On the regularization effect of space-time white noise on quasi-linear parabolic partial differential equations

Summary We show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi-linear parabolic equation in one space dimension, even if the nonlinearity is only measurable and not even locally bounded.Research supported by the Hungarian National Foundation of Scientific Research No. 2290. Université de Provence (Aix-Marseille I), Mathématiques Case 64, Place Victor Hugo, 13331 Marseille, Cedex 3 (for the academic year 1991/92)Partially supported by DRET under contract 901636/A000/DS/SR     

White noise driven quasilinear SPDEs with reflection

Summary We study reflected solutions of the heat equation on the spatial interval [0, 1] with Dirichlet boundary conditions, driven by an additive space-time white noise. Roughly speaking, at any point (x, t) where the solutionu(x, t) is strictly positive it obeys the equation, and at a point (x, t) whereu(x, t) is zero we add a force in order to prevent it from becoming negative. This can be viewed as an extension both of one-dimensional SDEs reflected at 0, and of deterministic variational inequalities. An existence and uniqueness result is proved, which relies heavily on new results for a deterministic variational inequality.INRIAPartially supported by DRET under contract 901636/A000/DRET/DS/SR     

Discretization and simulation of stochastic differential equations

We discuss both pathwise and mean-square convergence of several approximation schemes to stochastic differential equations. We then estimate the corresponding speeds of convergence, the error being either the mean square error or the error induced by the approximation on the value of the expectation of a functional of the solution. We finally give and comment on a few comparative simulation results.     

Extinction time and the total mass of the continuous-state branching processes with competition

Consider a general continuous-state branching process with additional interaction, which destroys the branching property. We give precise conditions on the interaction term, in order to decide whether the extinction time of the process remains or not bounded as the initial value tends to infinity, and similarly for the total mass of the process.     

Malliavin calculus for the stochastic 2D Navier—Stokes equation

We consider the incompressible, two‐dimensional Navier‐Stokes equation with periodic boundary conditions under the effect of an additive, white‐in‐time, stochastic forcing. Under mild restrictions on the geometry of the scales forced, we show that any finite‐dimensional projection of the solution possesses a smooth, strictly positive density with respect to Lebesgue measure. In particular, our conditions are viscosity independent. We are mainly interested in forcing that excites a very small number of modes. All of the results rely on proving the nondegeneracy of the infinite‐dimensional Malliavin matrix. © 2006 Wiley Periodicals Inc.     

Generalized BSDEs and nonlinear Neumann boundary value problems

Summary. We study a new class of backward stochastic differential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial differential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations. Received: 27 September 1996 / In revised form: 1 December 1997     

Singular homogenization with stationary in time and periodic in space coefficients

We study the homogenization problem for a random parabolic operator with coefficients rapidly oscillating in both the space and time variables and with a large highly oscillating nonlinear potential, in a general stationary and mixing random media, which is periodic in space. It is shown that a solution of the corresponding Cauchy problem converges in law to a solution of a limit stochastic PDE.     

Intégrales Hilbertiennes anticipantes par rapport à un processus de Wiener cylindrique et calcul stochastique associé

We define the Skorohod integral of an operator-valued process with respect to a cylindrical Hilbertian Wiener process. We study the resulting process, and establish a generalized Itô formula. We define also a Stratonovitch integral, and establish the corresponding chain rule. Our work is inspired by the finite-dimensional results in [10].     

Malliavin Calculus for White Noise Driven Parabolic SPDEs

We consider the parabolic SPDE

Backward doubly stochastic differential equations and systems of quasilinear SPDEs

Summary We introduce a new class of backward stochastic differential equations, which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDE's.The research of this author was partially supported by DRET under contract 901636/A000/DRET/DS/SRThe research of this author was supported by a grant from the French Ministère de la Recherche et de la Technologie, which is gratefully acknowledged