- #1

Tyler_D

- 6

- 0

- Homework Statement
- Calculate the auto-correlation of ##x(t)=e^{-\beta W(t)}W(t)##

- Relevant Equations
- ##\langle x(t)x(t+\tau)\rangle##, ##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 dW(t)\int_0^{t+\tau}dW(t))\rangle \\

&= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})\int_0^t dW(t)(\int_0^{t}dW(t)+\int_t^{\tau}dW(t))\rangle

\end{align}

$$

From here on I am quite lost on what to do with the exponentials. Do I Taylor-expand them, or what is the best thing to do?

$$

\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 dW(t)\int_0^{t+\tau}dW(t))\rangle \\

&= \langle (e^{-\beta W(t)}e^{-\beta W(t+\tau)})\int_0^t dW(t)(\int_0^{t}dW(t)+\int_t^{\tau}dW(t))\rangle

\end{align}

$$

From here on I am quite lost on what to do with the exponentials. Do I Taylor-expand them, or what is the best thing to do?