What Is the Integral Formulation of the Chapman-Kolmogorov Formula?

[eljose]

Hello..where i could find information about the Chapmann-Kolmogorov formula for continuous probability..i have hear something when taking a course of QM...something about this...if you want to go from A point to B point with a certain probability crossing a point C then:

$$P(A,B)=P(A,C)P(B,C)$$

My question is what is the Integral or differential formulation of this law?..considering we know all the probability distributions..thanks.

 

[iStealth]

the Chapman-Kolmogorov equation
$$p(\mathbf{x}_{k}|\mathbf{z}_{1:k-1})= \int p(\mathbf{x}_{k}|\mathbf{x}_{k-1})p(\mathbf{x}_{k-1}|\mathbf{z}_{1:k-1})d\mathbf{x}_{k-1}$$as an example, from "A Tutorial on Particle Filters for On-line Non-linear/Non-Gaussian Bayesian Tracking (2001)"

 

[tacman]

Hello! The Chapman-Kolmogorov formula is a fundamental concept in the field of probability theory. It is used to describe the probability of a system transitioning from one state to another over a certain period of time. This formula is widely used in various fields, including quantum mechanics, as you mentioned.

To answer your question, the integral or differential formulation of this law can be expressed as follows:

Let X(t) be a continuous-time Markov process with state space S and transition probability function P(t, x, A). Then, for any t1 < t2 < ... < tn and any states x1, x2, ... , xn in S, the probability of the process transitioning from state x1 at time t1 to state xn at time tn can be calculated using the following integral:

P(x1, t1; xn, tn) = ∫ P(tn, xn, ds) ∫ P(sn, xn, dxn) ... ∫ P(t2, x2, dx2) P(t1, x1, dx1)

Where the integrals are taken over all possible values of the intermediate states sn, sn-1, ... , s1.

In simpler terms, this integral formulation represents the sum of all possible paths that the process can take from state x1 at time t1 to state xn at time tn, weighted by the transition probabilities at each step along the way.

I hope this helps clarify the integral formulation of the Chapman-Kolmogorov formula for continuous probability. For more information, I suggest consulting a textbook on probability theory or searching for resources online. Best of luck in your studies!