Related papers: On the Langevin equation with variable friction
We consider the small mass asymptotics (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The limit of the solution in the classical sense does not exist in this case. We study a…
We consider the small mass asymptotic (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a…
A system of stochastic differential equations describing diffusive phenomena, which has arbitrary friction depending on both state and distribution is investigated. The Smoluchowski-Kramers approximation is seen to describe dynamics in the…
This note provides a simple derivation of the overdamped approximation for kinetic (or underdamped) equilibrium Langevin dynamics, in cases where certain coefficients depend on the position variable. The equivalent small-mass limit of these…
A class of Langevin stochastic differential equations is shown to converge in the small-mass limit under very weak assumptions on the coefficients defining the equation. The convergence result is applied to physically realizable examples…
The small mass limit is derived for a McKean-Vlasov equation subject to environmental noise with state-dependent friction. By applying the averaging approach to a non-autonomous stochastic slow-fast system with the microscopic and…
We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent…
The small mass limit of the Langevin equation perturbed by $\alpha$-stable L\'{e}vy noise is considered by rewriting it in the form of slow-fast system, and spliting the fast component into three parts, where $\alpha\in(1,2)$. By exploring…
We consider small mass asymptotics of the motion of a charged particle in a noisy force field combined with a variable magnetic field. The Smoluchowski-Kramers approximation does not hold in this case. We show that after a regularization of…
We study the validity of the so-called Smoluchowski-Kramers approximation for a two dimensional system of stochastic partial differential equations, subject to a constant magnetic field. As the small mass limit does not yield to the…
We discuss here the validity of the small mass limit (the so-called Smoluchowski-Kramers approximation) on a fixed time interval for a class of semi-linear stochastic wave equations, both in the case of the presence of a constant friction…
The dynamical evolution of a Brownian particle in an inhomogeneous medium with spatially varying friction and temperature field is important to understand conceptually. It requires to address the basic problem of relative stability of…
We consider systems of damped wave equations with a state-dependent damping coefficient and perturbed by a Gaussian multiplicative noise. Initially, we investigate their well-posedness, under quite general conditions on the friction.…
We study the validity of a Smoluchowski-Kramers approximation for a class of wave equations in a bounded domain of $\mathbb{R}^n$ subject to a state-dependent damping and perturbed by a multiplicative noise. We prove that in the small mass…
We study the asymptotics for the Ablowitz-Ladik equation. By taking appropriate continuum limits, it can be shown that the behavior of the equation near degenerate frequencies is well approximated by a complex modified Korteweg-de Vries…
An approximation is derived for a Langevin equation with distribution-dependent potential and state-dependent, randomly fast oscillation. By some estimates and a diffusion approximation the limiting equation is shown to be…
According to the Smoluchowski-Kramers approximation, the solution of the equation ${\mu}\ddot{q}^{\mu}_t=b(q^{\mu}_t)-\dot{q}^{\mu}_t+{\Sigma}(q^{\mu}_t)\dot{W}_t, q^{\mu}_0=q, \dot{q}^{\mu}_0=p$ converges to the solution of the equation…
We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study…
We get fractional symmetric Fokker - Planck and Einstein - Smoluchowski kinetic equations, which describe evolution of the systems influenced by stochastic forces distributed with stable probability laws. These equations generalize known…
We study the validity of a large deviation principle for a class of stochastic nonlinear damped wave equations, of Klein-Gordon type, in the joint small mass and small noise limit. The friction term is assumed to be state dependent.