Abstract:We study the regularity of the invariant density of 1D piecewise-deterministic Markov processes. Such densities are regular away from the zeros of the driving vector fields, but may become singular at these points, a phenomenon so far understood only when a single field vanishes. In the case of several simultaneously vanishing vector fields, we identify a new threshold on the jump rates separating continuity from divergence, which emerges from the interplay of all vanishing fields. Away from the zeros of the vector fields, we turn to distribution theory and the Kolmogorov forward equation. We show $C^r$ regularity of the density on intervals where the vector fields are $C^r$ and do not vanish, which is optimal, allowing $C^{r-1}$ position-dependent jump rates, as well as resetting contributing a $C^{r-1}$ source term to the Kolmogorov forward equation. For $r = 0$, it suffices that the jump rates be measurable and the resetting source be absolutely continuous with respect to Lebesgue measure. As an application, we study the shape transition of two run-and-tumble particle systems whose invariant measures cannot be computed explicitly, showing that shape transition does not depend on exact solvability.
From: Leo Hahn [view email]
[v1]
Mon, 2 Jun 2025 21:32:57 UTC (31 KB)
[v2]
Thu, 26 Jun 2025 15:06:17 UTC (32 KB)
[v3]
Fri, 21 Nov 2025 12:41:34 UTC (45 KB)
[v4]
Wed, 19 Aug 2026 10:08:48 UTC (33 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。