## Abstract

It is shown that equations like du = (a^{ij}u_{xixj} + b^{i}u_{xi} + cu + f) dt + (σ^{ik}u_{xi} + ν^{k}u + g^{k}) dw^{k}_{t}, t > 0, with variable random coefficients and with zero initial condition have unique solutions in the Sobolev spaces W^{2}_{p}, p ∈ [2, ∞), under natural ellipticity condition and under conditions that (i) a is uniformly continuous with respect to x, (ii) σ, ν have bounded first derivatives in x and all other coefficients are bounded, (iii) f ∈ L_{p}, g ∈ W^{1}_{p}. A corresponding result in the spaces of Bessel potentials H^{n}_{p} is proved, which implies that better differentiability properties of the coefficients and free terms of the equations lead to the better regularity of solutions. Applications to equations with space-time white noise are given.

Original language | English (US) |
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Pages (from-to) | 313-340 |

Number of pages | 28 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1996 |

## Keywords

- Bessel potentials
- Cylindrical white noise
- Stochastic equations