Abstract
Stochastic processes of evolving shapes are used in applications including evolutionary biology, where morphology changes stochastically as a function of evolutionary processes. Due to the non-linear and often infinite-dimensional nature of shape spaces, the mathematical construction of suitable stochastic shape processes is far from immediate. We define and formalize properties that stochastic shape processes should ideally satisfy to be compatible with the shape structure, and we link this to Kunita flows that, when acting on shape spaces, induce stochastic processes that satisfy these criteria by their construction. We couple this with a survey of other relevant shape stochastic processes and show how bridge sampling techniques can be used to condition shape stochastic processes on observed data thereby allowing for statistical inference of parameters of the stochastic dynamics.
Abstract (translated)
在包括进化生物学在内的各种应用中，会使用到形状演化的随机过程。在这种情况下，形态作为进化过程中函数的输出会发生随机变化。由于形状空间具有非线性和通常无限维的特性，构建合适的随机形状过程从数学上来说是相当复杂的。 我们定义并正式化了理想中的随机形状过程应具备的性质，以确保这些过程与形状结构相兼容，并且我们将此概念与Kunita流联系起来，当这种流作用于形状空间时，会构造出满足上述条件的随机过程。此外，我们将介绍其他相关的形状随机过程，并展示如何使用桥抽样技术来根据观察到的数据对形状随机过程进行条件化处理，从而允许对这些动态系统的参数进行统计推断。
URL
https://arxiv.org/abs/2512.11676
