Functions solving Chapman-Kolmogorov

[friend]

How many functions solve the Chapman-Kolmogorov equation? The Chapman-Kolmogorov equation is defined as:

$$f({x_n}|{x_m}) = \int_{ - \infty }^{ + \infty } {f({x_n}|{x_o})} f({x_o}|{x_m})d{x_o}$$

And I've seen it proved that the following function will satisfy the above condition:

$$f({x_n}|{x_o}) = {\left( {\frac{\lambda }{{2\pi ({t_n} - {t_0})}}} \right)^{{\raise0.5ex\hbox{\scriptstyle 1}<br /> \kern-0.1em/\kern-0.15em<br /> \lower0.25ex\hbox{\scriptstyle 2}}}}{e^{\frac{{\lambda {{({x_n} - {x_0})}^2}}}{{2({t_n} - {t_0})}}}}$$

But I have to wonder if there are any other functions that satisfy it. Any help is appreciated.

 

[friend]

How many functions solve the Chapman-Kolmogorov equation? The Chapman-Kolmogorov equation is defined as:

$$f({x_n}|{x_m}) = \int_{ - \infty }^{ + \infty } {f({x_n}|{x_o})} f({x_o}|{x_m})d{x_o}$$

And I've seen it proved that the following function will satisfy the above condition:

$$f({x_n}|{x_o}) = {\left( {\frac{\lambda }{{2\pi ({t_n} - {t_0})}}} \right)^{{\raise0.5ex\hbox{\scriptstyle 1}<br /> \kern-0.1em/\kern-0.15em<br /> \lower0.25ex\hbox{\scriptstyle 2}}}}{e^{\frac{{\lambda {{({x_n} - {x_0})}^2}}}{{2({t_n} - {t_0})}}}}$$

But I have to wonder if there are any other functions that satisfy it. Any help is appreciated.

I did find this solution to the Chapman-Kolmogorov equation:

http://www.pims.math.ca/files/monahan_2b_0.pdf

Although, I could not find any reference to the author. And I don 't know how general or particular his solution is. Maybe a more experienced person would like to take a look at it and advice. Thanks.

 

[chiro]

I don't really know all that much about DE's (ODE's or PDE's) but that equation for joint probabilities looks pretty accurate.

 

[Steve Zissou]

I believe just about any transition density for a markovian process should satisfy the C-K equation.
The solution listed above was for a "raw" brownian motion. You can see that any arithmetic or geometric process will result in a similar process. The post above had a link to show that an Ornstein-Uhlenbeck (Langevin) process can satisfy it.