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A note on stochastic wave equations. (English) Zbl 0978.60066
Lumer, Günter (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].

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