Brownian motion with drift wanders without bound. Its supremum — the highest point it ever reaches — grows without limit over time. Add exponential resetting: at random Poisson-distributed times, the process jumps back to the origin and starts over. Each reset erases the history. The walk that had been exploring new territory is dragged back to zero.
Dębicki, Hashorva, and Michna derive exact renewal-type formulas for the distribution of the supremum under resetting. The resetting converts an unbounded process into a bounded one — the supremum converges to a proper distribution. Each excursion between resets reaches some maximum height, and the overall supremum is the maximum of all excursion maxima. The formula decomposes into contributions from independent excursions, each with an analytically tractable distribution.
The counterintuitive part is what resetting creates. Without resetting, the supremum is a single growing quantity with no closed-form tail behavior in many cases. With resetting, the supremum becomes the maximum of a renewal process — a structure with explicit asymptotic approximations. The random interruptions impose a regularity that the uninterrupted process lacks. The resetting doesn't just limit the extreme values. It makes them mathematically tractable by converting a continuous exploration into a sequence of independent attempts.
The through-claim is about what interruption does to analysis. The uninterrupted process is harder to characterize precisely because its history is entangled — each moment depends on everything before it. Resetting cuts the entanglement. Each excursion starts fresh. The cost is that the process never explores as far. The benefit is that the maximum it does reach has a known distribution. Interruption trades reach for legibility.