Abstract
This survey is essentially based on our papers [13] and [14]. In the first section, we shall present our definition of symmetric multiparameter Markov processes and introduce the capacities associated with them. This definition is the starting point of our study, but it is not sufficient to develop the whole theory. Rather than give explicitly the other assumptions which are used in the proofs, we shall present in the second section the fundamental examples for which the main results that we shall give in the third section, are valid.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
J. Bauer, Multiparameter processes associated with Ornstein-Uhlenbeck semi-groups, in: Classical and Modern Potential Theory and Applications, Kluwer, 1994, 41–56.
R. Cairoli and J. B. Walsh, Stochastic integrals in the plane, Acta Mathematica, 134 (1975), 111–183.
E. B. Dynkin, Markov systems and their additive junctionals, Ann. Probab., 5 (1977), 653–677.
E. B. Dynkin, Additive Junctionals of several time-reversible Markov processes, J. Funct. Anal, 42 (1981), 64–101.
S. N. Evans, Potential theory for a family of several Markov processes, Ann. Inst. Henri Poincaré, 23 (1987), 499–530.
D. Feyel and A. de La Pradelle, On infinite dimensional sheets, Potential Analysis, 4 (4) (1995), 345–359.
P. J. Fitzsimmons and T. S. Salisbury, Capacity and energy for multiparameter Markov processes, Ann. Inst. Henri Poincaré, 25 (3) (1989), 325–350.
F. Hirsch, Représentation du processus d’Ornstein-Uhlenbeck à n paramètres, in: Séminaire de Probabilités XXVII, Springer, 1993, 302–303.
F. Hirsch, Potential theory related to some multiparameter processes, Potential Analysis, 4 (3) (1995), 245–267.
F. Hirsch, Quasi-sure analysis of non negative Brownian martingales, in: Dirichlet Forms and Stochastic Processes, de Gruyter, 1995, 191–197.
F. Hirsch, Theory of capacity on the Wiener space, in: Stochastic Analysis and Related Topics V: The Silivri Workshop, 1994, Birkhäuser, 1996, 69–98.
F. Hirsch and S. Song, Inequalities for Bochner’s subordinates of two-parameter symmetric Markov processes, Ann. Inst. Henri Poincaré, 32 (5) (1996), 589–600.
F. Hirsch and S. Song, Markov properties of multiparameter processes and capacities, Proba. Th. Rei. Fields, 103 (1) (1995), 45–71.
F. Hirsch and S. Song, Symmetric Skorohod topology on n-variable functions and hierarchical Markov properties of n-parameter processes, Proba. Th. Rei. Fields, 103 (1) (1995), 25–43.
K. Itô and H. P. McKean, Diffusion Processes and their Sample Paths, Springer, second printing, 1974.
H. Korezlioglu, P. Lefort and G. Mazziotto, Une propriété markovienne et diffusions associées, in Processus aléatoires à deux indices, Springer, 1981, 245–274.
Y. Le Jan, Quasi-continuous functions and Hunt processes, J. Math. Soc. Japan, 35 (1983), 37–42.
D. Nualart, The Malliavin Calculus and Related Topics, Springer, 1995.
D. Nualart and M. Sanz, A Markov property for two-parameter Gaussian processes, Stochastica, 3 (1979), 1–16.
N. Privault, A transfer principle from Wiener to Poisson space and applications, J. Funct. Anal., 132 (2) (1995), 335–360.
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 1991.
S. Song, Processus d’Omstein-Uhlenbeck et ensembles W 2,2 -polaires, Potential Analysis, 2 (2) (1993), 171–186.
S. Song, Inégalités relatives aux processus d’Ornstein-Uhlenbeck à n paramètres et capacité gaussienne c n ,2, in: Séminaire de Probabilités XXVII, Springer, 1993, 276–301.
S. Song, Construction d’un processus à deux paramètres à partir d’un semi-groupe à un paramètre, in: Classical and Modem Potential Theory and Applications, Kluwer, 1994, 419–451.
J. B. Walsh, An introduction to stochastic partial differential equations, in: École d’été de probabilités de Saint-Flour XIV-1984, Springer, 1986, 266–437.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Basel AG
About this paper
Cite this paper
Hirsch, F., Song, S. (1999). Multiparameter Markov Processes and Capacity. In: Dalang, R.C., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications. Progress in Probability, vol 45. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8681-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8681-9_12
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9727-3
Online ISBN: 978-3-0348-8681-9
eBook Packages: Springer Book Archive