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Comparison for measure valued processes with interactions. (English) Zbl 1045.60046
The authors describe stochastic processes, which will be solutions to certain martingale problem. Suppose $$\{X_t: t\in [0,T]\}$$ is an adapted process defined on a filtered probability space $$(\Omega,{\mathcal F},{\mathcal F}_t,P)$$ that has continuous paths with values in $${\mathcal M}$$. The terminal time $$T$$ will be fixed throughout the paper. Let $${\mathcal P}$$ be the predictable sets for this probability space. Let $$\sigma: [0,T]\times\Omega\times E$$ be $${\mathcal P}\otimes\varepsilon$$ measurable. In this case $$\{X_t\}$$ is a solution to the martingale problem $$M(a,\sigma)$$ if for all $$\Phi\in D(A)$$ the process $Z_t(\Phi)= (X_t,\Phi)- (X_0,\Phi)- \int^t_0 (X_s, A\Phi)\,ds\tag{1}$ is an $${\mathcal F}_t$$ local martingale for $$t\in [0,T]$$. The solutions for a constant branching rate $$\sigma^c_t= c$$ are called Dawson-Watanabe processes and have been extensively studied by D. A. Dawson [in: Ecole d’Été de probabilités de Saint-Flour XXI-1991. Lect. Notes Math. 1541, 1–260 (1993; Zbl 0799.60080)].
The authors estabish a comparison principle for expectations $$E(\Phi(X_t))$$ of certain functionals $$\Phi: {\mathcal M}\to [0,\infty)$$. When the branching rate satisfies $$\sigma\geq c$$, they find conditions on $$\Phi$$ which ensure that the expectation $$E(\Phi(X_t))$$ is greater than the corresponding expectation for the Dawson-Watanabe process with constant branching rate $$c$$. The analogous problem of comparing functionals of processes with different drift terms, for example, with the term $$+\int^t_0 (b_s,\Phi)\,ds$$ added to the martingale problem (1), can be treated via pathwise comparison results. These allow one to couple two processes one of which has a larger drift than the other from which one can deduce comparisons between the expectations of increasing functionals. These are various pathwise comparison arguments for SPDES [see S. Assing, Stochastic Processes Appl. 82, No. 2, 259–282 (1999; Zbl 0997.60064) and its bibliography]. For measure-valued branching processes if the drift terms come from immigration or from mass creation and annihilation terms, i.e. if they are of the form $$\int^t_0 (X,b,\Phi)\,ds,$$ a coupling can be constructed via a “thinning” procedure [see M. T. Barlow, S. N. Evans and E. A. Perkins, Can. J. Math. 43, No. 5, 897–938 (1991; Zbl 0765.60044), Theorem 5.1, for a related result]. Applications to hitting estimates and regularity of solutons are discussed. The result is established via the martingale-optimality principle of stochastic control theory.

MSC:
 60G57 Random measures 49K27 Optimality conditions for problems in abstract spaces 60J55 Local time and additive functionals 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60H99 Stochastic analysis 93C25 Control/observation systems in abstract spaces
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References:
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