We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak Hörmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish d...
Published in Russian Journal of Mathematical Physics
We give a review of some of our works which generalize stochastic analysis tools for semigroups where there is no stochastic process.
Published in Communications in Mathematical Physics
We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrödinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker–Planck type equation with the wave vector performing a Brownian motion on th...
Published in Communications in Mathematical Physics
In this paper, we study the loss of coherence of a wave propagating according to the Schrödinger equation with a time-dependent random potential. The random potential is assumed to have slowly decaying correlations. The main tool to analyze the decoherence phenomena is a properly rescaled Wigner transform of the solution of the random Schrödinger e...
Published in Resonance
This section features conversations with personalities related to science, highlighting the factors and circumstances that guided them in making the career choice to be a scientist.
Published in Ukrainian Mathematical Journal
For a solution of a reflection problem on a half-line similar to the Skorokhod reflection problem but with possible jump-like exit from zero, we obtain an explicit formula and study its properties. We also construct a Wiener process on a half-line with Wentzell boundary condition as a strong solution of a certain stochastic differential equation.
Published in Communications in Mathematical Physics
We consider a finite region of a lattice of weakly interacting geodesic flows on manifolds of negative curvature and we show that, when rescaling the interactions and the time appropriately, the energies of the flows evolve according to a nonlinear diffusion equation. This is a first step toward the derivation of macroscopic equations from a Hamilt...
Published in Ukrainian Mathematical Journal
We propose an approach to the proof of the weak convergence of a semi-Markov process to a Markov process under certain conditions imposed on local characteristics of the semi-Markov process.
Published in Journal of Pseudo-Differential Operators and Applications
We define the path integral associated to a big order generator of a convolution semi-group. We show that this path integral is the solution of a generalized martingale problem. We define the sheet associated to it. By this work we generalize in distributional sense some basic tools defined in stochastic analysis for the Brownian motion.
Les "équations de Schrödinger stochastiques" sont des équations différentielles stochastiques de type non classique qui apparaissent dans le domaine de la mesure en mécanique quantique. Leurs solutions sont appelées "trajectoires quantiques" et décrivent l'évolution de petits systèmes quantiques ouverts soumis à une mesure continue de type indirect...
