# On laws exhibiting universal ordering under stochastic restart

## TL;DR

This paper characterizes the laws of random times that are unaffected or dominated by various restart strategies, revealing universal ordering properties and differences among reset types.

## Contribution

It provides a comprehensive characterization of laws invariant under different stochastic and deterministic restart mechanisms, including partial results for reset with branching.

## Key findings

- Deterministic and arbitrary stochastic restart share the same laws of invariance.
- Exponential (constant-rate) reset behaves differently from deterministic and stochastic resets.
- Partial results extend the analysis to reset with branching.

## Abstract

For each of (i) arbitrary stochastic reset, (ii) deterministic reset with arbitrary period, (iii) reset at arbitrary constant rate, and then in the sense of either (a) first-order stochastic dominance or (b) expectation (i.e. for each of the six possible combinations of the preceding), those laws of random times are precisely characterized that are rendered no bigger [rendered no smaller; left invariant] by all possible restart laws (within the classes (i), (ii), (iii), as the case may be). Partial results in the same vein for reset with branching are obtained. In particular it is found that deterministic and arbitrary stochastic restart lead to the same characterizations, but this equivalence fails to persist for exponential (constant-rate) reset.