I would recommend obtaining a copy of the text

H. Cramer and M. R. Leadbetter (1967), Stationary and Related Stochastic Processes, John Wiley & Sons, Inc.

which is available as a 2004 Dover reprint, available, e.g., at the link above.

The whole book is a delight, but I think you will be particularly interested in the later chapters, for example,

• Chapter 10: "Crossing" problems and related topics
• Chapter 11: Properties of streams of crossings
• Chapter 12: Limit theorems for crossings
• Chapter 13: Nonstationary normal processes. Curve crossing problems

As a small taste of results from the text:

Lemma: Let $\xi(t)$ be a zero-mean stationary Gaussian process with almost-surely continuous sample paths. Let $C_u(t)$ denote the number of level crossings of the level $u \in \mathbb R$ in the interval $[0,t]$. Then, $$ \mathbb E C_u(t) = \frac{t}{\pi} \sqrt{\frac{\lambda_2}{\lambda_0}} e^{-u^2/2 \lambda_0}\\,, $$ where $\lambda_{2n}$, $n \geq 0$ denotes the $2n$th spectral moment $$ \lambda_{2n} = \int_0^\infty \lambda^{2n} \\,\mathrm dF(\lambda) $$ and $F$ is the spectral density function.

We can then relate this to the probability of a crossing via the following.

Lemma: Let $\xi(t)$ be a strictly stationary process with continuous one-dimensional distribution and almost-surely continuous sample paths. Suppose that $\mu := \mathbb E C_u(1) < \infty$. Let $q(t)$ denote the probability that at least one crossing occurs in time $t$. Then, $$ q(t) = \mu t + o(t) \\,, $$ as $t \to 0$.

In this way we see that the number of level crossings in one sense behaves roughly like a Poisson process with intensity $\mu$.

This only (barely) scratches the surface. The book covers much more and has additional (obviously somewhat dated) references as well. In particular, you may be interested in following classical work, which you may already be familiar with.

S. O. Rice (1945), Mathematical analysis of random noise, Bell System Tech. J., vol 24, pp. 46–156.

I would recommend obtaining a copy of the text

H. Cramer and M. R. Leadbetter (1967), Stationary and Related Stochastic Processes, John Wiley & Sons, Inc.

which is available as a 2004 Dover reprint, available, e.g., at the link above.

The whole book is a delight, but I think you will be particularly interested in the later chapters, for example,

• Chapter 10: "Crossing" problems and related topics
• Chapter 11: Properties of streams of crossings
• Chapter 12: Limit theorems for crossings
• Chapter 13: Nonstationary normal processes. Curve crossing problems

As a small taste of results from the text:

Lemma: Let $\xi(t)$ be a zero-mean stationary Gaussian process with almost-surely continuous sample paths. Let $C_u(t)$ denote the number of level crossings of the level $u \in \mathbb R$ in the interval $[0,t]$. Then, $$ \mathbb E C_u(t) = \frac{t}{\pi} \sqrt{\frac{\lambda_2}{\lambda_0}} e^{-u^2/2 \lambda_0}\,, $$ where $\lambda_{2n}$, $n \geq 0$ denotes the $2n$th spectral moment $$ \lambda_{2n} = \int_0^\infty \lambda^{2n} \,\mathrm dF(\lambda) $$ and $F$ is the spectral density function.

We can then relate this to the probability of a crossing via the following.

Lemma: Let $\xi(t)$ be a strictly stationary process with continuous one-dimensional distribution and almost-surely continuous sample paths. Suppose that $\mu := \mathbb E C_u(1) < \infty$. Let $q(t)$ denote the probability that at least one crossing occurs in time $t$. Then, $$ q(t) = \mu t + o(t) \,, $$ as $t \to 0$.

In this way we see that the number of level crossings in one sense behaves roughly like a Poisson process with intensity $\mu$.

This only (barely) scratches the surface. The book covers much more and has additional (obviously somewhat dated) references as well. In particular, you may be interested in following classical work, which you may already be familiar with.

S. O. Rice (1945), Mathematical analysis of random noise (broken link), Bell System Tech. J., vol 24, pp. 46–156.