Measure-valued diffusions and stochastic equations with Poisson process (Q1763049)

Let \(\{X(t)\}\) be an \((A,b,c)\)-superprocess, \(N(ds; dx, du,dw)\) a Poisson random measure on \([0,\infty)\times E\times [0,\infty)\times W\) with intensity \(ds\,m(dx) duQ^x(dw)\), where \(E\) is a locally compact metric space, \(W\) the set of Wiener excursion paths starting at \(0\). Assuming that \(X\) and \(N\) are independent, it is proved that the equation \[ Y(t)= X(t)+ \int_{[0,t]} \int_E \int_{[0,q(Y(s), x)]} \int_W w(t-s)\,N(ds; dx, du, dw),\qquad t\geq 0, \] has a pathwise unique solution, the distribution of which solves the martingale problem given by \[ \begin{gathered} M_t(f)= w_t(f)- \mu(f)- \int_{[0,t]} w_s(Af- bf)\,ds- \int_{[0,t]} m(q(w_s,\cdot)\,ds,\\ \langle M(f)\rangle_t= \int_{[0,t]} w_s(cf^2)\,ds,\qquad t\geq 0.\end{gathered} \] Here \(q\) is a nonnegative Borel function on \(M(E)\times E\), \(M(E)\) the space of finite Borel measures equipped with the topology of weak convergence, and \(q\) can be interpreted as an interactive immigration rate relative to \(m\).