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More colloquially, a first passage time in a stochastic system, is the time taken for a state variable to reach a certain value. Understanding this metric allows one to further understand the physical system under observation, and as such has been the topic of research in very diverse fields, from economics to ecology.
For a finite ergodic Markov chain [6] with transition matrix P and invariant distribution π, write m ij for the mean first passage time from state i to state j (denoting the mean recurrence time for the case i = j). Then = is a constant and not dependent on i. [7]
First passage percolation is one of the most classical areas of probability theory. It was first introduced by John Hammersley and Dominic Welsh in 1965 as a model of fluid flow in a porous media. [1] It is part of percolation theory, and classical Bernoulli percolation can be viewed as a subset of first passage percolation.
This distribution appears to have been first derived in 1900 by Louis Bachelier [6] [7] as the time a stock reaches a certain price for the first time. In 1915 it was used independently by Erwin Schrödinger [4] and Marian v. Smoluchowski [5] as the time to first passage of a Brownian motion.
A distribution on {,,,...} is a discrete phase-type distribution if it is the distribution of the first passage time to the absorbing state of a terminating Markov chain with finitely many states. Characterization
This is the smallest time after the initial time t 0 that y(t) is equal to one of the critical values forming the boundary of the interval, assuming y 0 is within the interval. Because y(t) proceeds randomly from its initial value to the boundary, τ(y 0) is itself a random variable. The mean of τ(y 0) is the residence time, [1] [2]
The first exit time (from A) is defined to be the first hit time for S \ A, the complement of A in S. Confusingly, this is also often denoted by τ A. [1] The first return time is defined to be the first hit time for the singleton set {X 0 (ω)}, which is usually a given deterministic element of the state space, such as the origin of the ...
The lifetime distribution function, conventionally denoted F, ... the survival event density function is known as the first passage time density.