On multiple Poisson stochastic integrals and associated Markov semigroups. (English) Zbl 0548.60058

Summary: Multiple stochastic integrals (m.s.i.) \(q^{(n)}(f)=\int_{X^ n}f(x_ 1,...,x_ n)q(dx_ 1)...q(dx_ n)\), \(n=1,2,...\), with respect to the centered Poisson random measure q(dx), \(E[q(dx)]=0\), \(E[(q(dx))^ 2]=m(dx)\), are discussed, where (X,m) is a measurable space. A ”diagram formula” for evaluation of products of (Poisson) m.s.i. as sums of m.s.i. is derived. With a given contraction semigroup \(A_ t,t\geq 0\), in \(L^ 2(X)\) we associate a semigroup \(\Gamma (A_ t),t\geq 0\), in \(L^ 2(\Omega)\) by the relation. \[ \Gamma (A_ t)q^{(n)}(f_ 1{\hat\otimes }...{\hat\otimes }f_ n)=q^{(n)}(A_ tf_ 1{\hat\otimes }...{\hat\otimes }A_ tf_ n) \] and prove that \(\Gamma (A_ t)\), \(t\geq 0\), is Markov if and only if \(A_ t\), \(t\geq 0\), is doubly sub-Markov; the corresponding Markov process can be described as time evolution (with immigration) of the (infinite) system of particles, each moving independently according to \(A_ t,t\geq 0\).


60H05 Stochastic integrals
60K35 Interacting random processes; statistical mechanics type models; percolation theory
47D07 Markov semigroups and applications to diffusion processes