Narrow Results

An incompressible ideal fluid in the two-dimensional torus (i.e. the Euler equation in a rectangle with periodic boundary conditions) is considered. The flow for a vorticity field concentrated in any finite number of points is analyzed. A compound Poisson measure Π, invariant for this flow, is introduced. The Hilbert space ℒ²(Π) and the properties of the corresponding ℒ²-flow are investigated. In particular it is proven that the corresponding generator is Markov unique.

In the semigroup approach to stochastic evolution equations, the fundamental issue of uniqueness of mild solutions is often "reduced" to the much easier problem of proving uniqueness for strong solutions. This reduction is usually carried out in a formal way, without really justifying why and how one can do that. We provide sufficient conditions for uniqueness of mild solutions to a broad class of semilinear stochastic evolution equations with coefficients satisfying a monotonicity assumption.

We first study the existence of stationary weak solutions of stochastic 3D Navier–Stokes equations involving jumps, and the associated Galerkin stationary probability measures for this case. Then we present a comparison between the Galerkin stationary probability measures for the case driven by Lévy noise and the one driven by Wiener processes.