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Smith's Markov Process Theorem

Consider

P_2(y_1,t_1|y_3,t_3)=intP_2(y_1,t_1|y_2,t_2)P_3(y_1,t_1;y_2,t_2|y_3,t_3)dy_2.
(1)

If the probability distribution is governed by a Markov process, then

P_3(y_1,t_1;y_2,t_2|y_3,t_3)=P_2(y_2,t_2|y_3,t_3)
(2)
=P_2(y_2|y_3,t_3-t_2).
(3)

Assuming no time dependence, so t_1=0,

P_2(y_1|y_3,t_3)=intP_2(y_1|y_2,t_2)P_2(y_2|y_3,t_3-t_2)dy_2.
(4)

See also

Markov Process

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Cite this as:

Weisstein, Eric W. "Smith's Markov Process Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SmithsMarkovProcessTheorem.html

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