Recent content by Tyler_D

Thanks Fred. But this is just a generic solution to the Ornstein-Uhlenbeck equation as far as I can tell? The question is, how do I go from this coupled system of SDEs I have above to an SDE in a single variable that I can solve. Or am I misunderstanding you? Hope you can say a bit more about...

There's no ##dy/dt## in the second equation. I could of course divide by ##dt##, but I don't believe we are allowed to as ##dW## is not differentiable due to it being Wiener noise (?)

In order to solve for ##x##, I need to re-write the equation for ##dx## so it is independent of ##y## and ##dy##. However, I am having some issues with this. Can someone give me a push in the right direction?

That's also a typo, thanks for spotting it. The upper limit in the last integral should naturally be ##\tau+t##, not just ##\tau##.

It is the Wiener-integral/Brownian motion (my bad, should've written it in the OP): ## W(t) = \int_0^t dW(s) ##

Here is my attempt: $$ \begin{align} \langle x(t)x(t+\tau) \rangle &= \langle (e^{-\beta W(t)}W(t))(e^{-\beta W(t+\tau)}W(t+\tau))\rangle \\ &= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})(W(t)W(t+\tau))\rangle \\ &= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})(\int_0^t...