Abstract
For a random element X of a nuclear space of distributions on Wiener space C([0,1],R d), the localization problem consists in “projecting” X at each time t∈[0,1] in order to define an S′(R d)-valued process X={X(t),t∈[0,1]}, called the time-localization of X. The convergence problem consists in deriving weak convergence of time-localization processes (in C([0,1],S′(R d)) in this paper) from weak convergence of the corresponding random distributions on C([0,1],R d). Partial steps towards the solution of this problem were carried out in previous papers, the tightness having remained unsolved. In this paper we complete the solution of the convergence problem via an extension of the time-localization procedure. As an example, a fluctuation limit of a system of fractional Brownian motions yields a new class of S′(R d)-valued Gaussian processes, the “fractional Brownian density processes”.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler, R.J., Feldman, R.E. and Lewin, M.: 'Intersection local times for infinite systems of Brownian motions and for the Brownian density process', Ann. Probab. 19 (1991), 192–220.
Billingsley, P.: Convergence of Probability Measures Wiley, 1968.
Bojdecki, T. and Gorostiza, L.G.: 'A nuclear space of distributions on Wiener space and application to weak convergence', in K. Kunita and H.H. Kuo (eds.), U.S.-Japan Bilateral Seminar on Stochastic Analysis in Infinite Dimensions Pitman Res. Notes in Math. 310, Longman, 1994, pp. 12–25.
Bojdecki, T. and Gorostiza, L.G.: 'Self-intersection local time for Gaussian & ′ (ℝ d)-processes: Existence, path continuity and examples', Stochastic Process. Appl. 60 (1995), 1991–226.
Bojdecki, T. and Gorostiza, L.G.: 'Fractional Brownian motion via fractional Laplacian', Statist. Probab. Lett. 44 (1999), 107–108.
Bojdecki, T., Gorostiza, L.G. and Nualart, D.: 'Time-localization of random distributions on Wiener space', Potential Anal. 6 (1997), 183–205.
Boulicaut, P.: 'Convergence cylindrique et convergence étroite d'une suite de probabilités de Radon', Z. Wahrsch. Verw. Geb. 28 (1973), 43–52.
Deuschel. J.-D. and Wang, K.: 'Large deviations of the occupation time functional of a Poisson system of independent Brownian particles', Stochastic Process. Appl. 52 (1994), 183–209.
Fernández, B.: 'Markov properties of the fluctuation limit of a particle system', J. Math. Anal. Appl. 149 (1990), 160–179.
Fernique, X.: 'Intégrabilité des vecteurs Gaussians', C. R. Acad. Sci. Paris Ser. A 270(25) (1970), 1698–1699.
Gorostiza, L.G.: 'Limites gaussiennes pour les champs aléatoires ramifiés supercritiques', in Aspects Statistiques et Aspects Physiques des Processus Gaussiens Editions du CNRS, Paris, 1981, pp. 385–398.
Gorostiza, L.G.: 'Distributions on Wiener space and fluctuation limits of Poisson systems of Brownian bridges', in C. Houdré and V. Pérez-Abreu (eds.), Chaos Expansions, Multiples Wiener-Itô Integrals and Their Applications Probab. Statist. Series, CRC Press, 1994, pp. 337-347.
Gorostiza, L.G. and Nualart, D.: 'Nuclear Gelfand triples on Wiener space and applications to trajectorial fluctuations of particle systems', J. Funct. Anal. 125 (1994), 37–66.
Gzyl, H.: 'Multidimensional extension of Faà di Bruno's formula', J. Math. Anal. Appl. 116 (1986), 450–455.
Holley, R. and Stroock, D.W.: 'Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motions', Publ. Res. Inst. Math. Sci. 14 (1981), 741–788.
Itô, K.: 'Distribution valued processes arising from independent Brownian motions', Math. Z. 182 (1983), 17–33.
Itô, K.: Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces SIAM, 1984.
Kolmogorov, A.N.: 'Wiener's spiral and some other interesting curves in Hilbert space', Dokl. Akad. Nauk SSSR 26 (1940), 115–118.
Mandelbrot, B.B.: The Fractal Geometry of Nature Freeman, San Francisco, 1982.
Mandelbrot, B.B. and Van Ness, J.W.: 'Fractional Brownian motions, fractional noises and applications', SIAM Rev. 10 (1968), 422–437.
Martin-Löf, A.: 'Limits theorems for the motion of a Poisson systems of independent Markovian particles with high density', Z. Wahrsch. Verw. Geb. 34 (1976), 205–223.
Meyer, P.-A.: 'Le théorème de continuité de P. Lévy sur les espaces nucléaires', Sem. Bourbaki 18e année, No. 311, 1965/66.
Mitoma, I.: 'On the sample continuity of {′-1}-processes', J. Math. Soc. Japan 35 (1983), 629–636.
Mitoma, I.: 'Tightness of probabilities on C([0 1] & ′) and D([0 1]& ′)', Ann. Probab. 11 (1983), 989–999.
Novikov, A. and Valkeila, E.: 'On some maximal inequalities for fractional Brownian motions', Statist. Probab. Lett. 44 (1999), 47–54.
Nualart, D. and Zakai, M.: 'Multiple Wiener-Itô integrals possessing a continuous extension', Probab. Theory Related Fields 85 (1990), 131–145.
Sznitmann, A.-S.: 'A fluctuation result for nonlinear diffusions', in Infinite-Dimensional Analysis and Stochastic Processes Res. Notes in Math. 124, Pitman, 1985, pp. 145–160.
Tanaka, H.: 'Limit theorems for certain diffusion processes with interaction', in Proceedings, Taniguchi International Symposium on Stochastic Analysis, Katata and Kyoto North-Holland, 1984.
Taqqu, M.S.: 'A bibliographical guide to self-similar processes and long range dependence', in E. Eberlein and M.S. Taqqu (eds.), Dependence in Probability and Statistics, A survey of Recent Results Progress in Probability and Statistics 11, Birkhäuser, 1986, pp. 137–161.
Treves, F.: Topological Vector Spaces, Distributions and Kernels Academic Press, 1967.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bojdecki, T., Gorostiza, L.G. Time-Localization of Random Distributions on Wiener Space II: Convergence, Fractional Brownian Density Processes. Potential Analysis 17, 267–291 (2002). https://doi.org/10.1023/A:1016172720335
Issue Date:
DOI: https://doi.org/10.1023/A:1016172720335