Killing operations in super-diffusions by Brownian snakes (Q2752170)

This paper establishes a connection between a Brownian snake modified by killing operations and the superdiffusion corresponding to NEWLINE\[NEWLINE\partial u/\partial t= Lu - c(u)(u^2+\theta(x)u).NEWLINE\]NEWLINE This was motivated by a work of \textit{J. Warren} [in: Séminaire de probabilités XXXI. Lect. Notes Math. 1655, 1-15 (1997; Zbl 0884.60081)] where -- through killing -- an analogous relationship between a reflected Brownian motion and a CB-diffusion was investigated. NEWLINENEWLINENEWLINESince there is no obvious picture of particles in the usual formulation of super-processes as measure-valued processes, the essential tool for applying the killing operation is the snake description of super-diffusions introduced by Le Gall. After introducing such description, the measure process \(\mu(t)\) is constructed through NEWLINE\[NEWLINE\langle\mu(t),f\rangle=\int_0^{l^{-1}(\gamma,0)} f(\langle X_s\rangle)\kappa(s)l(ds,t),NEWLINE\]NEWLINE where the measure process integrated against the test function is equal to the value of \(f\) at the endpoint \(\langle X_s\rangle\) of the snake \(X_s\) integrated with the local time. The killing function \(\kappa\) is the indicator function of the zero set of a nonnegative and continuous process which depends on \(\theta\), which is assumed bounded, nonnegative and continuous.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00044].