Markov Process In Space Research Articles

High Temperature Behaviour of Quantum Mechanical Thermal Functional

We derive a representation of thermal functionals as expectation with respect to a Markov process in phase space of classical quantities. We prove also that the above quantities have a classical behaviour for high temperature.

Decay of correlations and master equation in Kac' lattice model

We discuss on a simple discrete time spin model how a deterministic classical evolution may be transformed into a stochastic process. Simple averaging over a population of fixed scatterers yields a non-markovian microscopic evolution and is not compatible with a macroscopic description in terms of populations. Coupling each scatter-er to an independent reservoir leads to a Markov process in phase space, which can be projected on a macroscopic birth and death process. Some limiting cases (continuous time interpolation, ...) are considered.

Kinetics of crystalline configurations: I. General theory; Kinetics of homogeneous short range order

A simple and exact way of coarse graining the master equation for a Markov process in configuration space is shown to exist. The coarse grained master equation is applied to the Ising model of a homogeneous binary interstitial alloy and to a “magnetic” Ising model. Using an approximation analogous to the quasi-chemical approximation, for both models the macroscopic rate equations for the establishment of short range order and the Fokker-Planck equation governing the fluctuations are derived.

Numerical methods for infinite Markov processes

The estimation of steady state probability distributions of discrete Markov processes with infinite state spaces by numerical methods is investigated. The aim is to find a method applicable to a wide class of problems with a minimum of prior analysis. A general method of numbering discrete states in infinite domains is developed and used to map the discrete state spaces of Markov processes into the positive integers, for the purpose of applying standard numerical techniques. A method based on a little used theoretical result is proposed and is compared with two other algorithms previously used for finite state space Markov processes.

The fluctuation-dissipation theorem for contracted descriptions of Markov processes

It is shown that if the Onsager-Casimir relations and the fluctuationdissipation theorem are valid for a stationary, Gaussian, Markov process in anN-dimensional space, then these relations are valid when the process is projected into a subspace of the original space. Both time-reversal-even and time-reversal-odd variables are allowed. Previous derivations of the fluctuation-dissipation theorem for Brownian motion from fluctuating hydrodynamics are special cases of the present result. For the Brownian motion problem, the fluctuation-dissipation theorem is proven for the case of a compressible, thermally conducting fluid with a nonlocal equation of state. Arbitrary slip boundary conditions are considered as well.

MARKOV Processes in General State Spaces. (Part III)

MARKOV Processes in General State Spaces. (Part III)

MARKOV Processes in General State Spaces (Part II)

Mathematische NachrichtenVolume 82, Issue 1 p. 191-203 Article MARKOV Processes in General State Spaces (Part II) H. J. Engelbert, H. J. Engelbert Friedrich-Schiller-Universität Sektion Mathematik DDR – 69 Jena UniversitätshochhausSearch for more papers by this author H. J. Engelbert, H. J. Engelbert Friedrich-Schiller-Universität Sektion Mathematik DDR – 69 Jena UniversitätshochhausSearch for more papers by this author First published: 1978 https://doi.org/10.1002/mana.19780820117Citations: 6AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume82, Issue11978Pages 191-203 RelatedInformation

MARKOV Processes in General State Spaces. (Part V)

MARKOV Processes in General State Spaces. (Part V)

MARKOV Processes in General State Spaces

MARKOV Processes in General State Spaces

MARKOV Processes in General State Spaces (Part IV)

MARKOV Processes in General State Spaces (Part IV)

MARKOV Processes in General State Spaces (Part VI)

MARKOV Processes in General State Spaces (Part VI)

MARKOV Processes in General State Spaces (Part I)

MARKOV Processes in General State Spaces (Part I)

Wave Propagation in Certain One-Dimensional Random Media

In this paper we investigate the stochastic ordinary differential equation u″+k02[1+εy(t)]u=0 with y(t) a random process. Two specific types of process y(t) are considered. Both of these arise from a bounded mapping y(t) = f(x(t)) of a countable state space Markov process x(t). Exact equations are derived for the statistical moments of u(t), and the behavior of the first two moments is discussed in the limit of small ε. A description of the layered media to which our results apply is given and a comparison of our exact results with certain perturbation methods is made.

Moments of Solutions of a Class of Stochastic Differential Equations

This paper is concerned with the exact calculation of moments of solutions of the stochastic, ordinary differential equation d2udz2+β02 [1+nf(M(z))]u=0, where M(z) is an arbitrary, finite state space Markov process and f(·) is an arbitrary, real, single‐valued function of its argument. The process f(M(z)) is in general not a Markov process. The calculation of the moments is reduced to the solution of a system of ordinary, first‐order differential equations. In the special case where the process M(z) has a stationary transition mechanism, these systems of equations have constant coefficients, so that the moments must consist of sums of exponentials. A specific example of such an equation with M(z) stationary is analyzed in some detail. The results obtained provide an important nontrivial check of some useful approximate techniques.

Some ratio limit Theorems for a general state space Markov Process

Consider a discrete time parameter Markov Process with stationary probability functions, a general state spaceX and the Harris recurrence condition. This then implies the existence and essential uniqueness of a sigma-finite stationary measureπ. It is also assumed that the class of measurable sets ∇ contains single point sets. LetP (m)(x, S) denote them-step transition probability fromx toS∃∇ andp (m)(x, ·), the component ofP (m)(x, ·) which is absolutely continuous with respect toπ. Let ℒ=C: C∃∇, for some $$\mathop {\inf }\limits_{x,y \in C} p^{(n)} (x,y) > 0\} $$ and $$ = \{ n:\mathop {\inf ,}\limits_{x,y \in C} p^{(n)} (x,y) > 0,C \in $$ ℒ}. The paper here presented contains theorems of which the following is typical:Theorem: LetS∃ℐ withπ(S)>0, measurableB⊃S, π(B)>0 andq∃B with $$\mathop {\lim }\limits_{m \to \infty } \int\limits_B {f(x)P^{(m + 1)} (y,dx)/P^{(m)} (q,B)} = \int\limits_B {f(x)\pi } (dx)/\pi ({\rm B})$$ uniformly iny, y∃B for all non-negative measurable f. Then for all measurableA⊃S withπ(A)>0,k=0,±1, ±2,... $$\mathop {\lim }\limits_{m \to \infty } P^{(m + k)} (x,A)/P^{(m)} (q,B) = \pi ({\rm A})/\pi ({\rm B})$$ in measureπ onS. If the g.c.d. (ℐ)=1 and π′ ≪π withπ′(X)<∞ then the above limit holds in measureπ′ onX.