A stochastic representation for mean curvature type geometric flows
A stochastic representation for mean curvature type geometric flows
复制标题
平均曲率型几何流的随机表示
DOI：
10.1214/aop/1055425773
复制
发表时间：
2003
影响因子：
2.3
通讯作者：
H. Soner
中科院分区：
文献类型：
--
作者：
N. Touzi;H. Soner
抛物几何流的光滑解{Γ(T)}t∈[0，T]⊂\Rd被刻画为随机目标问题的可达集。在这个控制问题中，控制器试图以概率1将状态过程引导到给定的确定集T_c。目标问题的可达性集合V(T)是所有初始数据x的集合，其中状态过程xx(T)∈是某个控制过程ν的状态过程。通过研究Γ(T)的平方距离函数，证明了这一表象。对于余维k平均曲率流，状态过程为dx(T)=2√pdw(T)，其中W(T)为d维布朗运动，控制矩阵P为(d−k)维平面上的任意投影矩阵.给出了逆平均曲率流的光滑解和非光滑解的讨论。
A smooth solution {Γ(t)}t∈[0,T]⊂\Rd of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem. In this control problem the controller tries to steer the state process into a given deterministic set \Tc with probability one. The reachability set, V(t), for the target problem is the set of all initial data x from which the state process \xx(t)∈\Tc for some control process ν. This representation is proved by studying the squared distance function to Γ(t). For the codimension k mean curvature flow, the state process is dX(t)=2√PdW(t), where W(t) is a d-dimensional Brownian motion, and the control P is any projection matrix onto a (d−k)-dimensional plane. Smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given.