Local-Time Sensitivity and Burst Instability for Threshold Functionals of One-Dimensional Diffusions

Let X=Xt0≤t≤T be a real-valued continuous process. For a threshold a, the sub-threshold time set ET(a)={t∈[0,T]:Xt≤a} encodes several different threshold observables. The most elementary one is the cumulative occupation time AT(a)=∫0T1{Xt≤a}dt. For a regular one-dimensional diffusion, the classical occupation density formula gives AT(a)=∫−∞aLTy(X)σ2(y)dy, and hence ∂AT∂a(a)=LTa(X)σ2(a). Thus additive threshold occupation admits a local-time sensitivity calculus. In the terminology of barrier contracts, this additive clock is the cumulative, non-resetting Parisian clock, also called the Parasian clock. The purpose of this paper is to contrast this additive/Parasian regime with the behavior of resetting Parisian burst functionals. The connected components of ET(a) represent sub-threshold episodes. We study in particular the longest burst MT(a)=sup{|I|:IisaconnectedcomponentofET(a)}. While AT is locally controlled by local time, MT is governed by the connectivity of the sub-threshold time set. We prove that MT is monotone, that its supremum is attained, and that the weak-sublevel version is right-continuous with left limits, while the strict-sublevel version is its left-continuous regularization. The jump at a level is the increase in the maximal connected-component length produced by adjoining the level set. This gives a deterministic càdlàg/càglàd calculus for longest-burst profiles. For regular one-dimensional diffusions, this yields a sharp structural contrast. At deterministic levels which are almost surely not local-extreme values, the weak and strict longest bursts agree almost surely. Whenever the path has a unique interior maximum, the level-indexed longest-burst profile has a positive jump at the maximum level and is therefore not absolutely continuous. Brownian motion satisfies this criterion almost surely. We further identify the deterministic mechanism behind this instability: small threshold increases may fill short temporal bridges and merge large sub-threshold components. Finally, we show that the longest burst is exactly a one-sided continuous Parisian functional. This yields an exact Laplace-transform representation of its Brownian law through the Chesney–Jeanblanc-Picqué–Yor Parisian transform, and an excursion-measure formulation in which local time enters only as the Itô excursion intensity. We also discuss smoothed burst statistics, moving thresholds, and diffusion examples. The paper is intended as a threshold-sensitivity comparison: local time controls cumulative Parasian occupation, whereas resetting Parisian burst observables are controlled by component mergers and excursion structure.