On generalized Malliavin calculus

AbstractThe Malliavin derivative, the divergence operator (Skorokhod integral), and the Ornstein–Uhlenbeck operator are extended from the traditional Gaussian setting to nonlinear generalized functionals of white noise. These extensions are related to the new developments in the theory of stochastic PDEs, in particular elliptic PDEs driven by spatial white noise and quantized nonlinear equations

Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200

Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity

The strong existence and the pathwise uniqueness of solutions with (Formula presented.) -vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved

Well-posedness of the transport equation by stochastic perturbation

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.Comment: Addition of new part