Stochastic modification of Newtonian dynamics and induced potential—Application to spiral galaxies and the dark potential

Using the formalism of stochastic embedding developed by [J. Cresson, D. Darses, J. Math. Phys. 48, 072703 (2007)], we study how dynamics classical Newton equation for a force deriving from potential is deformed under assumption that this can admit processes as solutions. We focus on two denitions Newton's called dierential and variational. rst prove virial theorem which natural generalization case. The stochasticity modies relation adding term induced corresponds in quantum mechanics to Bohm potential. Moreover, naturally provides an action functional sat-ises Hamilton-Jacobi equation. real part with extra corresponding already observed theorem. has explicit form depending density solutions It proved satises nonlinear Schr{\"o}dinger Applying Kepler potential, one proves coincides ad-hoc ''dark potential'' used recover at rotation curve spiral galaxies. then discuss application previous context galaxies following proposal computations given [D. Da Rocha L. Nottale, Chaos, Solitons Fractals, 16(4):565-595, 2003] where emergence seen consequence fractality space Scale relativity theory.