$d$-dimensional pressureless Gas equations

Let $x\in R^d\to u(x,0)$ be a continuous bounded function and $\rho(dx,0)$ a probability measure on $R^d$. For all random variables $X_0$ with probability distribution $\rho(dx,0)$, we show that the stochastic differential equation (SDE)
$$ X_t = X_0 + \int_0^t E\big[u(X_0,0)\,|\, X_s\big]\,ds,\qquad t\ge 0, $$
has a solution which is a $\sigma(X_0)$-measurable Markov process. We derive a weak solution for the pressureless gas equation for $d \ge 1$, with initial distribution of masses $\rho(dx,0)$ and initial velocity $u(\cdot,0)$. We show for $d = 1$ the existence of a unique Markov process $(X_t)$ solution of our SDE.