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.


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


Zbl 0943.60047
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