Comparison for measure valued processes with interactions.

*(English)*Zbl 1045.60046The 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.

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.

Reviewer: G. G. Vrănceanu (Bucureşti)

##### 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 |

##### References:

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