Stochastics of shapes and Kunita flows
This work addresses a foundational mathematical problem for researchers in evolutionary biology and shape analysis, though it appears incremental as it builds on existing Kunita flow theory.
The paper tackles the challenge of constructing mathematically rigorous stochastic processes for evolving shapes in applications like evolutionary biology, where shape spaces are nonlinear and infinite-dimensional. It defines compatibility criteria for such processes, links them to Kunita flows to satisfy these criteria, and demonstrates how bridge sampling enables statistical inference from observed data.
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.