Forced Burgers equation with sticky impulsion source

We consider the inviscid Burgers equation with force \(\partial _t u+\partial _x(u^2/2)=\nu \), where the discontinuities of initial datum \(u_0\) are interpreted as force sources. Thence, \(\nu \) is the force of shocks in a sticky dynamics of (paradoxically) non accelerated particles, whose the mass distribution field is \(\partial _xu\). The force has its own dynamics of density field \(\eta =u-w\) (the experienced impulsion), where w denotes the sticky particle velocity field. Along the sticky particle trajectory \(t\mapsto X_t\), the processes \(t\mapsto \eta (X_t,t),u(X_t,t),w(X_t,t)\) are backward martingales.