We consider Burgers solutionu=u(x,t,v); v> 0, satisfying a linear growth with respect tox. We associate with u the stochastic characteristic. Here B is the Brownian motion. We show that if (x,t) is a continuity point of the inviscid Burgers solutionu(x,t,0) then the stochastic characteristic converges uniformly inL2to the characteristic segment.Our study contains the case where the initial value is singular