Propagation and conditional propagation of chaos for pressureless gas equations (Q1408496)

The author proves that for \(d\geq 1\), \(\nu >0\), a continuous bounded map \(v:{\mathbb R}^d\to{\mathbb R}^d\) and any probability measure \(\rho_0\) on \({\mathbb R}^d\) there exists a weak solution of the equation \[ dX_t= E(v(X_0)\mid X_t)dt+\nu dB_t, \qquad {\mathcal L}(X_0)=\rho_0(dx), \] where \(B\) is a standard Brownian motion independent of \(X_0\). The main novelty is the general form of \(v\) and \(\rho_0\). A strong solution of this equation had been obtained by the author earlier under much more restrictive assumptions. For \(d=1\), the result is established with the help of the propagation of chaos method for the Burgers equation, and for \(d>1\) the conditional propagation of chaos approach is employed, as introduced by \textit{W. Zheng} [Ann. Probab. 23, 1389-1413 (1995; Zbl 0836.60053)]. Putting \(\rho(dx,t)=P(X_t\in dx)\), \(u(x,t)=E(v(X_0)\mid X_t=x)\), a unique solution (in an appropriate class) of the following viscous system: \[ \partial_t(\rho)+\sum_{j=1}^d\partial_{x_j}(u_j\rho)= { \nu^2\over 2}\Delta(\rho),\quad \partial_t(u_i\rho)+\sum_{j=1}^d\partial_{x_j}(u_iu_j\rho)= { \nu^2\over 2}\Delta(u_i\rho), \quad i=1,\dots,d, \] \[ \rho(dx,t)\to\rho_0(dx),\;u_i(x,t)\rho(dx,t)\to v_i(x)\rho_0(dx)\quad \text{ as} t\to 0 \] is obtained.