Stability of Densities for Perturbed Diffusions and Markov Chains

Optimal control problems for degenerate two-dimensional diffusion processes are considered. The processes could serve as models for wear processes. The objective is to make the controlled process leave the continuation region through a given part of the boundary. Explicit and exact solutions are obtained for important processes such as the geometric Brownian motion and the Ornstein-Uhlenbeck process.

In this chapter we express the classical Wigner function by a degenerate diffusion process [164] solving a randomly perturbed Newton equation. We discuss first some estimates on the solution of these equations and the long time behaviour of the diffusion process. We discuss a relation between the classical Wigner function and the quantum one. We approximate the large time behaviour of the density matrix by the classical Wigner function. In this way we obtain some estimates on the decoherence in space expressed by the correlation functions of the diffusion process.

LetY(t) be a one-dimensional diffusion process anddX(t)=Y(t)dt. The process (X(t), Y(t)) is considered in the second quadrant. First, the probability that (X(t), Y(t)) will hit thex-axis before they-axis is computed explicitly whenY(t) is a standard Brownian motion or a particular case of the Bessel process. Next, an exact expression is obtained for the average time it takes (X(t), Y(t)) to exit the regionC={(x, y) ∈ ℝ2 :x < 0,0 <d 1 <y <d 2 whenY(t) is a standard Brownian motion. The solution is expressed as a generalized Fourier series.

This study formulates an open problem on spatial localization for quasilinear parabolic equations of pLaplacian type on the half-space R+N, with smooth, nonnegative initial data and boundary conditions that grow unbounded over time. While localization has been widely studied under homogeneous or decaying boundary conditions, its persistence under unbounded boundary input remains unresolved. We introduce two forms of spatial localization: effective localization, where the solution remains within a finite spatial domain, and strict localization, where it vanishes beyond a fixed boundary. The problem examines how the structure of the nonlinearities A(v) and B(v), and the growth rate of the boundary function ϕ influence solution confinement or spread. The study extends classical localization theory to degenerate and singular diffusion processes with diffusion exponent p &gt; 1, offering new insight into spatial confinement under persistent external forcing.

Weak convergence of the Euler scheme for stochastic differential equations is established when coefficients are discontinuous on a set of Lebesgue measure zero. The rate of convergence is presented when coefficients are Hölder continuous. Monte Carlo simulations are also discussed.

The direct Lyapunov method for stability of dynamical systems is extended to the almost sure stability of Ito stochastic differential equations by means of semicontinuous Lyapunov functions satisfying a suitable system of partial differential inequalities in viscosity sense. The a.s. stabilizability of controlled degenerate diffusion processes is also treated. The key tools are some geometric characterizations of the invariance and viability properties of closed sets for controlled diffusions obtained by PDE-viscosity methods.

A CLT for degenerate diffusions with periodic coefficients, and application to homogenization of linear PDEs

Article Communication structure of discretized degenerate diffusion processes and approximation of Lyapunov exponents was published on January 1, 2001 in the journal Monte Carlo Methods and Applications (volume 7, issue 3-4).

This work is concerned with weak convergence of non-Markov random processes modulated by a Markov chain. The motivation of our study stems from a wide variety of applications in actuarial science, communication networks, production planning, manufacturing and financial engineering. Owing to various modelling considerations, the modulating Markov chain often has a large state space. Aiming at reduction of computational complexity, a two-time-scale formulation is used. Under this setup, the Markov chain belongs to the class of nearly completely decomposable class, where the state space is split into several subspaces. Within each subspace, the transitions of the Markov chain varies rapidly, and among different subspaces, the Markov chain moves relatively infrequently. Aggregating all the states of the Markov chain in each subspace to a single super state leads to a new process. It is shown that under such aggregation schemes, a suitably scaled random sequence converges to a switching diffusion process.

The gambler’s ruin problem for a Markov chain related to the Bessel process

In the field of large deviations for stochastic dynamics, the canonical conditioning of a given Markov process with respect to a given time-local trajectory observable over a large time-window has attracted a lot of interest recently. In the present paper, we analyze the following inverse problem: when two Markov generators are given, is it possible to connect them via some canonical conditioning and to construct the corresponding time-local trajectory observable? We focus on continuous-time Markov processes and obtain the following necessary and sufficient conditions: (i) for continuous-time Markov jump processes, the two generators should involve the same possible elementary jumps in configuration space, i.e. only the values of the corresponding rates can differ; (ii) for diffusion processes, the two Fokker–Planck generators should involve the same diffusion coefficients, i.e. only the two forces can differ. In both settings, we then construct explicitly the various time-local trajectory observables that can be used to connect the two given generators via canonical conditioning. This general framework is illustrated with various applications involving a single particle or many-body spin models. In particular, we describe several examples to show how non-equilibrium Markov processes with non-vanishing steady currents can be interpreted as the canonical conditionings of detailed-balance processes with respect to explicit time-local trajectory observables.

We analyze the large time behavior of a stochastic model for the lay down of fibers on a moving conveyor belt in the production process of nonwovens. It is shown that under weak conditions this degenerate diffusion process has a unique invariant distribution and is even geometrically ergodic. This generalizes results from previous works [M. Grothaus and A. Klar, SIAM J. Math. Anal., 40 (2008), pp. 968--983; J. Dolbeault et al., arXiv:1201.2156] concerning the case of a stationary conveyor belt, in which the situation of a moving conveyor belt has been left open.

A class of degenerate diffusion processes occurring in population genetics

In this chapter, we show that the degenerate points may separate the state space into different regions for multiple classes, and we discuss the optimization of multi-class degenerate diffusion processes. We also show that under some conditions, the performance function of finite-horizon optimization problems, or the potential function of the long-run average optimization problems, is semi-smooth at degenerate points and smooth at non-degenerate points. Thus, degenerate points coincide with semi-smooth points. Furthermore, there are some special features at the degenerate points: the local time at these points are zero, and the process can only move toward one direction. Therefore, the effect of semi-smoothness of a function can be ignored at these degenerate points in the Ito-Tanaka formula. With these special features in consideration, various optimization problems such as long-run average, finite-horizon, optimal stopping, and singular control, become simpler.

We develop a direct Lyapunov method for the almost sure open-loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes. The infinitesimal decrease condition for a Lyapunov function is a new form of Hamilton--Jacobi--Bellman partial differential inequality of second order. We give local and global versions of the first and second Lyapunov theorems, assuming the existence of a lower semicontinuous Lyapunov function satisfying such an inequality in the viscosity sense. An explicit formula for a stabilizing feedback is provided for affine systems with smooth Lyapunov function. Several examples illustrate the theory.