Wave Resistance for Stochastic Motion at Interfaces

Wave resistance is the drag generated by the wave radiation that a source moving at a fluid interface sustains. Under stochastic trajectories, the mean drag is controlled by the ensemble-averaged surface profile built from the trajectory history. We show that the result is a finite resistance below the deterministic radiation threshold and a regularization of the singular response at the minimum phase velocity of the capillary-gravity waves. We derive explicit scaling laws for drifted Brownian trajectories, including a universal high-diffusivity decay. For drifted L\'evy flight, we find the mean wave resistance in closed-form, extending wave-drag theory to non-Gaussian trajectories.