Can Monte Carlo Methods be Used to Solve Infinite-State Integral Equations?

[Karlisbad]

If we have the C-K equation in the form (wikipedia ):

$$p_{i_3;i_1}(f_3\mid f_1)=\int_{-\infty}^\infty p_{i_3;i_2}(f_3\mid f_2)p_{i_2;i_1}(f_2\mid f_1)df_2$$

this is the form of some kind of integral equation.. but is there any differential version of it?? (Chapman-Kolmogorov law into a differential form)

By the way i read that you could use a Markov chain (Particle with a finite number of transition states ) to solve by Montecarlo's method the system of equations

$$a_{j}+x_{j}=A_{i,j}x_{j}$$

where we must find the x_j and the a_j are known numbers..

the obvious question is..can it be generalized for an infinite number of states to solve Integral equations with K(x,y)=K(y,x):

$$f(x)+g(x)=\int_{-\infty}^{\infty}K(x,y)f(x)dx$$

in this last case i was thinking of a process with an infinite number of states, in the 2 cases:

a) the set of infinite states is numerable so $${f1,f2,f3,f4,...}$$

b) the set is Non-numerable.. (you can't label them)..

In these cases i would like to know if there're any applications of MOntecarlos method to solve systems or Integral equations..thanks. :shy:

 

[EnumaElish]

Do you have a link to the C-K entry in wikipedia? I could not find it.

Results of a Yahoo search on "continuous state markov" and "infinite state markov"

You might also find the following link useful: https://www.physicsforums.com/showthread.php?t=10583&highlight=markov

 

[fopc]

Regarding your last sentence, you might investigate the book by G.A.Mikhailov titled
"Optimization of Weighted Monte Carlo Methods".
The originial is in Russian, but I believe it's been translated to English.
I don't own the book so I can't say precisely what's in it, but it's my understanding that it
considers (among other topics) statistical models (in this case vector Monte Carlo algorithms) for
solving systems described by integral equations.