I have the stochastic process given by

$$ \dot{x} = ax + v$$

where $v$ is a zero-mean, Gaussian white noise signal. What is the autocorrelation of $x$?

I believe the process is to first find the solution of the ODE, which is

$$ x(t) = e^{at} x(0) + \int_0^t e^{a(t-\tau)} v(\tau)\, d\tau $$

Then, the autocorrelation function is

$$R_{xx} = \int x(t+\tau)x(t)\, dt$$

However, I think I need to use the statistical definition since I have a stochastic process

$$R_{xx} = E[x(t+\tau)x(t)]$$

But, I am not sure how to solve this.