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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 $$A_ t,t\geq 0$$.

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