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 .

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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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We introduce a notion of mild solution for a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset D ⊂ R d and driven by an infinite-dimensional fractional noise. We prove the existence of such a solution, establish its relation with the variational solution introduced by Nualart and Vuillermot (J Funct Anal 232: 2006) and the Hölder continuity of its sample paths when we consider it as an L 2 (D)-valued stochastic process. When h is an affine function, we also prove uniqueness. An immediate consequence of our results is the indistinguishability of mild and variational solutions in the case of uniqueness.

In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of non-autonomous linear parabolic initial-and final-boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two Itô equations for some suitably constructed Wiener processes, and from that analysis derive Feynman-Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a twodimensional disk.

In this article we investigate a class of non-autonomous, semilinear, parabolic systems of stochastic partial differential equations defined on a smooth, bounded domain O & R n and driven by an infinitedimensional noise defined from an L 2 ðOÞ-valued Wiener process; in the general case the noise can be colored relative to the space variable and white relative to the time variable. We first prove the existence and the uniqueness of a solution under very general hypotheses, and then establish the existence of invariant sets along with the validity of comparison principles under more restrictive conditions; the ORDER REPRINTS main ingredients in the proofs of these results consist of a new proposition concerning Wong-Zakaï approximations and of the adaptation of the theory of invariant sets developed for deterministic systems. We also illustrate our results by means of several examples such as certain stochastic systems of Lotka-Volterra and Landau-Ginzburg equations that fall naturally within the scope of our theory.