×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] ASSING, S. (1999). Comparison of sy stems of stochastic partial differential equations. Stochastic Process. Appl. 82 259-282. · Zbl 0997.60064 · doi:10.1016/S0304-4149(99)00031-9
[2] BARLOW, M. T., EVANS, S. N. and PERKINS, E. A. (1991). Collision local times and measure-valued processes. Canad. J. Math. 43 897-938. · Zbl 0765.60044 · doi:10.4153/CJM-1991-050-6
[3] COX, T., FLEISCHMANN, K. and GREVEN, A. (1996). Comparison of interacting diffusions and an application to their ergodic theory. Probab. Theory Related Fields 105 513-528. · Zbl 0853.60080 · doi:10.1007/BF01191911
[4] DAWSON, D. A. (1993). Measure-valued Markov processes. Ecole d’été de probabilités de Saint Flour XXI. Lecture Notes in Math. 1541 1-260. Springer, Berlin. · Zbl 0799.60080
[5] DAWSON, D. A., ISCOE, I. and PERKINS, E. A. (1989). Super-Brownian motion: Path properties and hitting properties. Probab. Theory Related Fields 83 135-205. · Zbl 0692.60063 · doi:10.1007/BF00333147
[6] ETHIER, S. N. and KURTZ, T. (1986). Markov Processes: Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[7] KURTZ, T. (2001). Particle representations for measure-valued population processes with spatially varying birth rates. · Zbl 0960.60069
[8] LOPEZ, M. M. (1996). Path properties and convergence of interacting superprocesses. Ph.D. thesis, Univ. British Columbia.
[9] MELEARD, S. and ROELLY, S. (1992). Interacting branching measure processes. In Stochastic Partial Differential Equations and Applications (G. Da Prato and L. Tubaro, eds.) 246-256. Longman, New York. · Zbl 0787.60106
[10] METIVIER, M. (1985). Weak convergence of measure-valued processes using Sobolev imbedding techniques. Stochastic Partial Differential Equations and Applications. Lecture Notes in Math. 1236 172-183. Springer, Berlin. · Zbl 0621.60003
[11] PERKINS, E. A. (1995). On the martingale problem for interactive measure-valued branching diffusions. Mem. Amer. Math. Soc. 115 1-89. · Zbl 0823.60071
[12] ROELLY-COPPOLETTA, S. (1986). A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17 43-65. · Zbl 0598.60088 · doi:10.1080/17442508608833382
[13] ROGERS, L. C. G. and WILLIAMS, D. (1994). Diffusions, Markov Processes and Martingales 1: Foundations, 2nd ed. Wiley. · Zbl 0826.60002
[14] WATANABE, S. (1968). A limit theorem of branching processes and continuous state branching. J. Math. Ky oto Univ. 8 141-167. · Zbl 0159.46201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.