Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part
Download PDF
Download PDF
  • Published: 04 November 2003

Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part

  • Gerald Trutnau1 

Probability Theory and Related Fields volume 127, pages 455–495 (2003)Cite this article

Abstract.

Let \({{\overline{{G}}\subset {{\mathbb R}}^d}}\) be a compact set with interior G. Let ρ∈L 1(G,dx), ρ>0 dx-a.e. on G, and m:=ρdx. Let A=(a ij ) be symmetric, and globally uniformly strictly elliptic on G. Let ρ be such that \({{{{{{\mathcal E}}}}^r(f,g)=\frac{{1}}{{2}}\sum_{{i,j=1}}^{{d}}\int_G a_{{ij}}\partial_i f \partial_j g\,dm}}\); f, \({{g\in C^{{\infty}}(\overline{{G}})}}\), is closable in L 2(G,m) with closure (ℰr,D(ℰr)). The latter is fulfilled if ρ satisfies the Hamza type condition, or ∂ i ρ∈L 1 loc (G,dx), 1≤i≤d. Conservative, non-symmetric diffusion processes X t related to the extension of a generalized Dirichlet form \({{ {{{{\mathcal E}}}}^r(f,g) -\sum_{{i=1}}^{{d}}\int_G \rho^{{-1}}\overline{{B}}_i\partial_i f\, g\, dm; f,g\in D({{{{\mathcal E}}}}^r)_b }}\) where \({{\rho^{{-1}}(\overline{{B}}_1,...,\overline{{B}}_d)\in L^2(G;{{\mathbb R}}^d,m)}}\) satisfies \({{ \sum_{{i=1}}^{{d}}\int_G \overline{{B}}_i \partial_i f\,dx =0\quad {{\rm{ for all}}} f\in C^{{\infty}}(\overline{{G}}), }}\) are constructed and analyzed. If G is a bounded Lipschitz domain, ρ∈H 1,1(G), and a ij ∈D(ℰr), a Skorokhod decomposition for X t is given. This happens through a local time that is uniquely associated to the smooth measure 1{ Tr (ρ)>0} dΣ, where Tr denotes the trace and Σ the surface measure on ∂G.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Albeverio, S., Röckner, M.: Classical Dirichlet forms on topological vector spaces – closability and a Cameron-Martin formula. J. Funct. Anal. 88, 395–436 (1990)

    MathSciNet  MATH  Google Scholar 

  2. Bass, R.F., Hsu, P.: The semimartingale structure of reflecting Brownian motion. Proc. Amer. Math. Soc. 108 (4), 1007–1010 (1990)

    MATH  Google Scholar 

  3. Bass, R.F., Hsu, P.: Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19 (2), 486–508 (1991)

    MATH  Google Scholar 

  4. Bouleau, N., Hirsch, F.: Dirichlet forms and Analysis on Wiener space. Walter de Gruyter, Berlin, 1991

  5. Chen, Z.-Q.: On reflected Dirichlet spaces. Probab. Theory Relat. Fields 94, 135–162 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Chen, Z.-Q.: On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Relat. Fields 94, 281–315 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Eberle, A.: Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators. LNM 1718, Springer, 1999

  8. Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Boca Raton: CRC Press, 1992

  9. Fradon, M.: Diffusions réfléchies réversibles dégénérées. Potential Analysis 6, 369–414 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fukushima, M.: A construction of reflecting barrier Brownian motions for bounded domains. Osaka J. Math. 4, 183–215 (1967)

    MATH  Google Scholar 

  11. Fukushima, M.: Regular representations of Dirichlet spaces. Trans. Amer. Math. Soc. 155, 455–473 (1971)

    MATH  Google Scholar 

  12. Fukushima, M.,Oshima, Y., Takeda, M.: Dirichlet forms and Symmetric Markov processes. Berlin-New York: Walter de Gruyter, 1994

  13. Fukushima, M., Tomisaki, M.: Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps. Probab. Theory Related Fields 106 (4), 521–557 (1996)

    Article  MATH  Google Scholar 

  14. Kuwae, K.: Reflected Dirichlet forms and the uniqueness of Silverstein’s extension. Potential Anal. 16 (3), 221–247 (2002)

    Article  MATH  Google Scholar 

  15. Kuwae, K., Uemura, T.: Weak convergence of symmetric diffusion processes. Prob. Theory Relat. Fields 109 (2), 159–182 (1997)

    Article  MATH  Google Scholar 

  16. Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Berlin: Springer, 1992

  17. Maz’ja, V.G.: Sobolev spaces. Berlin-Heidelberg, Springer, 1985

  18. Oshima, Y.: Lectures on Dirichlet spaces. Universität Erlangen-Nürnberg, 1988

  19. Pardoux, E., Williams, R.J.: Symmetric reflected diffusions. Ann. Inst. Henri Poincaré 30 (1), 13–62 (1994)

    MATH  Google Scholar 

  20. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, Berlin, 1985

  21. Revuz, D.: Mesures associées aux fonctionelles additives de Markov I. Trans. Amer. Math. Soc. 148, 501–531 (1997)

    MATH  Google Scholar 

  22. Reed, M., Simon, B.: Methods of modern mathematical physics II. Fourier Analysis. Academic Press, New York-San Francisco-London, 1975

  23. Silverstein, M.L.: Symmetric Markov Processes. LMN 426, (1974)

  24. Silverstein, M.L.: The reflected Dirichlet space. Illinois J. Math. 18, 310–355 (1974)

    MATH  Google Scholar 

  25. Stannat, W.: The theory of generalized Dirichlet forms and its applications in analysis and stochastics. Mem. Amer. Math. Soc. 142 (678), (1999)

  26. Stannat, W.: (Nonsymmetric) Dirichlet operators on L 1: Existence, uniqueness and associated Markov processes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1), 99–140 (1999)

    Google Scholar 

  27. Stannat, W.: Time-dependent diffusion operators on L 1. Preprint 00-080, Preprintreihe SFB 343 Universität Bielefeld, 2000, extended version 29.1.2002

  28. Trutnau, G.: Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions. Osaka J. Math. 37, 315–343 (2000)

    MathSciNet  MATH  Google Scholar 

  29. Trutnau, G.: On a class of non-symmetric diffusions containing fully non-symmetric distorted Brownian motions. Forum Math. 15 (3), 409–437 (2003)

    MATH  Google Scholar 

  30. Widder, D.V.: The Laplace transform. Princeton: Princeton University press, 1946

  31. Williams, R.J., Zheng, W.A.: On reflecting Brownian motion – a weak convergence approach. Ann. Inst. Henri Poincaré 26 (3), 461–488 (1990)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris 13, Département de Mathématiques, Institut Galilée, 99, av. Jean-Baptiste Clément, 93430, Villetaneuse, France

    Gerald Trutnau

Authors
  1. Gerald Trutnau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerald Trutnau.

Additional information

This research has been financially supported by TMR grant HPMF-CT-2000-00942 of the European Union.

Mathematics Subject Classification (2000): 60J60, 60J55, 31C15, 31C25, 35J25

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Trutnau, G. Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part. Probab. Theory Relat. Fields 127, 455–495 (2003). https://doi.org/10.1007/s00440-003-0296-9

Download citation

  • Received: 08 October 2002

  • Revised: 04 August 2003

  • Published: 04 November 2003

  • Issue Date: December 2003

  • DOI: https://doi.org/10.1007/s00440-003-0296-9

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Diffusion processes
  • Local time and additive functionals
  • Potential and capacities
  • Dirichlet spaces
  • Boundary value problems for second order elliptic operators
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature