Equivalent of Density of Stochastic Process

Question

Consider the case where $X$ is a random variable defined on a space of functions say $C[0,1]$. Then each sample $X(\omega)$ is a path of stochastic process on $[0,1]$ as opposed to the usual case where $X(\omega)$ is simply a number or vector. Is it possible to have an equivalent of probability density function for such $X$ so that we can kind of measure probability of $X$ taking certain paths? For example, $\int_Gf_X(g)dg\in \mathbb{R}$ would be probability of $X$ realizes to one of the collection of paths $G$ where each $g \in G$ is a path on $[0,1]$.

Answer 1

Talking about densities requires you to fix a reference measure that your density is against. You implicitly do this when you write an integral against $dg$. The problem is that it's not obvious what you mean by $dg$.

In finite dimensions we have a somewhat canonical choice of reference measure, the Lebesgue measure. However, there is no good analogue of the Lebesgue measure in infinite dimensions. Indeed, we have

Lemma: Let $X$ be an infinite dimensional normed space and let $\mu$ be a translation invariant measure on the Borel $\sigma$-algebra of $X$. Then $\mu(B(0,1)) = \infty$.

Unfortunately, this means that densities become less useful in the infinite dimensional setting.

Of course, we could fix some measure on $C[0,1]$ and ask if there is a density against that measure but then it's not obvious what kind of measure you'd want to fix and why that would be useful here.