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Joint continuity of the solutions to a class of nonlinear SPDEs
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  • Published: 15 March 2011

Joint continuity of the solutions to a class of nonlinear SPDEs

  • Zenghu Li1,
  • Hao Wang2,
  • Jie Xiong3,4 &
  • …
  • Xiaowen Zhou5 

Probability Theory and Related Fields volume 153, pages 441–469 (2012)Cite this article

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Abstract

For a one-dimensional superprocess in random environment, a nonlinear SPDE was derived by Dawson et al. (Ann Inst Henri Poincaré Probab Stat 36(2):167–180, 2000) for its density process. The time-space joint continuity of the density process was left as an open problem. In this paper we give an affirmative answer to this problem.

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References

  1. Barros-Neto J.: An Introduction to the Theory of Distributions. Marcel Dekker, New York (1973)

    MATH  Google Scholar 

  2. Dawson D.A., Li Z., Wang H.: Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab. 6, 1–33 (2001)

    Article  MathSciNet  Google Scholar 

  3. Dawson D.A., Vaillancourt J., Wang H.: Stochastic partial differential equations for a class of interacting measure-valued diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 36(2), 167–180 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Foondum M., Khoshnevisan D.: Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14, 548–568 (2009)

    MathSciNet  Google Scholar 

  5. Kallianpur G.: Stochastic Filtering Theory. Springer, Berlin (1980)

    MATH  Google Scholar 

  6. Konno N., Shiga T.: Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Relat. Fields 79, 201–225 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Krasnoselskii, M.A., Pustylnik, E.I., Sobolevski, P.E., Zabrejko, P.P.: Integral Operators in Spaces of Summable Functions. Nauka, Moscow (1966, in Russian); English translation: Noordhoff International Publishing, Leyden (1976)

  8. Krylov, N.V.: An analytic approach to SPDEs. Stochastic partial differential equations: six perspectives. In: Math. Surveys Monogr., vol. 64, pp. 185–242. Amer. Math. Soc., Providence (1999)

  9. Kurtz T., Xiong J.: Particle representations for a class of nonlinear SPDEs. Stoch. Process. Appl. 83, 103–126 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kurtz T., Xiong J.: A stochastic evolution equation arising from the fluctuation of a class of interacting particle systems. Commun. Math. Sci. 2, 325–358 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Lee K.J, Mueller C, Xiong J.: Some properties for superprocess over a stochastic flow. Ann. Inst. Henri Poincaré Probab. Stat. 45, 477–490 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li Z., Wang H., Xiong J.: Conditional log-Laplace functionals of superprocesses with dependent spatial motion. Acta Appl. Math. 88, 143–175 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Reimers M.: One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Relat. Fields 81, 319–340 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Skoulakis G., Adler R.J.: Superprocesses over a stochastic flow. Ann. Appl. Probab. 11, 488–543 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang H.: State classification for a class of measure-valued branching diffusions in a Brownian medium. Probab. Theory Relat. Fields 109, 39–55 (1997)

    Article  MATH  Google Scholar 

  16. Wang H.: A class of measure-valued branching diffusions in a random medium. Stoch. Anal. Appl. 16, 753–786 (1998)

    Article  MATH  Google Scholar 

  17. Xiong, J.: An introduction to stochastic filtering theory. In: Oxford Graduate Texts in Mathematics, vol. 18. Oxford University Press, Oxford (2008)

  18. Xiong J., Zhou X.: Superprocess over a stochastic flow with superprocess catalyst. Int. J. Pure Appl. Math. 17, 353–382 (2004)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Zenghu Li

  2. Department of Mathematics, University of Oregon, Eugene, OR, 97403-1222, USA

    Hao Wang

  3. Department of Mathematics, Hebei Normal University, Shijiazhuang, 050016, People’s Republic of China

    Jie Xiong

  4. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996-1300, USA

    Jie Xiong

  5. Department of Mathematics and Statistics, Concordia University, Montreal, QC, H3G 1M8, Canada

    Xiaowen Zhou

Authors
  1. Zenghu Li
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  2. Hao Wang
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  3. Jie Xiong
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  4. Xiaowen Zhou
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Corresponding author

Correspondence to Jie Xiong.

Additional information

Research of Z. Li was supported partially by NSFC (10525103 and 10721091) and CJSP, J. Xiong by NSF DMS-0906907, and X. Zhou by NSERC.

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Cite this article

Li, Z., Wang, H., Xiong, J. et al. Joint continuity of the solutions to a class of nonlinear SPDEs. Probab. Theory Relat. Fields 153, 441–469 (2012). https://doi.org/10.1007/s00440-011-0351-x

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  • Received: 22 May 2009

  • Revised: 22 February 2011

  • Published: 15 March 2011

  • Issue Date: August 2012

  • DOI: https://doi.org/10.1007/s00440-011-0351-x

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Keywords

  • Superprocess
  • Random environment
  • Stochastic partial differential equation

Mathematics Subject Classification (2000)

  • Primary: 60G57
  • 60H15
  • Secondary: 60J80
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