# Continuity of the Random Variable Defining the Occupation Measure of Gaussian Process

Suppose $$Z:\Omega \times [0,1] \to \mathbb{R}$$ is a continuous Gaussian process with mean $$\mu(t)$$ and covariance kernel $$C(t,s)$$. Consider the random variable

$$X_\alpha = \lambda( \{t \; : \; Z(t) > \alpha \})$$

where $$\lambda$$ is Lebesgue measure. $$X_\alpha$$ is a random variable taking values in $$[0,1]$$. Another thing to note is that for many Gaussian processes the distribution of $$X_\alpha$$ will have mass at 0 and 1. My question is: under what conditions on $$C$$ is $$P( X_\alpha = x ) = 0, \mbox{ for all } x \in (0,1)?$$ Clearly this holds for many well known Gaussian processes, e.g. Brownian motion, and does not hold for many others (for example, suppose $$Z(t) = N\sin(2\pi t)$$, where $$N\sim \mathcal{N}(0,1)$$). I hoped there would be some result in the literature giving guidance on this, but I cannot find anything.

• This question is about continuity, not absolute continuity Commented Feb 19, 2021 at 18:15
• Thanks for the comment! I have updated the title. Commented Feb 19, 2021 at 18:24