Probability density function of Gaussian noise I am wondering if a probability density function on a stochastic process can be defined. I've been searching around but what I've seen so far are only finite-dimensional distributions of a stochastic process. What's I'm interested in is something else -- can a probability density function be defined on an infinite-dimensional space such as the samples of a stochastic process?
Until recently, I came across a paper which says that if $\eta(t)$ is a Gaussian noise process (i.e. white noise) such that $E[\eta(t)] = 0$ and $E[\eta(t) \eta(t')] = D \delta(t-t')$ then the "formal" probability density for this process is given by
$P(\eta(t)) \propto \exp(-\frac{1}{2D} \int_{t_i}^{t_f} \eta^2 \,dt)$ for time points $t_i,t_f$. 
I cannot find a reference for this and am unsure if the authors quoted "Quantum Mechanics and Path Integrals" by Feynman and Hibbs because the reference citation appeared to allude to a previous statement.
In any case, it appears that this density function was formulated by physicists. Can anyone direct me to mathematical papers which discuss this concept and how to make sense of it, somewhat more rigorously?
 A: Notation in the given expression aside, here is an attempt at interpreting it in relation to infinite-dimensional distributions, as mentioned in your first paragraph. 
(Real) Gaussians with the given time-independent mean and delta-correlated covariance define a stationary process, i.e., independent of $t_i$. For any pair $(t_i,t_j)$, the two associated Gaussian variates $\eta_i=\eta(t_i)$ and $\eta_j\equiv\eta(t_j)$ are uncorrelated r.v. Hence, being Gaussians, they are mutually independent, so that their joint 2-dimensional probability density is their product
\begin{equation}
p(\eta_i,\eta_j;t_i,t_j) \propto \exp \left ( -\frac{\eta^2_1}{2\sigma^2_1} \right ) \cdot \exp \left  ( - \frac{\eta^2_2}{2\sigma^2_2} \right ) = \exp \left ( -\frac{\eta^2_1+\eta^2_2}{2D} \right )
\end{equation}
with $\sigma^2_1=\sigma^2_2 = E(\eta^2_1)=E(\eta^2_2) = D$ (taking $t=t^\prime$ above)  because $E(\eta_1)=E(\eta_2)=0$. 
Repeating and extending this for $3, 4, \ldots, n < \infty$ time points, one gets a (finite) $n$-dimensional hierarchy of $n$-dimensional joint Gaussian density functions. For continuous $t$, one can let $n$ approach infinite, thus producing an infinite-dimensional joint pdf (i.e., for all uncountable infinite $t$ within $[t_i,t_f]$ or $[-\infty,+\infty]$) and hence a stochastic process as a probability functional on $t$. In general, this requires all 2-, 3-, ..., $n$-point correlations to be specified and known (a serious difficulty in practice). For real Gaussians, only the two-point correlations are necessary and sufficient to define the process completely.
Returning to the formula in your post, since all $\eta_i$ are uncorrelated no matter how small the mutual distances $|t_i-t_j|$, one can take them arbitrarily close. Thus, $\eta^2_1+\eta^2_2+\ldots + \eta^2_n$ in the numerator of the exponential (in the extension of the expression above to $n$ dimensions) approaches a Riemannian sum and becomes an integral over $t$ when considering all points $t$ in a given interval $[t_i,t_f]$.
