Is it true that for any Gaussian process $x(t)$ governed by a stochastic differential equation, such as the Ornstein-Uhlenbeck (O-U) process, with mean zero, the expected value of any odd function $f(\Phi(L))$ of its integral $\Phi(L) = \int_0^L x(t) \, dt$ is also zero, where the expectation is taken over all possible realizations of such processes? Specifically, since $x(t)$ is symmetrically distributed around zero, does it follow that $\mathbb{E}[f(\Phi(L))] = 0$ for any odd function $f$?
As an example, the O-U process is defined by the stochastic differential equation $dx(t) = -\theta x(t) dt + \sigma dW(t)$, where $W(t)$ is a Wiener process, and it has a stationary distribution with mean zero. In the case where $f(\Phi(L)) = \sin \Phi(L)$, does this symmetry (same number of positive and negative values for each $t$) guarantee that $\mathbb{E}[\sin \Phi(L)] = 0$? More generally, can we conclude that for any mean-zero O-U process, the expectation of any odd function $f(\Phi(L))$ is always zero, as suggested by the example with $\sin \Phi(L)$?