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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

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