We extend Monte Carlo samplers based on piecewise deterministic Markov processes (PDMP samplers) by formally defining different boundary conditions such as sticky floors, soft and hard walls and teleportation portals. This allows PDMP samplers to target measures with piecewise-smooth densities relative to mixtures of Dirac and continuous components and measures supported on disconnected regions or regions which are difficult to reach with continuous paths. This is achieved by specifying the transition kernel which governs the behaviour of standard PDMPs when reaching a boundary. We determine a sufficient condition for the kernel at the boundary in terms of the skew-detailed balance condition and give concrete examples. The probabilities to cross a boundary can be tuned by introducing a piecewise constant speed-up function which modifies the velocity of the process upon crossing the boundary without extra computational cost. We apply this new class of processes to two illustrative applications in epidemiology and statistical mechanics.

翻译：我们通过正式定义不同边界条件（如粘性地板、软硬壁及瞬间传送门），对基于分段确定性马尔可夫过程（PDMP采样器）的蒙特卡洛采样器进行扩展。这使得PDMP采样器能够针对具有分段光滑密度的测度（这些测度由狄拉克分量与连续分量混合而成），以及支撑在由连续路径难以到达的不连通区域或区域上的测度进行采样。其实现方式是通过指定标准PDMP在抵达边界时的行为转移核。我们基于偏斜细致平衡条件推导了边界处转移核的充分条件，并给出了具体实例。通过引入分段常数加速函数（在穿越边界时调整过程速度且不增加额外计算成本），可以调节穿越边界的概率。我们将这类新型过程应用于流行病学与统计力学中的两个示范性案例。