The Burgers superprocess
The authors define the Burgers superprocess to be the solution of the stochastic partial differential equation \[ \partial_t u=\partial_x^2 u -\lambda u \partial_x u +\gamma \sqrt{u} \dot{W} \] for \(u(t,x)\) with \(t\geq 0\) and \(x\in(-\infty,\infty)\), and \(\dot{W}(t,x)\) being space-time white noise. The limiting cases are the Burgers equation for \(\gamma=0\) and the super-Brownian motion for \(\lambda=0\). The existence of a weak solution is shown and the problem of uniqueness is discussed. In the method of proof the authors rely on spatial discretization in the spirit of the numerical method of lines and tightness arguments, in order to construct a solution. This solution is Hölder-continuous, positive, and dies out in finite time.
stochastic partial differential equation
scientific article; zbMATH DE number 5128007
scientific article