Mean first passage time , probability times

For the solution of real tasks, depending on the concrete setup of the problem, either the forward or the backward Kolmogorov equation may be used. If the one-dimensional probability density with known initial distribution deserves needs to be determined, then it is natural to use the forward Kolmogorov equation. Contrariwise, if it is necessary to calculate the distribution of the mean first passage time as a function of initial state xo, then one should use the backward Kolmogorov equation. Let us now focus at the time on Eq. (2.6) as much widely used than (2.7) and discuss boundary conditions and methods of solution of this equation. 

Figure 10. Evolution of the survival probability for the potential Tf.v) — ax2 - for3 for different values of noise intensity the dashed curve denoted as MFPT (mean first passage time) represents exponential approximation with MFPT substituted into the factor of exponent.

Exercise. A Brownian particle obeys the diffusion equation (VIII.3.1) in the interval Lsplitting probabilities nL(X0) and (Xq) as functions of its starting point X0. Also the conditional mean first-passage times. 

Secondly consider the mean first-passage time xRytn. We suppose that on the left there is a reflecting boundary L, so that (2.4) holds and R is reached with probability 1. At t = 0 the particle sets out at m. In the next At it jumps to the right with probability gm At or to the left with probability rm At or it remains at m with probability 1 — gm At — rm At. One therefore has (fig. 33) ... 

Thirdly we take a one-step process with two exits L and R. There are two splitting probabilities nR m and nLtm obeying (2.2) with the appropriate boundary conditions for each. There are also two conditional mean first-passage times xRm, iL m. In order to compute them we introduce the products R,m = nR,m R,m and L,m = L,m L,m- In the same way as before one argues that... 

Exercise. A particle obeys the ordinary diffusion equation in the space between two concentric spheres. Find the splitting probability and the conditional mean first-passage times. 

Type (ii). The particle has a non-zero probability to reach the wall, but we show that it takes an infinite time to arrive. The conditional mean first-passage time at R — s is, according to (3.11),... 

The less ambitious approach in 2 through the adjoint equation aims only at the splitting probabilities and the mean first-passage times. We construct an equation analogous to (2.2) and (3.1). If at time t the value of Y equals x, it will have at t + At another value x with probability At JT(x x), or it has the same value x. The probability 7rK(x) that Y, starting at x, will exit through R obeys therefore the identity... 

First question How long does the metastable state survive That is, we want to calculate the probability per unit time, 1/t, for a macroscopic population to die by a fluctuation. We have to compute the mean first-passage time t0>m from some m near ns to the absorbing boundary n = 0. We cannot use (XII.2.11) as our psn vanishes for all n > 0. We can use (XII.2.10) when it is adjusted to the case of a lower boundary ... 

The probability p x, t) that the absorbing state is reached between t and t + dt is readily computed from G x,t) as p = —dtG x,i). Note that p is already normalized due to the initial condition G(x, 0). Hence, the mean first passage time T x) equals... 

When a large number of time constants is required to describe adequately the course of the reaction it is probably most useful to adopt the mean first passage time as a measure of the reaction rate. This somewhat digressive topic is discussed next, but in Section V the emphasis is again on the ordinary case in which Tj, and for which the mean first passage time reduces simply to k. ... 

Mean first passage time. Let Nij be the number of steps until the chain reaches state j given it is in state i at time 0. P Nij =n) = /H. Assuming that for a given i and j, the first passage probabilities form a probability distribution (their sum over all n equals 1), then iV j is a random variable and its mean is given by... 

One may ask the following question. Suppose the random walker starts out at site m at t = 0 how long does it take him to reach a given site R for the first time This first-passage time is, of course, different for the different realizations of his walk and is therefore a random quantity. Our purpose is to find its probability distribution, and in particular the average or mean first-passage time ]... 

If this is equal to unity, then fR,m(t) is the probability density for the first-passage time at R. In that case the mean first-passage time is [compare (VI.7.5)]... 

Since G(E(, r) is the probability that the first passage time of the particle (with initial dimensionless energy E0) is >t, the mean first passage time T (Eo) is given by... 

Figure 5. Cell division as a first-passage time (FPT) problem, (a) Schematic of stochastic cell size increase from a common initial condition. Between times t and T + At, the some growth tracks cross the threshold size, 9. Using probability conservation, the cumulative probability that the size is greater than 9 (above the black dotted horizontal line) must be equal to the complement of the cumulative probability that the FPT is less than or equal to T Oeft of blue dotted vertical line at t). (b) Scahng of the FPT distribution. The shape of the mean-rescaled division time distribution is timescale invariant, that is, independent of k, when there is a single timescale, 1 /if ocr, in the FPT dynamics. (See insert for color representation of the figure.)...

The probability distribution of the first passage time and its moments can be obtained from the Fokker-Planck equation for general situations of F x) and i/f (x), by following the standard procedures given in Section 6.6. In the present context, the average translocation time is the mean first passage time, which can be calculated by choosing the appropriate boundary conditions for P(x,t). 

Exercise. Write the corresponding equations for the case of a left exit point L < m. Exercise. Solve the first-passage problem for the simple symmetric random walk. Show that any site R is reached with probability nR m= 1, but that the mean first-passage time is infinite. 

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