At first, hi everyone. I'm having difficulties looking for information about the so-called 'characteristic functional' of a random process.
What I know about this concept is what follows: when you have a random variable, let it be $X$, it is possible to describe its statistics by means of the probability density function, the distribution function or the Fourier transform of the former (the characteristic function). Thus, we define the caracteristic function of $X$ as follows: $$ G_X(k) := \int e^{ikx} P_X(x) \, \textrm{d}x \, . $$
When you are dealing with stochastic processes, characteristic function is available, but characteristic functional is supposed to be a more natural way to describe the process. This object is defined as: $$ G_X[k(t)] := \left\langle e^{i\int k(t) X(t) \, \textrm{d}t} \right\rangle \, , $$ where the average is taken over the stochastic process's values (if X(t) is the process, x are its possible values), and $t$ is the 'deterministic' parameter of the process.
I have been searching more information (some formal definition, some example,...) for a while and I have not found nothing. I would be greatful if somebody could tell me about a book where I colud find out more about this concept or gave me an example of its use (something like get the functional of Wiener process). I'm also interested in finding out about Volterra series (functional series).