The Splitting Strain

Stochastic interpolation between probability distributions — the Schrödinger bridge — produces a velocity field that transports one distribution smoothly into another. When the target distribution is nonconvex or disconnected, the interpolating paths must split: a single particle cloud must divide into separate clusters at some point during transport.

arXiv:2603.16618 proposes using Jacobi fields to detect exactly when and where this splitting begins. The spatial Jacobian of the Eulerian velocity field reveals the local geometry of the posterior distribution. Its symmetric component functions as a strain tensor — the same object that measures deformation in continuum mechanics.

The spectral properties of this strain tensor quantify how perturbations amplify along trajectories. When the largest eigenvalue grows, nearby trajectories diverge — the flow is stretching. The location and time of maximal stretching identify where the distribution is about to split. The splitting is not a discrete event but a continuous stretching that reaches a critical threshold.

The connection to Jacobi fields is not metaphorical. In Riemannian geometry, Jacobi fields describe the infinitesimal separation between neighboring geodesics. In the Schrödinger bridge, the analogous object describes the infinitesimal separation between neighboring transport paths. Where geodesics diverge, the geometry is negatively curved. Where transport paths diverge, the target distribution has separated modes.

The method localizes both the spatial origin and the temporal onset of branching in complex target distributions. It transforms splitting detection from a global topological question — does the distribution become disconnected? — into a local differential question — is this strain tensor about to blow up?