In this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset D ⊂ R d and driven by an infinite-dimensional multiplicative fractional noise. We introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an L 2 (D)-valued fractional Wiener process W H with Hurst parameter H ∈ 1 +1 , 1 , whose covariance operator satisfies appropriate integrability conditions, and where ∈ (0, 1] denotes the Hölder exponent of the derivative of the nonlinearity in the stochastic term of the equations. We also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field. Our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of Faedo-Galerkin approximations, while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to W H .
The planar and spatial lattice versions of the Maier-Saupe model (1959, 1960) for a nematic liquid crystal are exactly solved for a one-dimensional lattice, without periodic boundary conditions. The two-molecule correlation functions are studied, and it is shown that these two models do not exhibit an order-disorder phase transition except at T=0, as is the case for the classical Heisenberg model in one dimension.
In this article we prove new results concerning the long-time behavior of random ®elds that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random ®elds eventually homogeneize with respect to the spatial variable and ®nally converge to a non-random global attractor which consists of two spatially and temporally homogeneous asymptotic states. More precisely, we prove that the random ®elds either stabilize exponentially rapidly with probability one around one of the asymptotic states, or that they set out to oscillate between them. In the ®rst case we can also determine exactly the corresponding Lyapunov exponents. In the second case we prove that the random ®elds are in fact recurrent in that they can reach every point between the two asymptotic states in a ®nite time with probability one. In both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random ®elds spend in small neighborhoods of the asymptotic states. Our methods of proof
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