Probabilistic Interpretation of Sticky Particle Model

Abstract

This work presents a construction of a solution for the nonlinear stochastic differential equation $X_t= X_0+ \int_0^t \mathbb{E}[u_0(X_0)|X_s]ds, t \geq 0$. The random variable $X_0$ with values in $\mathb{R}$ and the function $u_0$ are given. We denote by $P_t$ the probability distribution of $X_t$ and $u(x,t) = \mathbb{E}[u_0(X_0)|X_t= x]$. We prove that $(P_t,u(\cdot,t), t\geq 0)$ is a weak solution for a system of conservation laws arising in adhesion particle dynamics.