By Matt Scott
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Additional resources for Applied Stochastic Processes in Science and Engineering
Tn ), or density f (x1 , x2 , . . , xn ; t1 , t2 , . . , tn ), completely determines a stochastic process, this leads to a natural classification system. Assuming the time ordering t1 < t2 < . . < tn , we then identify several examples of stochastic processes: 1. e. f (x1 , . . , xn ; t1 , . . , tn ) = f (x1 , t1 ) · f (x2 , t2 ) · · · f (xn , tn ); in other words, all the information about the process is contained in the 1st -order density. g. from f (x1 , x2 , x3 ; t1 , t2 , t3 ) = f (x1 , t1 ) · f (x2 , t2 ) · f (x3 , t3 ), it follows that, f (x1 , x2 ; t1 , t2 ) = f (x1 , t1 ) · f (x2 , t2 ).
9) where w(x|z) is the transition probability per unit time, and a0 is the 58 Applied stochastic processes zeroth -jump moment, ∞ a0 (z) = w (x|z) dx. 10) −∞ The physical content of Eq. e. z = x) = the transition probability of moving from z to x during time τ ′ . This will be the case in systems where the fluctuations arise from approximating a discrete stochastic process by a continuous deterministic model – for example, models of chemical reaction kinetics, predator-prey dynamics, particle collisions, radioactive decay, etc..
The term seems to have a magical appeal, which invites its use in an intuitive sense not covered by the definition. In particular: 52 Applied stochastic processes • When a physicist talks about a “process,” a certain phenomenon involving time is usually what is being referred to. It is meaningless to say a “phenomenon” is Markovian (or not) unless one specifies the variables (especially the time scale) to be used for its description. • Eq. 1 is a condition on all the probability densities; one simply cannot say that a process is Markovian if only information about the first few of them is available.
Applied Stochastic Processes in Science and Engineering by Matt Scott