\Gamma \over\partial t}(t,\theta), \;u(0,\theta) = {\partial u\over\partial t}(0,\theta) = 0, \quad t>0, \;\theta\in T. \tag{1} \] Necessary and sufficient conditions on \(\Gamma\) for (1) to have a function-valued mild solution are found. More precisely: let \(H^\alpha\), \(\alpha\in\mathbb R\), be the standard scale of Sobolev spaces on \(T\) and denote by \((\gamma_{n})\) the Fourier coefficients of \(\Gamma\), \(\Gamma = \sum_{n\in\mathbb Z^{d}} \gamma_{n} e^{i(n,\cdot)}\) in \(\mathcal D^\prime(T)\). Then (1) has an \(H^{-\alpha+1}\)-valued solution if and only if \(\sum_{n\in\mathbb Z^{d}} (1+|n|^2)^{-\alpha}\gamma_{n} <\infty\).
For the entire collection see [Zbl 0957.00037].

60H15 Stochastic partial differential equations (aspects of stochastic analysis)