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SPDEs in \(L_q(\mskip-2mu(0,\tau], L_p)\) spaces. (English) Zbl 0963.60053

A linear second-order stochastic parabolic equation \[ du = \Biggl(\sum^{d}_{i,j=1} a^{ij}\frac{\partial^2 u} {\partial x_{i}\partial x_{i}} + f\Biggr) dt + \sum^\infty _{k=1}\Biggl(\sum^{d}_{i=1} \sigma^{ik}\frac{\partial u} {\partial x_{i}} + g^{k}\Biggr) dw^{k}_{t}, \tag{1} \] where \(t\geq 0\) and either \(x\in\mathbb R^{d}\) or \(x\in \mathbb R^{d}_{+} = \{(x_1,\widetilde x)\); \(x_1>0\), \(\widetilde x\in\mathbb R^{d-1}\}\), is studied. It is supposed that \(w^{k}\), \(k\geq 1\), are independent one-dimensional Wiener processes, \(f\), \(g^{k}\) are some functions of \((\omega,t,x)\), and \(a^{ij}\), \(\sigma^{ik}\) are predictable processes depending only on \((\omega,t)\), \(a^{ij} = a^{ji}\). The equation (1) is assumed to be parabolic in the following sense: for some \(\delta\in]0,1[\), \(\delta^{-1} |\lambda|^2\geq\sum_{ij} a^{ij}\lambda_{i}\lambda_{j} \geq \delta |\lambda|^2\) and \((1-\delta)\sum_{ij} a^{ij}\lambda_{i}\lambda_{j}\geq \frac 12\sum_{ij}\sum_{k} \sigma^{ik}\sigma^{jk}\lambda_{i}\lambda_{j}\) for all \(\omega\), \(t\) and \(\lambda\in\mathbb R^{d}\).
Recently, the author has developed in a series of papers a deep Sobolev space theory of (1), proving existence and uniqueness of solutions to (1) in suitable weighted Sobolev spaces of fractional or negative order [cf. e.g. the author and S. V. Lototsky, SIAM J. Math. Anal. 31, No. 1, 19-33 (1999; Zbl 0943.60047)]. In the paper under review these results are extended by allowing different powers of summability with respect to space and time variables.

MSC:

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

Citations:

Zbl 0943.60047
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