We construct a modified Arratia flow with mass and energy conservation. We suppose that particles have a mass obeying the conservation law, and their diffusion is inversely proportional to the mass. Our main result asserts that such a system exists under the assumption of the uniform mass distribution on an interval at the starting moment. We introduce a stochastic integral with respect to such a flow and obtain the total local time as the density of the occupation measure for all particles.
"A system of coalescing heavy diffusion particles on the real line." Ann. Probab. 45 (5) 3293 - 3335, September 2017. https://doi.org/10.1214/16-AOP1137