Probability of a Stochastic Markov process

  • #1
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Hi everyone! I'm approaching the physics of stochastic processes. In particular I am studying from "Handbook of stochastic processes - Gardiner". This book defines a stationary process like:
$$ p(x_1, t_1; x_2, t_2; ...; x_n, t_n) = p(x_1, t_1 + \epsilon; x_2, t_2 + \epsilon; ...; x_n, t_n + \epsilon) $$
and this means that the statistics of ## X(t) ## is equal to that of ## X(t + \epsilon) ##. Hence the probabilities are only function of ## t_i - t_j ##.
Then the book says the if the process is also Markovian, the only things I need to know are the conditionale probabilities like:
$$ p_s(x_1, t_1 - t_2| x_2, 0) $$
because all joint probabilities can be written as a product of conditional probabilities.
Here comes my question. Is it hence correct for a stationary stochastic Markov process to write for 3 values of ## X(t) ##, for instance:
$$ p_s(x_1, t_1; x_2, t_2; x_3, t_3) = p_s(x_1, t_1 - (t_2 - t_3)| x_2, t_2 - t_3 ) \ p_s(x_2, t_2 - t_3| x_3, 0) \ p_s(x_3) \ $$
or should the probability be written as:
$$ p_s(x_1, t_1; x_2, t_2; x_3, t_3) = p_s(x_1, t_1 - t_2| x_2, 0) \ p_s(x_2, t_2 - t_3| x_3, 0) \ p_s(x_3) \ ?$$
To me the first equation is more understandable.
 

Answers and Replies

  • #2
Stephen Tashi
Science Advisor
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Hi everyone! I'm approaching the physics of stochastic processes. In particular I am studying from "Handbook of stochastic processes - Gardiner".

(The second edition has a lot of corrections to the first edition and I think there is a 3rd edition.)

Is it hence correct for a stationary stochastic Markov process to write for 3 values of ## X(t) ##, for instance:
$$ p_s(x_1, t_1; x_2, t_2; x_3, t_3) = p_s(x_1, t_1 - (t_2 - t_3)| x_2, t_2 - t_3 ) \ p_s(x_2, t_2 - t_3| x_3, 0) \ p_s(x_3) \ $$

What justifies the factor ##p_s(x_1, t_1 - (t_2- t_3)| x_2,t_2 - t_3)##?

If we seek some quantity equivalent to ##p_s(x_1,t_1 | x_2,t_2)## we can translate by ##-t_2## giving ##p_s(x_1, t_1-t_2| x_2,0)##.

Suppose ##t_1 = 100, t_2 = 20, t_3 = 3##. Can you justify saying ##p_s(x_1, 100| x_2,20) = p_s(x_1, 83| x_2,17) ## ?
 

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