Uncertainty Propagation in Stochastic Hybrid Systems with Dimension-Varying Resets

TL;DR

This paper introduces a unified weak-form framework for modeling probability density evolution in stochastic hybrid systems with dimension-changing resets, overcoming limitations of traditional boundary-condition methods.

Contribution

It develops a novel weak-form formulation that captures probability flux across guards, including cases where resets change the dimension of the state space.

Findings

The framework accurately models probability transfer for dimension-varying resets.

It demonstrates the approach with a particle merging and splitting example.

The method captures interior source densities and singular sources on lower-dimensional sets.

Abstract

This paper studies probability density evolution for stochastic hybrid systems with reset maps that change the dimension of the continuous state across modes. Existing Frobenius--Perron formulations typically represent reset-induced probability transfer through boundary conditions, which is insufficient when resets map guard sets into the interior or onto lower-dimensional subsets of another mode. We develop a weak-form formulation in which reset-induced transfer is represented by the pushforward of probability flux across the guard, yielding a unified description for such systems. The proposed framework naturally captures both cases: when the reset decreases dimension, the transferred probability appears as an interior source density, whereas when the reset increases dimension, it generally appears as a singular source supported on a lower-dimensional subset. The approach is illustrated using a stochastic hybrid model in which two particles merge into one and later split back into two, demonstrating how dimension-changing resets lead to source terms beyond classical boundary-condition-based formulations.