A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

Regularity theory of Kolmogorov operator revisited

We consider Kolmorogov operator $-\Delta +b \cdot \nabla $ with drift b in the class of form-bounded vector fields (containing vector fields having critical-order singularities). We characterize quantitative dependence of the Sobolev and Hölder regularity of solutions to the corresponding elliptic equation on the value of the form-bound of b.

Moment estimates for invariant measures of stochastic Burgers equations

In this paper, we study moment estimates for the invariant measure of the stochastic Burgers equation with multiplicative noise. Based upon an a priori estimate for the stochastic convolution, we derive regularity properties on invariant measure. As an application, we prove smoothing properties for the transition semigroup by introducing an auxiliary semigroup. Finally, the m-dissipativity of the associated Kolmogorov operator is given.

A NEW A PRIORI ESTIMATE FOR THE KOLMOGOROV OPERATOR OF A 2D-STOCHASTIC NAVIER–STOKES EQUATION

We prove that the Kolmogorov operator L associated with a 2D-stochastic Navier–Stokes equation with periodic boundary conditions and space-time white noise is m-dissipative for sufficiently large viscosity on finitely based cylindrical test functions in the space L1 w.r.t. the Gaussian measure induced by the enstrophy. The proof is based on a new a priori estimate for the solution of the resolvent equation λF - LF = H.