Mean, variance and correlation function of Langevin equation

[PhoenixWright]

Homework Statement

I have simulated Langevin equation (numerically in Matlab) for some specific conditions, so I have obtained the solution $X(t)$.

But now, with the solution I have obtained, I have to calculate $<X(t)|x_0>, <X^2(t)|x_0>-(<X(t)|x_o>)^2$ and the conditional correlation function $<X(t)X(0)>$.

$X(0)=1$

The Attempt at a Solution

I know how to calculate the mean and variance, even for several variables. Nevertheless, I don't know how to calculate the mean and variance of $X$ conditioned to a single value $x_0$, nor conditional correlation function...
I would appreciate if someone could help me.
Thank you so much.

 

[bigfooted]

I assume that you have a certain volume filled with particles governed by the Langevin equation. The state of each particle is given by X(t). If you want to know the unconditional mean in the volume, you compute it using all particles in your volume. For a conditional mean you simply discard all particles that do not fulfill your condition. So if you want to know the mean velocity of all particles that have a mass of 1, your condition is $< U | m = 1>$

I'm not sure if this is what you are doing, so please give more details if I'm not describing your problem correctly.

Also note that the error in computing the variance depends much on the numerical method used for the stochastic part of the Langevin equation, see e.g. the book of Kloeden and Platen.

 

[PhoenixWright]

I assume that you have a certain volume filled with particles governed by the Langevin equation. The state of each particle is given by X(t). If you want to know the unconditional mean in the volume, you compute it using all particles in your volume. For a conditional mean you simply discard all particles that do not fulfill your condition. So if you want to know the mean velocity of all particles that have a mass of 1, your condition is $< U | m = 1>$

I'm not sure if this is what you are doing, so please give more details if I'm not describing your problem correctly.

Also note that the error in computing the variance depends much on the numerical method used for the stochastic part of the Langevin equation, see e.g. the book of Kloeden and Platen.

Thank you for your answer.
Actually, the problem doesn't explain the physical situation we are solving. It only asks us to solve a certain Langevin equation with the initial condition $X(0)=1$, and then to calculate $<X(t)|x_0>, <X^2(t)|x_0>-(<X(t)|x_o>)^2, <X(t)X(0)>$. So, I guess we can think a situation like the one you have explained. And that's more or less what I can't understand, as I have a very specific solution for $X(t)$ (I used Euler-Maruyama aproximation for solving it, by the way), but it's not specified what $x_0$ is, so I don't know which condition I should impose in order to calculate $<X(t)|x_0>, <X^2(t)|x_0>-(<X(t)|x_o>)^2, <X(t)X(0)>$...