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Annealed Law

An annealed law is the probability law obtained by averaging the conditional law of a stochastic process over its random environment. For an environment law P and quenched law P_omega, it is

P(B)=intP_omega(B)P(domega).

For a random walk in random environment, this averages both the environment and the path randomness. Even though the walk is a Markov chain under each quenched law, its annealed position process need not be Markovian, since its past provides information about the environment.

See also

Markov Chain, Probability Measure, Quenched Law, Random Walk in Random Environment

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References

Drewitz, A. and Ramírez, A. F. "Selected Topics in Random Walks in Random Environment." In Topics in Percolative and Disordered Systems. (Ed. A. F. Ramírez, G. Ben Arous, P. A. Ferrari, C. M. Newman, V. Sidoravicius, and M. E. Vares). New York: Springer, pp. 23-83, 2014. https://doi.org/10.1007/978-1-4939-0339-9_3.

Cite this as:

Weisstein, Eric W. "Annealed Law." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AnnealedLaw.html

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