Strong approximation of diffusion processes by transport processes (original) (raw)

An approximation of It\^ o diffusions based on simple random walks

The aim of this paper is to develop a sequence of discrete approximations to a one-dimensional Itô diffusion that almost surely converges to a weak solution of the given stochastic differential equation. Under suitable conditions, the solution of the stochastic differential equation can be reduced to the solution of an ordinary differential equation plus an application of Girsanov's theorem to adjust the drift. The discrete approximation is based on a specific strong approximation of Brownian motion by simple, symmetric random walks (the so-called "twist and shrink" method). A discrete Itô's formula is also used during the discrete approximation.

Hitting and martingale characterizations of one-dimensional diffusions

The main theorem of this report states that, for a large class of one-dimensional diffusions (i. e., strong Markov processes with continuous sample paths): If x(t) is a continuous stochastic process possessing the hitting probabilities and mean exit times of the given diffusion, then x(t) is Markovian, with the transition probabilities of the diffusion.

Strong approximations of Brownian sheet by uniform transport processes

Collectanea Mathematica

Many years ago, Griego, Heath and Ruiz-Moncayo proved that it is possible to define realizations of a sequence of uniform transform processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In this paper we extend their results to the multi parameter case. We begin constructing a family of processes, starting from a set of independent standard Poisson processes, that has realizations that converge almost surely to the Brownian sheet, uniformly on the unit square. At the end the extension to the d-parameter Wiener processes is presented. * X.

Strongly and weakly self-similar diffusion

Many dispersive processes have moments of displacements with large-t behavior |x| p ∼ t γ p . The study of γ p as a function of p provides a more complete characterization of the process than does the single number γ 2 . Also at long times, the core of the concentration relaxes to a self-similar profile, while the large-x tails, consisting of particles which have experienced exceptional displacements, are not self-similar. Depending on the particular process, the effect of the tails can be negligible and then γ p is a linear function of p (strong self-similarity). But if the tails are important then γ p is a non-trivial function of p (weak self-similarity). In the weakly self-similar case, the low moments are determined by the self-similar core, while the high moments are determined by the non-self-similar tails. The popular exponent γ 2 may be determined by either the core or the tails. As representatives of a large class of dispersive processes for which γ p , is a piecewise-linear function of p, we study two systems: a stochastic model, the "generalized telegraph model", and a deterministic area-preserving map, the "kicked Harper map". We also introduce a formula which enables one to obtain the moment |x| p from the Laplace-Fourier representation of the concentration. In the case of the generalized telegraph model, this formula provides analytic expressions for γ p .

Diffusion processes

This appendix provides a brief introduction to diffusion processes and to some of the mathematical techniques used in their description and analysis. More comprehensive and rigorous accounts are provided by Gardiner (1985), Gillespie (1992), Arnold (1974), and Karlin and Taylor (1981).

Comparison of interacting diffusions and an application to their ergodic theory

Probability Theory and Related Fields, 1996

A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals. Classification (1991): 60K35, 60J60, 60J15 is one-sided bounded. We show that clustering is universal in the diffusion coefficient. This had been conjectured in Cox, Greven and Shiga [CGS95a] (see also Shiga [Shi92]). On the way, we obtain a new proof, in the case where the state space of a component is compact, based on the interacting Fisher-Wright diffusion where a well-known duality is available.

The diffusion limit of transport equations derived from velocity-jump processes

2000

In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak-Keller-Segel-Alt model for chemotaxis.

DIFFUSION APPROXIMATION FOR TRANSPORT PROCESSES WITH GENERAL REFLECTION BOUNDARY CONDITIONS

Mathematical Models and Methods in Applied Sciences, 2006

Diffusion approximations are obtained for space inhomogeneous linear transport models with reflection boundary conditions. The collision kernel is not required to satisfy any balance condition and the scattering kernel on the boundary is general enough to include all examples of boundary conditions known to the authors (with conservation of the number of particles) and, in addition, to model the Debye sheath. The mathematical approach does not rely on Hilbert expansions, but rather on martingale and stochastic averaging techniques.

On the time a diffusion process spends along a line

Stochastic Processes and Their Applications, 1993

For an arbitrary diffusion process X with time-homogeneous drift and variance parameters μ(x) and σ2(x), let Vε be times the total time X(t) spends in the strip . The limit V as ε → 0 is the full halfline version of the local time of X(t) − a − bt at zero, and can be thought of as the time X spends along the straight line x = a + bt. We prove that V is either infinite with probability 1 or distributed as a mixture of an exponential and a unit point mass at zero, and we give formulae for the parameters of this distribution in terms of μ(·), σ(·), a, b, and the starting point X(0). The special case of a Brownian motion is studied in more detail, leading in particular to a full process V(b) with continuous sample paths and exponentially distributed marginals. This construction leads to new families of bivariate and multivariate exponential distributions. Truncated versions of such ‘total relative time’ variables are also studied. A relation is pointed out to a second order asymptotics problem in statistical estimation theory, recently investigated in Hjort and Fenstad (1992a, 1992b).