Expected Signature on a Riemannian Manifold and Its Geometric Implications

TL;DR

This paper demonstrates how the expected signature of Brownian paths on a Riemannian manifold encodes geometric information, enabling reconstruction of distances and curvature properties from stochastic data.

Contribution

It introduces a method to recover the Riemannian distance and curvature tensors from the asymptotics of expected signatures of Brownian motion and loops.

Findings

Distance function reconstructed from expected signature asymptotics.

Curvature properties derived from fourth level expected signature.

Intrinsic PDE for expected Brownian signature dynamics established.

Abstract

On a compact Riemannian manifold $M,$ we show that the Riemannian distance function $d(x,y)$ can be explicitly reconstructed from suitable asymptotics of the expected signature of Brownian bridge from $x$ to $y$. In addition, by looking into the asymptotic expansion of the fourth level expected signature of the Brownian loop based at $x\in M$, one can explicitly reconstruct both intrinsic (Ricci curvature) and extrinsic (second fundamental form) curvature properties of $M$ at $x$. As independent interest, we also derive the intrinsic PDE for the expected Brownian signature dynamics on $M$ from the perspective of the Eells-Elworthy-Malliavin horizontal lifting.