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

Let \(X = \left( X_{t} \right)_{0 \leq t \leq T}\) be a real-valued continuous process. For a threshold \(a\), the sub-threshold time set \[E_{T}(a) = \{ t \in \lbrack 0,T\rbrack:X_{t} \leq a\}\] encodes several different threshold observables. The most elementary one is the cumulative occupation time \[A_{T}(a) = \int_{0}^{T}\mathbf{1}_{\{ X_{t} \leq a\}}\, dt.\] For a regular one-dimensional diffusion, the classical occupation density formula gives \[A_{T}(a) = \int_{- \infty}^{a}\frac{L_{T}^{y}(X)}{\sigma^{2}(y)}\, dy,\] and hence \[\frac{\partial A_{T}}{\partial a}(a) = \frac{L_{T}^{a}(X)}{\sigma^{2}(a)}.\] 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 \(E_{T}(a)\) represent sub-threshold episodes. We study in particular the longest burst \[M_{T}(a) = \sup\{|I|:I\text{ is a connected component of }E_{T}(a)\}.\] While \(A_{T}\) is locally controlled by local time, \(M_{T}\) is governed by the connectivity of the sub-threshold time set. We prove that \(M_{T}\) 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 [1] 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.