Normalization of Brownian loop measure on higher-genus Riemann surfaces

Determine whether the normalization of Brownian loop measure on arbitrary Riemann surfaces can be defined so that the resulting restriction function has the same normalization as in the planar setting, including the required treatment of increasingly long loops.

Background

The paper extends the framework of restriction functions from planar domains to arbitrary Riemann surfaces. Although Brownian loop measure itself extends naturally to Riemann surfaces, its total mass can be infinite because of increasingly long loops, so an additional normalization or renormalization is required.

The authors explicitly state that it is unknown whether the planar normalization carries over to Riemann surfaces. Establishing such a normalization would support the construction of restriction-covariant loop measures in higher-genus settings.