Expectation value of exponential of a function - first two moments of function known

Question

I have a function $f(t)$ and know that $\langle f(t)\rangle=0$ and $\langle f(t)f(t')\rangle=C(t-t')$;

Now i want to calculate:

$$\left\langle\exp\left(\int\limits_{0}^t f(t') \, \mathrm{d}t'\right)\right\rangle$$

I tried to look at the sum definition of the exponential but could do anything...

Any help?

Answer 1

I think what you mean is that $f$ is a stochastic process (rather than a function) of mean $0$ and covariance $C(t - t')$. If it's a Gaussian process, that's all you need to determine it, but if it's non-Gaussian you really don't know enough to say anything about expectations of exponentials. They might not even exist.