Representations of the Brownian snake with drift

We consider a path-valued process which is a generalization of the classical Brownian snake introduced by Le Gall. More precisely we add a drift term b to the lifetime process, which may depends on the spatial process. Consequently, this introduces a coupling between the lifetime process and the spatial motion. This process can be obtained from the standard Brownian snake by Girsanov's theorem or by killing of the spatial motion. It can also be viewed as the limit of discrete snakes or, in some special cases, as conditioned Brownian snakes. We also use this process to describe the solutions of the non-linear partial differential equation j u =4 u 2 +4 bu .