R-D systems and branching Jirina processes (Q806189)

Let \(X_ t\) be a homogeneous Markov process on the Borel space and L be its strong infinitesimal operator. The author proves that the solution of the equation \[ \partial u_ t(x)/\partial t=Lu_ t(x)+f(u_ t(x)),\quad t\geq 0,\quad u_ t(x)=g(x)\geq 0,\quad f(u)=au-bu^ 2+\int^{\infty}_{0}(1-\exp \{-uy\}-uy)\Lambda (dy), \] where \(b>0\), \(\int^{\infty}_{0}(y^ 2\wedge y)\Lambda (dy)<\infty\), may be represented as the special functional of the Jirina branching process with an abstract set of types.