It\^o tracers: continuous-trajectory Lagrangian particles for Eulerian hydrodynamics

Lagrangian tracer particles have long been used to track the history of individual gas parcels in hydrodynamical codes. Particles advected by the cell-centered velocity carry no representation of underlying numerical diffusion, and thus exhibit systematic bias. The Monte-Carlo (MC) tracer resolves this with discrete probabilistic cell-to-cell, flux-based jumps, at the cost of trajectories that are discontinuous in time. We introduce the It\^o tracer, a continuous-time Lagrangian particle with moments matched to the advection, diffusion, and dispersion of the gas. A subgrid-scale variant (SGS-It\^o) replaces the numerical diffusion with a Smagorinsky--Lilly turbulent diffusivity, illustrating that the form of the diffusion matters less than its magnitude. We validate these methods with a 1D square-pulse advection test and 3D decaying turbulence at $\sigma_{\rm rms} = 15\,c_{\rm s}$. We compare the different tracer particle methods using several statistical tests. It\^o tracers largely reproduce or improve upon MC tracers statistics across column-density maps, joint density histograms, log-density-ratio PDFs, and density power spectra. In the turbulence test, It\^o tracers improve the correlation between tracers and gas over the MC tracers by >3\%, and reduce the width of the log-density ratio PDF by nearly 50\%. Relative to classical tracers, these improvements are $\gtrsim$30\% and 230\%, respectively. Because It\^o tracers follow a stochastic differential equation, the method maps onto other continuous-trajectory Lagrangian processes (e.g. dust grains, charged particles, cosmic rays), admits variance-reduction techniques, higher-order integrators, and GPU-friendly implementations -- all of which are unavailable to discrete-jump schemes.