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Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self-intersection local times. (English) Zbl 0659.60105
Les processus stochastiques, Coll. Paul Lévy, Palaiseau/Fr. 1987, Astérisque 157-158, 147-171 (1988).
Let \(\{\xi_ t,t\in \Delta =[0,T]\}\) be a Markov process on a probability space into a measurable space (S,\({\mathcal B})\) with the transition functions p(s,x;t,dy), \(s<t\), \(x\in S\), and let \(\{X_ t,t\in \Delta \}\) be another Markov process with values in a set \({\mathcal M}\) of finite measures on (S,\({\mathcal B})\). Then the X-process is termed a superprocess over the \(\xi\)-process if for all \(r<t\) in \(\Delta\), \(\mu\in M\), and \(B\in {\mathcal B}\), one has \(E_{r,\mu}(X_ t(B))=\int_{S}p(s,x;t,B)\mu (dx)\). Let \[ (T^ r_ tf)(x)=\int_{S}f(y)p(r,x;t,dy)=<f,p(r,x;t,\cdot)>. \] thus \(T^ s_ s=id.\), and set \(T^ r_ s=0\) for \(r>s\). Making the natural measurability assumptions, so that for instance \(E_{r,\mu}(<f,X_ t>)=<T^ r_ tf,\mu >\), the author proves the existence of superprocesses for a class of non-Gaussian measure valued Markov processes. The main tool in such a construction is the following theorem.
Let \(r<\min (t_ 1,...,t_ n)\in \Delta\). Then for any measurable \(f_ i:S\to {\mathbb{R}}^+\), \[ E_{r,\mu}(<f_ 1,X_{t_ 1}>...<f_ n,X_{t_ n}>)=\sum_{\Lambda_ 1,...,\Lambda_ k}\prod^{k}_{i=1}\int_{E}W_{\Lambda_ i}(r,x)\mu (dx), \] where the sum is taken over all partitions of \(\{\) 1,2,...,n\(\}\) into disjoint nonempty subsets \(\Lambda_ 1,...,\Lambda_ k\), \(k=1,2,...,n\), and where \[ W_{\Lambda}(r,x)=\prod_{i\in \Lambda}^*h^ i_ r(x),\quad h^ i_ r(x)=(T^ r_{t_ i}f_ i)(x), \] the symbol \(\Pi^*\) standing for the sum of certain *-products over all orders of factors and all orders of operations. The results have been applied for the existence of local times and self-intersection times for the X- process over classes of \(\xi\)-processes. The paper is technical and more precise details cannot be given here.
For the entire collection see [Zbl 0649.00017].
Reviewer: M.M.Rao

60J55 Local time and additive functionals
60J35 Transition functions, generators and resolvents
60H05 Stochastic integrals