Diffusion limit for the radiative transfer equation perturbed by a Markovian process

Diffusion limit for the radiative transfer equation perturbed by a Wiener process

The aim of this paper is the rigorous derivation of a stochastic non-linear diffusion equation from a radiative transfer equation perturbed with a random noise. The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The Hilbert expansion has to be done up to order 3 to overcome some diffculties caused by the random noise.

Perturbed Linear-Quadratic Control Problems and Their Probabilistic Representations

We consider some certain nonlinear perturbations of the stochastic linear-quadratic optimization problems and study the connections between their solutions and the corresponding Markovian backward stochastic diferential equations (BSDEs). Using the methods of stochastic control, nonlinear partial differential equations (PDEs) and BSDEs, we identify conditions for the solvability of the problem and obtain some regularity properties of the solutions.

混合慢-快系统鞅解的收敛性

Convergence of martingale solutions to the hybrid slow-fast system

本文研究 Markov 切换带跳慢-快系统鞅解的弱收敛，通过结合扰动测试函数与时间离散化克服直接方法的困难，并推广至无切换情形。

Radiative Transport Limit for the Random Schrödinger Equation with Long-Range Correlations

In this paper we study the asymptotic phase space energy distribution of solution of the Schrödinger equation with a time-dependent random potential. The random potential is assumed to be with slowly decaying correlations. We show that the Wigner transform of a solution of the random Schrödinger equation converges in probability to the solution of a radiative transfer equation. Moreover, we show that this radiative transfer equation with long-range coupling has a regularizing effect on its solutions. Finally, we give an approximation of this equation in term of a fractional Laplacian. The deri…

Diffusion-approximation for a kinetic spray-like system with random forcing

We study a kinetic toy model for a spray of particles immersed in an ambient fluid, subject to some additional random forcing given by a mixing, space-dependent Markov process. Using the perturbed test function method, we derive the hydrodynamic limit of the kinetic system. The law of the limiting density satisfies a stochastic conservation equation in Stratonovich form, whose drift and diffusion coefficients are completely determined by the law of the stationary process associated with the Markovian perturbation.

Radiative Transport Limit of Dirac Equations with Random Electromagnetic Field

This paper concerns the kinetic limit of the Dirac equation with random electromagnetic field. We give a detailed mathematical analysis of the radiative transport limit for the phase space energy density of solutions to the Dirac equation. Our derivation is based on a martingale method and a perturbed test function expansion. This requires the electromagnetic field to be a space-time random field. The main mathematical tool in the derivation of the kinetic limit is the matrix-valued Wigner transform of the vector-valued Dirac solution. The major novelty compared to the scalar (Schrödinger) cas…