Stochastics of shapes and Kunita flows

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.

💡 Research Summary

The paper “Stochastics of shapes and Kunita flows” addresses a fundamental challenge in modeling the stochastic evolution of morphological structures, a process central to fields such as evolutionary biology. The core difficulty lies in the intrinsic nature of “shape spaces,” which are characterized by non-linear, often infinite-dimensional, and non-Euclidean geometries. Standard stochastic processes, like Brownian motion in Euclidean space, fail to account for the complex topological and geometric constraints inherent in shape transformations.

To overcome this, the authors first establish a formal mathematical framework by defining the essential properties that a stochastic shape process must satisfy to remain compatible with the underlying shape manifold. The primary contribution of the paper is the introduction and application of “Kunita flows” to this problem. Kunita flows, which are stochastic flows of diffeomorphisms, provide a mathematically rigorous way to induce stochastic processes on shape spaces. By utilizing these flows, the authors demonstrate that one can generate stochastic trajectories that respect the geometric integrity of the shape space, ensuring that the transformations are smooth and structurally consistent.

Beyond the theoretical construction, the paper provides a comprehensive survey of existing stochastic shape processes, situating the proposed Kunata-flow-based approach within the broader mathematical landscape. A significant practical advancement presented in the work is the integration of “bridge sampling” techniques. This methodology allows for the conditioning of stochastic shape processes on empirical observations. In practical terms, if a researcher possesses morphological data from specific time points (such as fossilized remains or developmental stages), bridge sampling enables the reconstruction of the most probable stochastic paths between these observations.

This capability is crucial for statistical inference. By conditioning the process on observed data, the authors provide a framework for estimating the underlying parameters of the stochastic dynamics, such as the magnitude of morphological drift or the rate of evolutionary change. Consequently, the paper bridges the gap between high-level stochastic differential geometry and applied statistical inference. The implications of this research extend far beyond evolutionary biology, offering powerful new tools for any scientific discipline involving the analysis of time-varying, complex geometries, including computational anatomy, computer vision, and materials science.