Markov jump process

We assume that Yi, T, k>0 follows a Markov renewal process which generalizes the notion of Markov jump process. Then, the probability that the N components will step to state J f m state i in the time interval [Tj, T + Ar], given Tjj.,7ji,, k < n, is defined as follows ... 

Bladt, M. and Sorensen, M., 2001 Statistical inference for discretely observed Markov jump processes. Journal of the Royal Statistical Society Series B (Statistical Methodology), 67 395-410. 

In this section, we consider the description of Brownian motion by Markov diffusion processes that are the solutions of corresponding stochastic differential equations (SDEs). This section contains self-contained discussions of each of several possible interpretations of a system of nonlinear SDEs, and the relationships between different interpretations. Because most of the subtleties of this subject are generic to models with coordinate-dependent diffusivities, with or without constraints, this analysis may be more broadly useful as a review of the use of nonlinear SDEs to describe Brownian motion. Because each of the various possible interpretations of an SDE may be defined as the limit of a discrete jump process, this subject also provides a useful starting point for the discussion of numerical simulation algorithms, which are considered in the following section. 

In the finite grid point method [87, 88], the Markov operator is represented by a matrix W( l, ft) whose elements give the transition rates between discrete sites of ft. The values of the transition rates depend upon the model used to describe the motion. For the intramolecular dynamics such as tram-gauche isoTnenz tion or ring flips (see Fig. 4) a random jump process is assumed. Consequently [90]... 

Fig. 5.11 Interrelations between different types of processes (MPP, Markov population process CCR, complex chemical reaction SBD, simple birth and death process S, V, simple Markovian jump processes and models of reactions in the scalar and vector case respectively).

Contents A Historical Introduction. - Probability Concepts. -Markov Processes. - The Ito Calculus and Stochastic Differential Equations. - The Fokker-Planck Equatioa - Approximation Methods for Diffusion Processes. - Master Equations and Jump Processes. - Spatially Distributed Systems. - Bistability, Metastability, and Escape Problems. - Quantum Mechanical tokov Processes. - References. - Bibliogr hy. - Symbol Index. - Author Index. - Subject Index. 

Counting process Equations for response statistical moments Jump processes Markov processes Non-Poisson processes Point process Probability density Random impulses Random vibrations Renewal processes... 

DetaUed Integrodifferential Equations for the Joint Probability Density of the State Vector for Renewal Impulse Process Driven by Two Independent Poisson Processes The jump process Z(t) driven by two independent Poisson processes Eq. 90 is tantamount to a two-state Markov chain 5(0, such that 5(0 = 1 when Z(t) = 0 and 5(0 = 2 when Z(t) = 1 (Figs. 6 and 7). 

Non-Poisson Impulse Processes, Fig. 7 Markov chain for a two-state jump process driven by two independent Poisson processes... 

The oldest and best known example of a Markov process in physics is the Brownian motion.510 A heavy particle is immersed in a fluid of light molecules, which collide with it in a random fashion. As a consequence the velocity of the heavy particle varies by a large number of small, and supposedly uncorrelated jumps. To facilitate the discussion we treat the motion as if it were one-dimensional. When the velocity has a certain value V, there will be on the average more collisions in front than from behind. Hence the probability for a certain change AV of the velocity in the next At depends on V, but not on earlier values of the velocity. Thus the velocity of the heavy particle is a Markov process. When the whole system is in equilibrium the process is stationary and its autocorrelation time is the time in which an initial velocity is damped out. This process is studied in detail in VIII.4. 

Remark. Consider a Markov process that can be visualized as a particle jumping back and forth among a finite number of sites m, with constant probabilities per unit time. Suppose it has a single stationary distribution psn, with the property (5.3). After an initial period it will be true that, if I pick an arbitrary t, the probability to find the particle at n is ps . That implies that psn is the fraction of its life that the particle spends at site n, once equilibrium has been reached. This fact is called ergodicity. For a Markov process with finitely many sites ergodicity is tantamount to indecom-posability. ) In (VII.7.13) a more general result for the times spent at the various sites is obtained. 

Many stochastic processes are of a special type called birth-and-death processes or generation-recombination processes . We employ the less loaded name one-step processes . This type is defined as a continuous time Markov process whose range consists of integers n and whose transition matrix W permits only jumps between adjacent sites,... 

Solving this equation means determining the probability to find at t > 0 the system in (i, r) when it was at t = 0 in (i0, r0). This problem decomposes into two successive steps first find how the molecule jumps among the levels regardless of r, and subsequently add on the behavior in r. This is the reason why we use the name composite Markov process for any random process obeying a master equation of the type (7.4). ... 

We have introduced the Fokker-Planck equation as a special kind of M-equation. Its main use, however, is as an approximate description for any Markov process Y(t) whose individual jumps are small. In this sense the linear Fokker-Planck equation was used by Rayleigh 0, Einstein, Smoluchowskin), and Fokker, for special cases. Subsequently Planck formulated the general nonlinear Fokker-Planck equation from an arbitrary M-equation assuming only that the jumps are small. Finally Kolmogorov8 provided a mathematical derivation by going to the limit of infinitely small jumps. 

Exercise. The two-state Markov process has jumps up and down. Compute the functions (in obvious notation) /2(Ui, t2i) and t2t)-... 

In this review we show that there are two main sources of memory. One of them correspond to the memory responsible for Anderson localization, and it might become incompatible with a representation in terms of trajectories. The fluctuation-dissipation process used here to illustrate Anderson localization in the case of extremely large Anderson randomness is an idealized condition that might not work in the case of correlated Anderson noise. On the other hand, the non-Poisson renewal processes generate memory properties that may not be reproduced by the stationary correlation functions involved by the projection approach to the GME. Before ending this subsection, let us limit ourselves to anticipating the fundamental conclusion of this review The CTRW is a correct theoretical tool to address the study of the non-Markov processes, if these correspond to trajectories undergoing unpredictable jumps. 

Levy diffusion is a Markov process corresponding to the conditions established by the ordinary random walk approach with the random walker making jumps at regular time values. To explain why the GME, with the assumption of Eq. (112), yields Levy diffusion, we notice [50] that the waiting time distribution is converted into a transition probability n(x) through... 

Surprisingly, M.C.Escher, the greatest graphic artist (1898-1972), probably unfamiliar with Markov processes, has already demonstrated in 1931 the same situation in his woodcut Frog [10, p.231]. This is reproduced in Fig.0-1. As time goes by, the frog, system, jumps from one lily pad, state, to another according to... 

For translational long-range jump diffusion of a lattice gas the stochastic theory (random walk, Markov process and master equation) [30] eventually yields the result that Gg(r,t) can be identified with the solution (for a point-like source) of the macroscopic diffusion equation, which is identical to Pick s second law of diffusion but with the tracer (self diffusion) coefficient D instead of the chemical or Fick s diffusion coefficient. 

Another form of stochastic analysis is known as Markov Simulation, named after the nineteenth-century Russian mathematician. A Markov model shows all the possible system states, then goes through a series of jumps or transitions. Each jump represents a unit of time or a step in batch process. At each transition the system either stays where it is or moves to a new state. 

Because a penetrant is Kkely to explore a sorption state for a long time before jumping, it is reasonable at low concentrations to assume that individual penetrant jumps are uncoupled from one another and that the sequential visiting of states is a Markov process [103]. Since each jump occurs independently, the probabihty density p ii) that a time t elapses before the next jump occurs (the waiting time) is distributed according to a continuous-time Poisson process [103],... 

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