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 angels 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.

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.

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 angels 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: $\delta(t-\pi/4) + \delta(t +\pi/4)$, 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.

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 angels 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.