×

Coalescing and noncoalescing stochastic flows in \(R_ 1\). (English) Zbl 0536.60016

Two kinds of Brownian stochastic flows on the real line are identified and studied. In one kind the constituent random mappings are all homeomorphisms. The other uses random mappings whose ranges are locally finite points under the flow then ”coalesce”. Conditions are given which imply one or the other kind of behaviour; these conditions depend on the differentiability at zero of a covariance function used to characterise the flow. Whether other kinds of behaviour are possible is left as an open question, as is the matter of whether coalescence can occur from Brownian flows in the plane or space. Various properties are investigated, and the case of exponential covariance is shown to lead to a spatial Markov property.
Reviewer: Wilfrid S.Kendall

MSC:

60B99 Probability theory on algebraic and topological structures
60J60 Diffusion processes
60G99 Stochastic processes
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Arratia, R., Coalescing Brownian motions on the line, ()
[2] Arratia, R., Coalescing Brownian motions on R and the voter model on Z, (1981), Preprint
[3] Athreya, K.; Ney, P., Branching processes, (1972), Springer Berlin, New York · Zbl 0259.60002
[4] Baxendale, P., Stochastic flows and malbavin calculus, (1981), Summary of lecture at a conference in San Diego
[5] Baxendale, P., Brownian motions in the diffeomorphism group I, (1982), Preprint
[6] Bismut, J.-M.; Michel, D., Diffusions conditionnelles, VI, (1980), University of Paris, To appear in J. Functional Analysis
[7] Bismut, J.-M., A generalized formula of ito and some other properties of stochastic flows, Z. wahrsch. verw. geb., 55, 331-350, (1981) · Zbl 0456.60063
[8] Bucan, G., On the mixed product of non-homogeneous stochastic semigroups, Theory of prob. applic., 24, 166-175, (1979), (Russian).
[9] Cramér, H.; Leadbetter, M., Stationary and related processes, (1966), Wiley New York · Zbl 0162.21102
[10] Dynkin, E., Markov processes, Vol. 1, (1965), Springer, Berlin New York · Zbl 0132.37901
[11] Elworthy, K., Stochastic dynamical systems and their flows, (1978), Mathematics Institute, University of Warwick Coventry · Zbl 0439.60065
[12] Feller, W., Two singular diffusion problems, Ann. math., 54, 173-182, (1951) · Zbl 0045.04901
[13] Feller, W., Generalized second order differential operators and their lateral conditions, Illinois J. math., 1, 459-504, (1957) · Zbl 0077.29102
[14] Friedman, A., Stochastic differential equations and applications, Vol. 1, (1975), Academic Press New York
[15] Harris, T., Random measures and motions of point processes, Z. wahrsch. verw. geb., 18, 85-115, (1971) · Zbl 0194.49204
[16] Harris, T., Brownian motions on the homeomorphisms of the plane, Ann. prob., 9, 232-254, (1981) · Zbl 0457.60013
[17] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1981), North-Holland Amsterdam, Oxford, New York · Zbl 0495.60005
[18] Ito, K.; McKean, H., Diffusion processes and their sample paths, (1965), Springer Berlin, New York · Zbl 0127.09503
[19] Kunita, H., Stochastic flows and their infinitesimal generators, (1983), a Preprint
[20] Lee, W., Random stirrings of the real line, Ann. prob., 2, 580-592, (1974) · Zbl 0288.60094
[21] Le Jan, Y., Flota de diffusion dans \(R\)_{d} 07, No. 21, 697-699, (1982), C.R. Paris 294, Ser. I
[22] Le Jan, Y.; Watanabe, S., Stochastic flows of diffeomorphisms, (1983), Preprint
[23] Lipster, R.; Shiryayev, A., Statistics of random processes, Vol. 1, (1977), Springer Berlin, New York
[24] McKean, H., Stochastic integrals, (1969), Academic Press New York · Zbl 0191.46603
[25] Stroock, D.; Varadhan, S., Multidimensional diffusion processes, (1979), Springer, Berlin New York · Zbl 0426.60069
[26] Yamada, T.; Ogura, Y., On the strong comparison theorems for solutions of stochastic differential equations, Z. wahrsch. verw. geb., 56, 3-19, (1981) · Zbl 0468.60056
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.