Persistence and ball exponents for Gaussian stationary processes

Consider a real Gaussian stationary process (Formula presented.), indexed on either (Formula presented.) or (Formula presented.) and admitting a spectral measure (Formula presented.). We study (Formula presented.), the persistence exponent of (Formula presented.). We show that, if (Formula presented.) has a positive density at the origin, then the persistence exponent exists; moreover, if (Formula presented.) has an absolutely continuous component, then (Formula presented.) if and only if this spectral density at the origin is finite. We further establish continuity of (Formula presented.) in (Formula presented.), in (Formula presented.) (under a suitable metric) and, if (Formula presented.) is compactly supported, also in dense sampling. Analogous continuity properties are shown for (Formula presented.), the ball exponent of (Formula presented.), and it is shown to be positive if and only if (Formula presented.) has an absolutely continuous component. [Correction added on 26 June 2025, after first online publication: In the previous sentence, in the definition, “inf” was corrected to “sup”.].