Persistence and ball exponents for Gaussian stationary processes

Consider a real Gaussian stationary process , indexed on either or and admitting a spectral measure . We study , the persistence exponent of . We show that, if has a positive density at the origin, then the persistence exponent exists; moreover, if has an absolutely continuous component, then if and only if this spectral density at the origin is finite. We further establish continuity of in , in (under a suitable metric) and, if is compactly supported, also in dense sampling. Analogous continuity properties are shown for , the ball exponent of , and it is shown to be positive if and only if has an absolutely continuous component.