The correlated linking numbers of a Brownian loop with two arbitrary curves

The standard kinetic path integral for all spatially closed Brownian paths (loops) of duration t weighted by the product mn is evaluated, where m and n are the linking numbers of the Brownian loop with two arbitrary curves in 3D space. The path integral thus indicates the extent to which these two linking numbers are correlated, ranging from the value zero for far apart curves when it is unlikely that the Brownian loop links with both, to ±infinity for nearly coincident curves. The result takes a form that loosely resembles that for the mutual inductance of two current carrying circuits in magnetostatics, a double line integral, but is also dependent on a single extra parameter, the duration t of the path. The result for the equivalent 2D problem was given previously (Hannay 2018).