The Burgers superprocess

We define the Burgers superprocess to be the solution of the stochastic partial differential equation ∂ ∂ t u ( t , x ) = Δ u ( t , x ) − λ u ( t , x ) ∇ u ( t , x ) + γ u ( t , x ) W ( d t , d x ) , where t ≥ 0 , x ∈ R , and W is space-time white noise. Taking γ = 0 gives the classic Burgers equation, an important, non-linear, partial differential equation. Taking λ = 0 gives the super-Brownian motion, an important, measure valued, stochastic process. The combination gives a new process which can be viewed as a superprocess with singular interactions. We prove the existence of a solution to this equation and its Hölder continuity, and discuss (but cannot prove) uniqueness of the solution.