zbMATH — the first resource for mathematics

A weighted occupation time for a class of measure-valued branching processes. (English) Zbl 0555.60034
A weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on \(R^ d\). Considered as a process in its own right, the first and second order asymptotics are found as time \(t\to \infty\). Specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d, and when infinite, a central limit type renormalization is considered, yielding Gaussian or asymmetric stable generalized random fields in the limit. In one Gaussian case the results are contrasted in high versus low dimensions.

60G60 Random fields
60G50 Sums of independent random variables; random walks
60J55 Local time and additive functionals
Full Text: DOI
[1] Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968 · Zbl 0169.49204
[2] Choquet, G.: Lectures in analysis: I. Amsterdam: Benjamin 1969
[3] Chorin, A., Hughes, T., McCraken, M., Marsden, J.: Product formulas and numerical algorithms. Commun. Pure Appl. Math. XXXI, 205-256 (1978) · Zbl 0369.65025
[4] Dawson, D.: The critical measure diffusion process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 125-145 (1977) · Zbl 0343.60001
[5] Dawson, D., Hochberg, K.: The carrying dimension of a stochastic measure diffusion. Ann. Probab. 7, 693-703 (1979) · Zbl 0411.60084
[6] Dawson, D., Ivanoff, G.: Branching diffusions and random measures. Advances in probability and related Topics, Vol. 5, 61-103. Joffe, A., Ney, P. (eds.) 1978 · Zbl 0407.60087
[7] Dynkin, F.B.: Markov processes, Vol. 1. Berlin-Heidelberg-New York: Springer 1965 · Zbl 0132.37901
[8] Feller, W.: Introduction to probability theory and its applications, Vol. II. New York: Wiley 1966 · Zbl 0138.10207
[9] Fowler, R.: Further studies of Emden’s and similar differential equations. Q.J. Math. Ser. 22, 259-288 (1931) · Zbl 0003.23502
[10] Hida, T.: Brownian motion. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0423.60063
[11] Holley, R.A., Stroock, D.W.: Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian Motions. RIMS Kyoto Univ. 14, 741-788 (1978) · Zbl 0412.60065
[12] Iscoe, I.: The man-hour process associated with measure-valued branching random motions in R d. Ph.D. thesis. Carleton University, Ottawa 1980
[13] Iscoe, I.: On the supports of measure-valued critical branching Brownian motion (Technical report) · Zbl 0685.60087
[14] Jagers, P.: Aspects of random measures and point processes. Adv. Probab. 3, 179-238 (1974) · Zbl 0333.60059
[15] Jirina, M.: Stochastic branching processes with continuous state space. Czech. Math. J. 8, 292-313 (1958) · Zbl 0168.38602
[16] Kuelbs, J.: A representation theorem for symmetric stable processes and stable measures on H. Z. Wahrscheinlichkeitstheor. Verw. Geb. 26, 259-271 (1973) · Zbl 0253.60011
[17] Ladyzhenskaya, O., Ural’tseva, N.: Linear and quasilinear elliptic equations. London-New York: Academic Press 1968 · Zbl 0164.13002
[18] Liggett, T.M.: The stochastic evolution of infinite systems of interacting particles. Lect. Notes Math. 598. Berlin-Heidelberg-New York: Springer 1978 · Zbl 0363.60109
[19] Reed, M., Simon, B.: Methods of mathematical physics. I: Functional analysis. London-New York: Academic Press 1972 · Zbl 0242.46001
[20] Sawyer, S., Fleischman, J.: Maximum geographic range of a mutant allele considered as a subtype of a Brownian branching random field. Proc. Nat. Acad. Sci. U.S.A. 76, No. 2, 872-875 (1979) · Zbl 0404.92011
[21] Simon, B.: Distributions and their Hermite expansions. J. Math. Phys. 12, 140-148 (1971) · Zbl 0205.12901
[22] Watanabe, S.: A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8-1, 141-167 (1968) · Zbl 0159.46201
[23] Yosida, K.: Functional Analysis. Berlin-Heidelberg-New York: Springer 1968 · Zbl 0152.32102
[24] Gel’fand, I.M., Vilenkin, N.Ya.: Generalized functions, Vol. 4. London-New York: Academic Press 1964
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.