We consider a Piecewise Deterministic Markov Process given by random switching between finitely many vector fields vanishing at [Formula: see text]. It has been shown recently that the behavior of this process is mainly determined by the signs of Lyapunov exponents. However, results have only been given when all these exponents have the same sign. In this paper, we consider the degenerate case where the process leaves invariant some face and results are stated when the Lyapunov exponents are of opposite signs. Our results enable in particular to close a gap in a discussion on random switching between two Lorenz vector fields by Bakhtin and Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity 25 (2012) 2937–2952.

中文翻译：

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在公共不变面上共享零的随机切换向量场

我们考虑通过在[公式：见文本]处消失的有限多个向量场之间的随机切换给出的分段确定性马尔可夫过程。最近已经表明，这个过程的行为主要由李雅普诺夫指数的符号决定。然而，只有当所有这些指数都具有相同的符号时，才会给出结果。在本文中，我们考虑了退化的情况，即当 Lyapunov 指数的符号相反时，过程留下不变的一些面和结果。我们的结果尤其能够弥补 Bakhtin 和 Hurth 在关于随机切换两个洛伦兹矢量场之间的讨论中的空白，具有随机切换的动态系统的不变密度，非线性 25 (2012) 2937-2952。