Uncertainty Propagation in Stochastic Hybrid Systems with Dimension-Varying Resets
This work addresses a theoretical gap in stochastic hybrid systems modeling for researchers in control theory and applied mathematics, though it appears incremental as it extends existing formulations to handle dimension-varying resets.
The paper tackles the problem of modeling probability density evolution in stochastic hybrid systems where reset maps change the state dimension across modes, developing a weak-form formulation that unifies cases of dimension decrease and increase by representing reset-induced transfer as a pushforward of probability flux. The result is demonstrated through a model where particles merge and split, showing how dimension-changing resets lead to source terms not captured by existing boundary-condition methods.
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.