We present a wavelet-based algorithm to identify dwarf galaxies in the Milky Way in ${it Gaia}$ DR2 data. Our algorithm detects overdensities in 4D position--proper motion space, making it the first search to explicitly use velocity information to search for dwarf galaxy candidates. We optimize our algorithm and quantify its performance by searching for mock dwarfs injected into ${it Gaia}$ DR2 data and for known Milky Way satellite galaxies. Comparing our results with previous photometric searches, we find that our search is sensitive to undiscovered systems at Galactic latitudes~$lvert brvert>20^{circ}$ and with half-light radii larger than the 50% detection efficiency threshold for Pan-STARRS1 (PS1) at (${it i}$) absolute magnitudes of =$-7<M_V<-3$ and distances of $32$ kpc $< D < 64$ kpc, and (${it ii}$) $M_V< -4$ and $64$ kpc $< D < 128$ kpc. Based on these results, we predict that our search is expected to discover $5 pm 2$ new satellite galaxies: four in the PS1 footprint and one outside the Dark Energy Survey and PS1 footprints. We apply our algorithm to the ${it Gaia}$ DR2 dataset and recover $sim 830$ high-significance candidates, out of which we identify a gold standard list of $sim 200$ candidates based on cross-matching with potential candidates identified in a preliminary search using ${it Gaia}$ EDR3 data. All of our candidate lists are publicly distributed for future follow-up studies. We show that improvements in astrometric measurements provided by ${it Gaia}$ EDR3 increase the sensitivity of this technique; we plan to continue to refine our candidate list using future data releases.
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