On the long-time behavior of a perturbed conservative system with degeneracy

We consider in this work a model conservative system subject to dissipation and Gaussian-type stochastic perturbations. The original conservative system possesses a continuous set of steady states, and is thus degenerate. We characterize the long-time limit of our model system as the perturbation parameter tends to zero. The degeneracy in our model system carries features found in some partial differential equations related, for example, to turbulence problems.Comment: Revised version. We added a Section 6 on the connection with the Euler-Arnold equation. To appear at Journal of Theoretical Probabilit

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

Random perturbations of dynamical systems with reflecting boundary and corresponding PDE with a small parameter

We study the asymptotic behavior of a diffusion process with small diffusion in a domain $D$. This process is reflected at $\partial D$ with respect to a co-normal direction pointing inside $D$. Our asymptotic result is used to study the long time behavior of the solution of the corresponding parabolic PDE with Neumann boundary condition.Comment: 17 pages, 1 figure, comments are welcom

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.Comment: 23 pages, 3 figures, comments are welcom

Large deviations and averaging for systems of slow–fast reaction–diffusion equations

We study a large deviation principle for a system of stochastic reaction--diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the fast process and the associated control that arises from the weak convergence method decouple from each other. We show that in this decoupling case one can use the weak convergence method to characterize the limiting process via a "viable pair" that captures the limiting controlled dynamics and the effective invariant measure simultaneously. The characterization of the limit of the controlled slow-fast processes in terms of viable pair enables us to obtain a variational representation of the large deviation action functional. Due to the infinite--dimensional nature of our set--up, the proof of tightness as well as the analysis of the limit process and in particular the proof of the large deviations lower bound is considerably more delicate here than in the finite--dimensional situation. Smoothness properties of optimal controls in infinite dimensions (a necessary step for the large deviations lower bound) need to be established. We emphasize that many issues that are present in the infinite dimensional case, are completely absent in finite dimensions.First author draf

Large deviations and averaging for systems of slow--fast stochastic reaction--diffusion equations

We study a large deviation principle for a system of stochastic reaction--diffusion equations (SRDEs) with a separation of fast and slow components and small noise in the slow component. The derivation of the large deviation principle is based on the weak convergence method in infinite dimensions, which results in studying averaging for controlled SRDEs. By appropriate choice of the parameters, the fast process and the associated control that arises from the weak convergence method decouple from each other. We show that in this decoupling case one can use the weak convergence method to characterize the limiting process via a "viable pair" that captures the limiting controlled dynamics and the effective invariant measure simultaneously. The characterization of the limit of the controlled slow-fast processes in terms of viable pair enables us to obtain a variational representation of the large deviation action functional. Due to the infinite--dimensional nature of our set--up, the proof of tightness as well as the analysis of the limit process and in particular the proof of the large deviations lower bound is considerably more delicate here than in the finite--dimensional situation. Smoothness properties of optimal controls in infinite dimensions (a necessary step for the large deviations lower bound) need to be established. We emphasize that many issues that are present in the infinite dimensional case, are completely absent in finite dimensions

AcF 706 : assessing default risk of a public company

The dissertation presents the determinants of credit spread, evolution of credit risk modeling and empirically evidence over the period, as well as models based on accounting information. The study explores performance of the firm with accounting and share price information. It also evaluates the predictive of two credit risk models: Merton (1974) and Leland (1994), using accounting and market variables. The finding is that both models tend to underestimate credit risk spreads, though most of the previous literature points out that Leland model usually overestimates credit spread. Further research may focus on market and industrial component of models

Asymptotic Problems in Stochastic Processes and Differential Equations

We study some asymptotic problems in stochastic processes and in differential equations. We consider Smoluchowski-Kramers approximations with variable and vanishing friction. We also consider the problem around second order elliptic equations with a small parameter

On the Global Convergence of Continuous-Time Stochastic Heavy-Ball Method for Nonconvex Optimization

We study the convergence behavior of the stochastic heavy-ball method with a small stepsize. Under a change of time scale, we approximate the discrete method by a stochastic differential equation that models small random perturbations of a coupled system of nonlinear oscillators. We rigorously show that the perturbed system converges to a local minimum in a logarithmic time. This indicates that for the diffusion process that approximates the stochastic heavy-ball method, it takes (up to a logarithmic factor) only a linear time of the square root of the inverse stepsize to escape from all saddle points. This results may suggest a fast convergence of its discrete-time counterpart. Our theoretical results are validated by numerical experiments.Comment: accepted at IEEE International Conference on Big Data in 201