Expected Signature Kernels for Rough Paths

The expected signature kernel is a similarity measure of probability laws on path space and enters directly into signature maximum mean discrepancies. Building on [P. K. Friz, P. P. Hager, and N. Tapia, Forum Math. Sigma, 10 (2022), e42] and [M. Lemercier, T. Lyons, and C. Salvi, SIAM J. Numer. Anal., 64 (2026), pp. 1095-1125], we characterize the expected signature kernel of group-valued inhomogeneous Lévy processes, aka Lévy rough paths, as the scalar component of the solution to an extended Goursat-type system. We give an original description, first in the continuous case where the finite-dimensional system is exact. In the presence of jumps, this system is inherently infinite-dimensional; we prove its well-posedness and derive finite-dimensional truncations with explicit error control. This permits direct evaluation of the reference-dependent terms in one-sample signature-MMD tests without an auxiliary reference sample. Finally, for conditionally Gaussian martingales, conditioning the usual unbiased empirical squared signature-MMD on the sampled quadratic variations yields an unbiased statistic for the same quantity with no larger variance. We derive computable finite-grid versions and illustrate the resulting variance reduction and gain in power.