论文标题
根据其第一义属性来调节两个扩散过程
Conditioning two diffusion processes with respect to their first-encounter properties
论文作者
论文摘要
我们考虑了两个独立的相同扩散过程，它们会在开会时歼灭，以研究其初次遇到特性的条件。对于有限的地平线$ t <+\ infty $，最大条件是强加$ p^*（x，y，t）$的概率，即两个粒子在$ x $ x $ x $ t $ t $上存活的概率$ t $，以及概率$γ^*（z，t）$ z $ z $ z $ z $ t]的概率$γ^*（z，t）$ t] $ t] $ t]。通过优化适当的相对熵相对于无条件的过程，分析了对各种条件约束的适应。对于无限范围$ t =+\ infty $的情况，最大条件在于在所有有限时间$ z $ at posite $γ^*（z，t）$在所有有限时间$ t \ in [0，+\ infty [$ in in++h infty [$ in+z $）的正常化$ [1- s^*（\ infty] $ n n in n n herfty time $ t \ in n in [0+\ infty] $ s $ s $ s。永远存在。然后将该一般框架应用于明确的情况，无条件的过程分别是两个布朗动作，两个Ornstein-Uhlenbeck过程，或两个Tanh-Drift过程，以产生满足各种类型的调理约束的随机轨迹。最后，通过在存在条件约束的情况下，通过优化2.5级的动力大偏差来描述与随机控制理论的联系。
We consider two independent identical diffusion processes that annihilate upon meeting in order to study their conditioning with respect to their first-encounter properties. For the case of finite horizon $T<+\infty$, the maximum conditioning consists in imposing the probability $P^*(x,y,T ) $ that the two particles are surviving at positions $x$ and $y$ at time $T$, as well as the probability $γ^*(z,t) $ of annihilation at position $z$ at the intermediate times $t \in [0,T]$. The adaptation to various conditioning constraints that are less-detailed than these full distributions is analyzed via the optimization of the appropriate relative entropy with respect to the unconditioned processes. For the case of infinite horizon $T =+\infty$, the maximum conditioning consists in imposing the first-encounter probability $γ^*(z,t) $ at position $z$ at all finite times $t \in [0,+\infty[$, whose normalization $[1- S^*(\infty )]$ determines the conditioned probability $S^*(\infty ) \in [0,1]$ of forever-survival. This general framework is then applied to the explicit cases where the unconditioned processes are respectively two Brownian motions, two Ornstein-Uhlenbeck processes, or two tanh-drift processes, in order to generate stochastic trajectories satisfying various types of conditioning constraints. Finally, the link with the stochastic control theory is described via the optimization of the dynamical large deviations at Level 2.5 in the presence of the conditioning constraints that one wishes to impose.
