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Wenqing Hu

Publications and source records attributed to Wenqing Hu.

30 records · Page 2Linked to original sources
arXiv2017

Hypoelliptic multiscale Langevin diffusions: Large deviations, invariant measures and small mass asymptotics

We consider a general class of non-gradient hypoelliptic Langevin diffusions and study two related questions. The first one is large deviations for hypoelliptic multiscale diffusions. The second one is small mass asymptotics of the invariant measure corresponding to hypoelliptic Langevin operators and of related hypoelliptic Poisson equations. The invariant measure corresponding to the hypoelliptic problem and appropriate hypoelliptic Poisson equations enter the large deviations rate function due to the multiscale effects. Based on the small mass asymptotics we derive that the large deviations behavior of the multiscale hypoelliptic diffusion is consistent with the large deviations behavior of its overdamped counterpart. Additionally, we rigorously obtain an asymptotic expansion of the solution to the related density of the invariant measure and to hypoelliptic Poisson equations with respect to the mass parameter, characterizing the order of convergence. The proof of convergence of invariant measures is of independent interest, as it involves an improvement of the hypocoercivity result for the kinetic Fokker-Planck equation. We do not restrict attention to gradient drifts and our proof provides explicit information on the dependence of the bounds of interest in terms of the mass parameter.

math.PR↗
arXiv2013

On second order elliptic equations with a small parameter

The Neumann problem with a small parameter $$(\dfrac{1}εL_0+L_1)u^ε(x)=f(x) \text{for} x\in G, .\dfrac{\partial u^ε}{\partial γ^ε}(x)|_{\partial G}=0$$ is considered in this paper. The operators $L_0$ and $L_1$ are self-adjoint second order operators. We assume that $L_0$ has a non-negative characteristic form and $L_1$ is strictly elliptic. The reflection is with respect to inward co-normal unit vector $γ^ε(x)$. The behavior of $\lim\limits_{ε\downarrow 0}u^ε(x)$ is effectively described via the solution of an ordinary differential equation on a tree. We calculate the differential operators inside the edges of this tree and the gluing condition at the root. Our approach is based on an analysis of the corresponding diffusion processes.

math.PR↗
arXiv2013

On diffusion in narrow random channels

We consider in this paper a solvable model for the motion of molecular motors. Based on the averaging principle, we reduce the problem to a diffusion process on a graph. We then calculate the effective speed of transportation of these motors.

math.PR↗
arXiv2012

Small mass asymptotic for the motion with vanishing friction

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 regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and boundary condition of this limiting Markov process and prove the convergence.

math.PR↗
arXiv2012

On stochasticity in nearly-elastic systems

Nearly-elastic model systems with one or two degrees of freedom are considered: the system is undergoing a small loss of energy in each collision with the "wall". We show that instabilities in this purely deterministic system lead to stochasticity of its long-time behavior. Various ways to give a rigorous meaning to the last statement are considered. All of them, if applicable, lead to the same stochasticity which is described explicitly. So that the stochasticity of the long-time behavior is an intrinsic property of the deterministic systems.

math.PR↗
arXiv2012

On perturbations of generalized Landau-Lifshitz dynamics

We consider deterministic and stochastic perturbations of dynamical systems with conservation laws in $\R^3$. The Landau-Lifshitz equation for the magnetization dynamics in ferromagnetics is a special case of our system. The averaging principle is a natural tool in such problems. But bifurcations in the set of invariant measures lead to essential modification in classical averaging. The limiting slow motion in this case, in general, is a stochastic process even if pure deterministic perturbations of a deterministic system are considered. The stochasticity is a result of instabilities in the non-perturbed system as well as of existence of ergodic sets of a positive measure. We effectively describe the limiting slow motion.

math.PR↗
arXiv2012

Smoluchowski-Kramers approximation in the case of 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 modification of the Smoluchowski-Kramers approximation. Some applications of the Smoluchowski-Kramers approximation to problems with fast oscillating or discontinuous coefficients are considered.

math.PR↗
arXiv2012

On metastability in nearly-elastic systems

We consider a nearly-elastic model system with one degree of freedom. In each collision with the "wall", the system can either lose or gain a small amount of energy due to stochastic perturbation. The weak limit of the corresponding slow motion, which is a stochastic process on a graph, is calculated. A large deviation type asymptotics and the metastability of the system are also considered.

math.PR↗