Diffusive behaviour of the Wigner function and decoherence

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.

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.

We study the sensitivity of the densities of non degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coefficients. Natural applications of these results appear in models with misspecified coefficients or for the investigation of the weak error of the Euler scheme with irregular coefficients.

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.

The Weyl correspondence for obtaining quantum operators from functions of classical coordinates and momenta is known to be incorrect. To calculate quantum-mechanical expectation values as phase-space averages with the Wigner density function, one cannot use classical functions but must use Weyl transforms. These transforms are defined and their properties derived from quantum mechanics. Their properties are expressed in terms of a Hermitian operator Δ(Q, K) whose Weyl transform is a δ function. The Wigner function is the transform of the density operator. Every Weyl transform is exhibited as a difference of two functions which are nonnegative on the phase spece. Weyl transforms do not obey the algebra of classical functions. In the classical limit ℏ → 0, Weyl transforms become classical functions, the Wigner function becomes nonnegative throughout the phase space, and the Hilbert space is spanned by an orthonormal set of vectors which are simultaneous eigenkets of the commuting coordinate and momentum operators.

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

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.

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).

We demonstrate that the Wigner function of a pure quantum state is a wave\nfunction in a specially tuned Dirac bra-ket formalism and argue that the Wigner\nfunction is in fact a probability amplitude for the quantum particle to be at a\ncertain point of the classical phase space. Additionally, we establish that in\nthe classical limit, the Wigner function transforms into a classical\nKoopman-von Neumann wave function rather than into a classical probability\ndistribution. Since probability amplitude need not be positive, our findings\nprovide an alternative outlook on the Wigner function's negativity.\n

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.

A minimization principle behind the diffusion bridge of diurnal fish migration.

The renewal problem for degenerate diffusion processes (increasing coefficients)