Title: The most likely evolution of diffusing and vanishing particles: Schrodinger Bridges with unbalanced marginals Show Chinese title

Title: 扩散和消失粒子最可能的演化：边缘不平衡的薛定谔桥

Abstract: Stochastic flows of an advective-diffusive nature are ubiquitous in physical sciences. Of particular interest is the problem to reconcile observed marginal distributions with a given prior posed by E. Schrodinger in 1932/32 and known as the Schrodinger Bridge Problem (SBP). Due to its fundamental significance, interest in SBP has in recent years enticed a broad spectrum of disciplines. Yet, while the mathematics and applications of SBP have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention; the problem to interpolate between unbalanced marginals has been approached by introducing source/sink terms in an Adhoc manner. Nevertheless, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schrodinger's dictum; that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated law represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this work is to develop such a natural generalization of the SBP for stochastic evolution with losses, whereupon particles are "killed" according to a probabilistic law. Through a suitable embedding, we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism, given a prior law that allows for losses, we ask for the most probable evolution of particles along with the most likely killing rate as the particles transition between the specified marginals. Our approach differs sharply from previous work involving a Feynman-Kac multiplicative reweighing of the reference measure: The latter, as we argue, is far from Schrodinger's quest. Show Chinese abstract

Abstract: 具有平流-扩散性质的随机流在物理科学中无处不在。 特别感兴趣的问题是调和观察到的边缘分布与E. Schrodinger于1932年提出的先验问题，这个问题被称为Schrodinger桥问题（SBP）。 由于其根本的重要性，近年来SBP引起了广泛的学科兴趣。 然而，尽管SBP的数学和应用发展迅速，但考虑质量不相等的边缘分布却很少受到关注；在非平衡边缘之间插值的问题通过以Adhoc方式引入源/汇项来解决。 然而，损失在许多物理过程中是固有的，因此，可能也需要根据Schrodinger的教条来调整考虑损失运输的模型，即调整粒子轨迹的概率，包括那些未能到达终端观测点的粒子，以便更新后的定律表示粒子在某个中间点被运输或消失的最可能方式。 因此，本工作的目的是开发一种自然的SBP广义形式，用于具有损失的随机演化，在这种情况下，粒子会根据概率定律“死亡”。 通过适当的嵌入，我们将问题转化为Schrodinger桥问题的一个扩展，涉及同时具有扩散和跳跃特性的随机过程。 然后，遵循大偏差形式主义，在允许损失的先验律下，我们寻求粒子最可能的演化路径以及在指定边缘之间过渡时最可能的死亡率。 我们的方法与以前涉及参考测度Feynman-Kac乘法重加权的工作截然不同：正如我们所论证的那样，后者远非Schrodinger的追求。