Introduction to stochastic processes

The count of atoms limited, what follows Is just this kind of tossing, being tossed, 

Never allowed to join in peace, to dwell In peace, estranged from amity and growth... 

Lucretius (c.99-c.55 bce,) The way things are translated by Rolfe Humphries, Indiana University Press, 1968 

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random flmction z t) are made at discrete time 0 ti t2, , tn T. In this case the sequence z(ti) is a discrete sample of the stochastic process z(t). 

---

D. Kenan, An Introduction to Stochastic Processes. North-Holland, Amsterdam, 1979. 

Chiang, C., Introduction to Stochastic Processes and Their Applications, Krieger, New York, 1980. 

With turbulent combustion viewed as a random (or stochastic) process, mathematical bases are available for addressing the subject. A number of textbooks provide introductions to stochastic processes (for example, [55]). In turbulence, any stochastic variable, such as a component of velocity, temperature, or the concentration of a chemical species, which we might call v, is a function of the continuous variables of space x and time t and is, therefore, a stochastic function. A complete statistical description of a stochastic function would be provided by a probability-density functional, tf, defined by stating that the probability of finding the function in a small range i (x, t) about a particular function v(x, t) is [t (x, t)]<3t (x, t) ... 

Hoel, P, Port, S and Stone, C. (1972) Introduction to Stochastic Processes, Waveland Press, Prospect Heights, Illinois. 

E. Cinlar, Introduction to Stochastic Processes, Prentice-Hall, Inc., 1975. Moshik, Painted on Weekdays, Zmora, Bitan, Modan - Publisher, 1981. S.Whitfield, Magritte, The South Bank Center, London, 1992. 

Hoel PG, Port SC, Stone Q (1972) Introduction to stochastic processes. Houghton Mifflin, Boston... 

---