Given a possibly discontinuous, bounded function \(f:{{\mathbb {R}}}\mapsto {{\mathbb {R}}}\), we consider the set of generalized flows, obtained by assigning a probability measure on the set of Carathéodory solutions to the ODE \(\dot{x} = f(x)\). The paper provides a complete characterization of all such flows which have a Markov property in time. This is achieved in terms of (i) a positive, atomless measure supported on the set \(f^{-1}(0)\) where f vanishes, (ii) a countable number of Poisson random variables, determining the waiting times at points in \(f^{-1}(0)\), and (iii) a countable set of numbers \(\theta _k\in [0,1]\), describing the probability of moving up or down, at isolated points where two distinct trajectories can originate.

给定一个可能不连续的有界函数\（f：{{\ mathbb {R}}} \ mapsto {{\ mathbb {R}}} \），我们考虑通过在上分配概率测度获得的广义流集ODE \（\ dot {x} = f（x）\）的Carathéodory解集。本文提供了具有马尔科夫性质的所有此类流的完整表征。这可以通过以下方式实现：（i）f消失的集合\（f ^ {-1}（0）\）上支持的正无原子量度；（ii）数量可观的泊松随机变量，确定等待时间在\（f ^ {-1}（0）\）中的点，和（iii）可数的一组数字\（\ [0,1] \中的theta _k \），描述了在两个不同的轨迹可能起源的孤立点处向上或向下移动的可能性。