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:

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

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:

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

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.

 

[blue_raver22]

I can say that there are potentially an infinite number of functions that can solve the Chapman-Kolmogorov equation. The equation is a fundamental part of stochastic processes and is used to describe the evolution of a system over time. It is a general equation that can be applied to a wide range of systems, so the specific functions that satisfy it will depend on the specific system being studied.

The function you have mentioned is a common solution to the Chapman-Kolmogorov equation and is known as the transition probability density function. It describes the probability of a system transitioning from one state to another in a given time interval. However, there may be other functions that can accurately describe the behavior of a system and satisfy the Chapman-Kolmogorov equation.

To find other potential functions, one could use mathematical techniques such as Fourier analysis or numerical methods to approximate solutions. Additionally, experimental data can be used to fit different functions to the system and see if they satisfy the equation.

Ultimately, the number of functions that solve the Chapman-Kolmogorov equation will depend on the complexity of the system being studied and the accuracy required for the solution. It is an ongoing area of research and there may be new functions discovered in the future that can satisfy the equation.