Distribution of zeros and angles of a function with additive coloured noise

Question

Let us consider some real-variable function $$ f(t) = f_0(t) + \xi(t), $$ where $f_0$ - some "regular" (a continuously differentiable function without any noise [one can consider $f_0 = \text{const}$ for simplicity]), and $\xi(t)$ - Gaussian continuous noise with zero mean and variance $\sigma^2$. We also know the autocorrelation function $$ \langle \xi(t) \xi(t') \rangle = C(t-t'). $$ The function $C(t-t')$ can be pretty general, but let us assume it has the first derivative. We can also assume that the amplitude of the noise is relatively small: $\xi/f_0 \ll 1$.

Let us consider some interval $t \in (t_1, t_2) = T$. I am interested in two aspects.

1. What is the density of zeros of the function $f(t)$ in $T$? I.e. how many times it crosses the abscissa axis on average.
2. What is the distribution of the angles crossing the abscissa axis on average?

If my formulation is a bit confusing I will try to give an intuitive example. Let us consider a function which is almost a $\sin(t)$ but with very weak additive noise, i.e. we can think about it as a sine in general. Thus, I would expect the following answers to the questions above

1. $1/\pi$,
2. something like: $\frac{\delta(t-\pi/4) + \delta(t +\pi/4)}{2}$, or some sort of the similar result.

P.S. I am not familiar with this topic. Thus, if it is a well-known story, please let me know where I can read about this. I also thought about filtration and analysing the filtered function, but I am unsure if it is valid in this case.

Answer 1

since I don't see any other answers, I will turn comments into answer since they address the density issue.

For just general continuous stationary Gaussian process, there might not be any density because the zero sets can be fractal and singular to Lebesgue measure eg. see "The Exact Hausdorff Measure of the Zero Set of Certain Stationary Gaussian Processes".

If you add further regularity assumptions such C1 smoothness of the noise, then the zero set is discrete eg. "Zeros of smooth stationary Gaussian processes" and again we get singular.

But if you still want low regularity for the continuous noise, then I would suggest using Local time and average density techniques eg. "Holder conditions for the local times and the Hausdorff-measure of the level sets of Gaussian random fields"