Autocorrelation function of stochastic process

Question

Suppose that we have the full linear stochastic equation $$dx=-\frac{a^2}{2}xdt+axdW$$ where $a>0$ and $W(t)$ a Wiener process. Solving this equation with a change in variables $y=ln(x)$ i get $$x(t)=e^{aW(t)}$$ The autocorrelation function is $$g(τ)=<x(t)x(t+τ)>=<(e^{aW(t)})(e^{aW(t+t)})>=<e^{a(W(t)+W(t+τ))}>$$ I assumed that since the variables are independent and the last 2 terms are deterministic $$g(τ)=<e^{aW(t)}><e^{aW(t+τ)}>$$ I know that $$<e^{aW(t)}>=e^{at/2}$$ but can we assume that $<e^{aW(t+τ)}>=e^{a(t+τ)/2}$? Also, in the limit of $t{\rightarrow}{\infty}$ i should get a steady state, right?