Hitting probabilities and potential theory for the brownian path-valued process

Jean-François Le Gall

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 1, page 277-306
  • ISSN: 0373-0956

Abstract

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We consider the Brownian path-valued process studied in [LG1], [LG2], which is closely related to super Brownian motion. We obtain several potential-theoretic results related to this process. In particular, we give an explicit description of the capacitary distribution of certain subsets of the path space, such as the set of paths that hit a given closed set. These capacitary distributions are characterized as the laws of solutions of certain stochastic differential equations. They solve variational problems in the space of probability measures on the path space. We also investigate some special classes of polar sets for the path-values process. These results are closely related to the polarity questions for super Brownian motion recently investigated by Dynkin and others. They are also related to removable singularities for the nonlinear partial differential equation Δ u = u 2 .

How to cite

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Le Gall, Jean-François. "Hitting probabilities and potential theory for the brownian path-valued process." Annales de l'institut Fourier 44.1 (1994): 277-306. <http://eudml.org/doc/75059>.

@article{LeGall1994,
abstract = {We consider the Brownian path-valued process studied in [LG1], [LG2], which is closely related to super Brownian motion. We obtain several potential-theoretic results related to this process. In particular, we give an explicit description of the capacitary distribution of certain subsets of the path space, such as the set of paths that hit a given closed set. These capacitary distributions are characterized as the laws of solutions of certain stochastic differential equations. They solve variational problems in the space of probability measures on the path space. We also investigate some special classes of polar sets for the path-values process. These results are closely related to the polarity questions for super Brownian motion recently investigated by Dynkin and others. They are also related to removable singularities for the nonlinear partial differential equation $\Delta u=u^2$.},
author = {Le Gall, Jean-François},
journal = {Annales de l'institut Fourier},
keywords = {path-valued process; super Brownian motion; potential-theoretic results; capacitary distributions; variational problems; partial differential equation},
language = {eng},
number = {1},
pages = {277-306},
publisher = {Association des Annales de l'Institut Fourier},
title = {Hitting probabilities and potential theory for the brownian path-valued process},
url = {http://eudml.org/doc/75059},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Le Gall, Jean-François
TI - Hitting probabilities and potential theory for the brownian path-valued process
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 1
SP - 277
EP - 306
AB - We consider the Brownian path-valued process studied in [LG1], [LG2], which is closely related to super Brownian motion. We obtain several potential-theoretic results related to this process. In particular, we give an explicit description of the capacitary distribution of certain subsets of the path space, such as the set of paths that hit a given closed set. These capacitary distributions are characterized as the laws of solutions of certain stochastic differential equations. They solve variational problems in the space of probability measures on the path space. We also investigate some special classes of polar sets for the path-values process. These results are closely related to the polarity questions for super Brownian motion recently investigated by Dynkin and others. They are also related to removable singularities for the nonlinear partial differential equation $\Delta u=u^2$.
LA - eng
KW - path-valued process; super Brownian motion; potential-theoretic results; capacitary distributions; variational problems; partial differential equation
UR - http://eudml.org/doc/75059
ER -

References

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  1. [AL]R. ABRAHAM, J.F. LE GALL, La mesure de sortie du super mouvement brownien, Probab. Th. Rel. Fields, to appear. Zbl0801.60040
  2. [AP]D.R. ADAMS, J.C. POLKING, The equivalence of two definitions of capacity, Proc. Amer. Math. Soc., 37 (1973), 529-534. Zbl0251.31005MR48 #6451
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  7. [Dy2]E.B. DYNKIN, A probabilistic approach to one class of nonlinear differential equations, Probab. Th. Rel. Fields, 89 (1991), 89-115. Zbl0722.60062MR92d:35090
  8. [Dy3]E.B. DYNKIN, Superprocesses and parabolic nonlinear differential equations, Ann. Probab., 20 (1992), 942-962. Zbl0756.60074MR93d:60124
  9. [FG]P.J. FITZSIMMONS, R.K. GETOOR, On the potential theory of symmetric Markov processes, Math. Ann., 281 (1988), 495-512. Zbl0627.60067MR89k:60110
  10. [GV]A. GMIRA, L. VÉRON, Boundary singularities of some nonlinear elliptic equations, Duke Math. J., 64 (1991), 271-324. Zbl0766.35015MR93a:35053
  11. [HW]R.A. HUNT, R.L. WHEEDEN, Positive harmonic functions in Lipschitz domains, Trans. Amer. Math. Soc., 147 (1970), 507-527. Zbl0193.39601MR43 #547
  12. [LG1]J.F. LE GALL, A class of path-valued Markov processes and its applications to superprocesses, Probab. Th. Rel. Fields, 95 (1993), 25-46. Zbl0794.60076MR94f:60093
  13. [LG2]J.F. LE GALL, A path-valued Markov process and its connections with partial differential equations, Proceedings of the First European Congress of Mathematics, to appear. Zbl0812.60058
  14. [Me]N.G. MEYERS, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand., 26 (1970), 255-292. Zbl0242.31006MR43 #3474
  15. [Pe]E.A. PERKINS, Polar sets and multiple points for super Brownian motion, Ann. Probab., 18 (1990), 453-491. Zbl0721.60046MR91i:60109
  16. [Sh]Y.C. SHEU, A characterization of polar sets on the boundary, preprint (1993) 

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