On L_p-theory of stochastic partial differential equations in the whole spaces

It is shown that equations like du = (a^iju_xixj + bⁱu_xi + cu + f) dt + (σ^iku_xi + ν^ku + g^k) dw^k_t, t > 0, with variable random coefficients and with zero initial condition have unique solutions in the Sobolev spaces W²_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 ∈ W¹_p. A corresponding result in the spaces of Bessel potentials Hⁿ_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.