Stated somewhat informally, the continuity equation or transport equation $\partial_t\rho_t = -\nabla\cdot(\rho_t v_t)$ describes the evolution of a density where each particle flows along a vector field $v_t$, that is $\rho_t=\text{Law}(X_t)$ where $dX_t=v_t(X_t)dt$. Is there an analogous particle-based description of the modified PDE $\partial_t\rho_t = -\nabla\cdot(((1-\lambda)\rho_t+\lambda\pi) v_t)$ for $\pi$ a fixed distribution and $\lambda\in (0,1)$?
Since the forward equation always multiplies the distribution $\rho_t$ inside the div term, I was wondering if it is possible to add a term that is independent of the current $\rho_t$ by modifying the particle dynamics. I have tried "contaminating" the distribution of the particles by adding in new particles distributed as $Y_t\sim\pi$ and seeing how the transport equation changes, but no luck so far. A discrete-time or other related modification would also be interesting.