Papers › Expected Signature on a Riemannian Manifold and Its Geometric Implications
Expected Signature on a Riemannian Manifold and Its Geometric Implications
Xi Geng, Hao Ni, Chaorui Wang
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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∈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.
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