# Differential Equation with Noise term

1. Oct 22, 2014

### MisterX

1. The problem statement, all variables and given/known data
$$(\partial_t - A\nabla^2 + B)f(x) = \eta(x, t)$$
So I have a homogeneous linear differential equation except for an added noise term $\eta(t)$. The noise is uncorrelated between times and has a Gaussian distribution with zero mean. That is we have Gaussian white noise.

The problem is to calculate the time dependence of f(x,t) and also the time dependence of $\left| F(k, t)\right|^2$, where F(k, t) is the Fourier transform of f(x, t).

2. Relevant equations

3. The attempt at a solution

If I know the specific function for the noise, I can solve using the Green's function - but I do not know $\eta(t)$.
What can I say about the behavior of the solution? If you like, the noise starts at t=0, and we are interested in some kind of transient response I suppose. I began to proceed with solving the Green's function in Fourier space (that is x -> k, but t remained unchanged) but I am not sure what to do next.

I am wishing I knew or remember more I guess.

2. Oct 22, 2014

### Simon Bridge

3. Oct 22, 2014

### MisterX

This is actually for a different course, we haven't been taught at all about random processes. However, years ago when I was getting my bachelors as an engineer I learned a little bit. I'd like to be able to do a nice job with this but I don't have time for a huge amount of extra learning.