GaussianProcess

The infinitesimal generator of a stochastic process is indeed closely related to its covariance function, and this could potentially provide a way to investigate its zeros.

Let's make clear that the self-adjoint infinitesimal generator is essentially the differential operator associated with the Gaussian process. Now, if a function is a zero of the Gaussian process, meaning that it is orthogonal to all other functions in the function space (in a probabilistic sense), then it should be annihilated by the infinitesimal generator.

Mathematically, this would mean that if the Gaussian process is defined by a stochastic differential equation (SDE) of the form:

$$dZ(t) = A(Z(t))dt + B(Z(t))dW(t)$$

where $dW(t)$ is a Wiener process, $A(Z(t))$ is the drift term, and $B(Z(t))$ is the diffusion term, then a zero $f(t)$ of the Gaussian process should satisfy:

$$A(f(t)) + B(f(t)) = 0$$

This is essentially saying that $f(t)$ is a deterministic path that is unaffected by the noise term $dW(t)$. This is a very strong condition, and not all Gaussian processes will have non-trivial solutions. But if they do, then these zeros would be real-valued functions, as they are solutions to a deterministic differential equation.

Again, this is quite a high-level explanation and the specifics will depend heavily on the nature of the Gaussian process, especially the form of the functions $A(Z(t))$ and $B(Z(t))$.

Finally, I would like to note that this topic is a very advanced one and a full treatment would be beyond the scope of this platform. If you need a more detailed explanation or if you are working with a specific Gaussian process, I would suggest consulting a specialist or referring to a detailed textbook or research article on the subject.

Non-tangency of Level Crossing Functionals of Continuously Differentiable Gaussian Processes

In the context of level crossings for Gaussian processes, for a Gaussian process $X(t)$ that is continuously differentiable, the phenomenon of the process becoming tangent to a specific level $x$ is noteworthy. Intuitively, if we consider the number of times the process crosses this level $x$ within a certain interval, the event where $X(t) = x$ and its rate of change $\dot{X}(t)$ is zero (indicating tangency) has a probability of zero. This can be understood as the unlikelihood of the process not only reaching a specific level but also having a derivative of zero at that exact moment in a continuous setting.

Given this context, the joint probability of the two events can be written as:

$$\mathbb{P} [X(s) = x \cap \dot{X}(s) = 0] = 0$$