论文标题
马尔可夫的解决方案
Markovian Solutions to Discontinuous ODEs
论文作者
论文摘要
鉴于可能是不连续的，有限的函数$ f：\ mathbb {r} \ mapsto \ mathbb {r} $，我们考虑了通过在carathéodorysolutions上为ode〜$ \ dot x = x = f（x）$分配一组概率度量来获得的广义流。该论文提供了所有此类流量具有及时属性的这些流量的完整表征。 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 $θ_k\in [0,1]$, describing the probability of moving up or down,在孤立的点，两个不同的轨迹可以起源。
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 $θ_k\in [0,1]$, describing the probability of moving up or down, at isolated points where two distinct trajectories can originate.
