Transition kernel

A series of probable transitions between states can be described with the Markov chain. A Markovian stochastic process is memoryless, and this is illustrated subsequently. We generate a sequence of random variables, (yo, yi, yi, ), so that each time t > 0, the next state yt+i would be sampled from a distribution P(y,+ily,), which would depend only on the current state of the chain, y,. Thus, given y, the next state y,+i would not depend additionally on the history of the chain (yo, yi, yi,---, y i). The name Markov chain is used to describe this sequence, and the transition kernel of the chain is i (.l.) does not depend on t if we assume that the chain is time homogeneous. A detail description of the Markov model is provided in Chapter 26. 

Figure 3. Functions for transition-rate calculations. Shown for a two-macrostate, onedimensional system characterized by pB are the ground-state ( 0) and first excited-state eigenfunctions of X, the transition function and the transition kernel p .

Equations (3.7) can be solved iteratively using Monte Carlo integrations to evaluate the numerator and denominator [25]. Since the main contributions to the v-2 integrals come from the central macrostate regions where = pB is maximal, they can be performed using kernels px and pY, which are determined as described in Section II.C. However, integration of HXY requires a special transition kernel... 

The main statistical characteristic of the two-component Markov chain X , T ) is the transition kernel... 

If the process starts in the state i, then the subsequent state j is determined by the transition kernel Q so that the process remains in state i some random time before making a transition to j. One can introduce the conditional waiting time distribution as... 

The standard continuous-time Markov chain is a special case of a semi-Markov process with the transition kernel... 

It is significant that in Eq. (21) the initial states in the transition kernel are always reactant states, never product states. It has already been noted that in a dissociation reaction, molecules A in product states (that is, dissociated molecules) necessarily fail to satisfy the binary collision conditions in their collisions with X, but this does not affect the validity of Eq. (21). On the other hand, it means that only in the case of an isomerization is the present theory applicable simultaneously to a reaction and to its inverse. 

For Markov chains with a continuous state space, there are too many states for us to use a transition probability function. Instead we define a transition kernel which measures the probability of going from each individual state to every measurable set of states. 

For a Markov chain with continuous state space, if the transition kernel is absolutely continuous, the Chapman-Kolmogorov and the steady state equations are written as integral equations involving the transition density function. 

The transition kernels we will encounter consist of two parts. The first part takes points in the parameter space to other points in the parameter space in a continuous way, similar to a probability density function. The second part takes each point in the parameter space to itself, with finite probability. 

We need to find a probability transition kernel P 0, A) that satisfies... 

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