probabilistic models, like probabilistic time automata .. it is used extensively for model checking purpose.

A stochastic process X is a random variable valued function of time (verifying some additional conditions depending on the precise situation). I.e. at every instant t, X(t) is a random variable having some probability distribution. (t can be real on integer, X(t) can be real or more complicated an object but the essential property is what I wrote above).

In general, the distribution of X(t) can depend on various things, eg can depend on the values the process took on instants st or many other complicated things.

A Markov stochastic process is a very particular stochastic process where the distribution of the random variable X(t) conditioned on the values it took at all times r

I'll be appreciated if Dr. Teneng send to me a source (e.g. articles or book) for using Levy statistics to apply it for radiative transfer problesm

An important non-makovian quantity is persistence. Persistence is simply the probability that the fluctuating nonequilibrium field does not change sign upto time t [1]. The problem of persistence in spatially extended nonequilibrium systems has recently generated a lot of interest both theoretically [2, 3, 4, 5] and experimentally [6,7]. In Ising spin systems Single spin persistence provides a natural counterpart to the survival probability in the realm of many-particle systems. In the context of reaction processes, persistence is equivalent to the survival of immobile impurities and therefore does not provide information about collective properties of the bulk [8]. In Ising model, in a zero temperature quench, persistence is simply the probability that a spin has not flipped till time t .

[1] For a review, see S. N. Majumdar, Curr. Sci. 77 370 (1999).

[2] B. Derrida, A.J. Bray, and C. Godrche, J. Phys. A 27, L357 (1994);

[3] D. Stauffer, J. Phys. A 27 5029 (1994).

[4] P. L. Krapivsky, E.Ben-Naim and S. Redner, Phys. Rev.E 50 2474

(1994).

[5] A. Watson, Science, 274, 919 (1996).

[6] M. Marcos-Martin, D. Beysens, J-P Bouchaud, C. Godr‘che, and I.

Yekutieli, Physica 214D, 396 (1995).

[7] W.Y. Tam, R. Zeitak, K.Y. Szeto, and J. Stavans, Phys. Rev. Lett.

78, 1588 (1997).

B. Yurke, A. N. Pargellis, S. N. Majumdar, and C. Sire, Phys Rev

E, 56 R40 (1997)

28

[8] P. L. Krapivsky and E. Ben-Naim, Phys Rev E, 56, 3788 (1997)

You can also consider semi_Markov processes or semimartingals. Authors: Anisimov; Koroliuk; Limnios; Shiryaev.

Best regards, Igor Malik.