math.PR — Probability

Curvature-Induced Killing at Stochastic Algebraic Folds

Contributed by Zhenyu Sun

We study the lifetime of a regular algebraic branch under continuous forcing and infinite-activity tangent jumps. For a parabolic fold, a first-order tangent kick decreases the discriminant by an exact quadratic amount. Symmetric stable marks of index $\alpha\in(0,2)$ therefore generate a one-sided normal jump kernel of index $q=\alpha/2$, with a coefficient determined by curvature. The normal diffusion and jump amplitudes depend on the evolving tangential state. We compute the full killed generator on powers of the discriminant and obtain occupation and terminal-jump bounds for this same joint process. Uniformly subcritical normal drift gives finite censored boundary lifetime and finite cumulative killing rate; continuous contact and jump exit both have positive probability. At and above the uniform critical threshold, a logarithmic nonlocal barrier excludes censored boundary contact, and the original branch terminates at a jump almost surely. A comparison after a stochastic time change yields an explicit stretched-exponential upper bound for survival and hence all positive lifetime moments. In the subcritical regime, an optimized upper bound for the probability of jump termination tends to zero as the curvature coefficient or the initial discriminant tends to zero. The analysis combines power estimates for the jump generator with a logarithmic barrier at the critical threshold. Numerical calculations illustrate the dependence of terminal causes on curvature.