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The goal of this paper is to establish the Lipschitz and W 2 , ∞ estimates for a second-order parabolic PDE ∂ t u (t , x) = 1 2 Δ u (t , x) + f (t , x) on R d with zero initial data and f satisfying a Ladyzhenskaya–Prodi–Serrin type condition. Following the theoretic result, we then give two applications. The first is to discuss the regularity of the stochastic heat equations, and the second is to discuss the Sobolev differentiability of strong solutions to a class of SDEs with singular drift coefficients. [ABSTRACT FROM AUTHOR]

We are going to study a kind of stochastic fractional partial differential equation driven by an impulsive noise, which is singular not only in time but also in space. We first study the existence and uniqueness of the solution and then investigate the regularity of the solution in its spatial variable which depends on the order of the fractional operator, and deeply relies on the precise analysis of the kernel generated by our operator. In addition, we also discuss the stochastic flow property of the solutions. [ABSTRACT FROM PUBLISHER]

We are interested in the control problem of a partially observable diffusion process, which is initialized at a fixed point of ℝ n , and we want to characterize the associated value function. To resort to the theory of viscosity solutions depends on the possibility to translate such a problem on Hilbert spaces like L 2 (ℝ n ), and so it can not be used here. Nevertheless, a result of N. Bouleau and F. Hirsch allows us to introduce a broadened problem which fulfills the condition. The fact remains to link these two control problems. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to transform a nonlinear stochastic partial differential equation of parabolic type with multiplicative noise into a random partial differential equation by using a bijective random process. A stationary conjugation is constructed, which is of interest for asymptotic problems. The conjugation is used here to prove the existence of the stochastic flow, the perfect cocycle property and the existence of the random attractor, all nontrivial properties in the case of multiplicative noise. [ABSTRACT FROM AUTHOR]