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Generalized Anderson Model with Time-Space Multiplicative Fractional Noise

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Abstract

We study the existence and the properties of the solution to a stochastic partial differential equation with multiplicative time-space fractional noise. The equation we consider involves a pseudo-differential operator that generates a stable-like process and it extends the standard heat equation. Our techniques are based on stochastic analysis, Malliavin calculus and Wiener-Itô chaos expansion.

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References

  1. Balan, R.M.: The stochastic wave equation with multiplicative fractional noise: a Malliavin calculus approach. Potential Anal. 36, 1–34 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Balan, R.M., Tudor, C.A.: Stochastic heat equation with multiplicative fractional-colored noise. J. Theor. Probab. 23, 834–870 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bass, R.F.: Uniqueness in law for pure jump type Markov processes. Probab. Theory Relat. Fields 79, 271–287 (1988)

    Article  MATH  Google Scholar 

  4. Bo, L., Jiang, Y., Wang, Y.: On a class of stochastic Anderson models with fractional noises. Stoch. Anal. Appl. 26, 256–273 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Hu, Y.: Chaos expansion of heat equations with white noise potentials. Potential Anal. 16, 45–66 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hu, Y.: Heat equations with fractional white noise potentials. Appl. Math. Optim. 43, 221–243 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hu, Y., Nualart, D.: Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields 143, 285–328 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jacob, N., Leopold, H.G.: Pseudo differential operators with variable order of differentiation generating Feller semigroups. Integral Equ. Oper. Theory 17, 544–553 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jacob, N., Potrykus, A., Wu, J.-L.: Solving a non-linear stochastic pseudo-differential equation of Burgers type. Stoch. Process. Appl. 120, 2447–2467 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jiang, Y., Shi, K., Wang, Y.: Stochastic fractional Anderson models with fractional noises. Chin. Ann. Math. 31B(1), 101–118 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jiang, Y., Wei, T., Zhou, X.: Stochastic generalized Burgers equations driven by fractional noises. J. Differ. Equ. 252, 1934–1961 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kikuchi, K., Negoro, A.: On Markov process generated by pseudo-differential operator of variable order. Osaka J. Math. 34, 319–335 (1997)

    MathSciNet  MATH  Google Scholar 

  13. Kolokoltsov, V.: Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. 80, 725–768 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Komatsu, T.: Markov processes associted with certain integro-differential operators. Osaka J. Math. 10, 271–303 (1973)

    MathSciNet  MATH  Google Scholar 

  15. Komatsu, T.: Continuity estimates for solutions of parabolic equations associated with jump type Dirichlet forms. Osaka J. Math. 25, 697–728 (1988)

    MathSciNet  MATH  Google Scholar 

  16. Komatsu, T.: Uniform estimates of fundamental solutions associted with non-local Dirichlet forms. Osaka J. Math. 32, 833–860 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Liu, J., Tudor, C.A.: Analysis of the density of the solution to a semilinear SPDE with fractional noise. Stochastics 88(7), 959–979 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Liu, J., Yan, L.: Solving a nonlinear fractional stochastic partial differential equation with fractional noise. J. Theor. Probab. 29, 307–347 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, J., Yan, L.: On a nonlinear stochastic pseudo-differential equation driven by fractional noise. Stoch. Dyn. (2017)

  20. Negoro, A.: Stable-like processes: construction of the transition density and the behavior of sample paths near \(t=0\). Osaka J. Math. 31(1), 189–214 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Nualart, D.: The Malliavin calculus and related topics, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  22. Walsh, J.B.: An introduction to stochastic partial differential equations. In: Ecole d’été de Probabilités de St. Flour XIV, Lecture Notes in Mathematics, vol. 1180, pp. 266–439. Springer, Berlin (1986)

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Authors and Affiliations

  1. Department of Statistics, Nanjing Audit University, 86 West Yushan Rd., Pukou, Nanjing, 211815, People’s Republic of China

    Junfeng Liu

  2. Laboratoire Paul Painlevé, Université de Lille 1, 59655, Villeneuve d’Ascq, France

    Ciprian A. Tudor

  3. Romanian Academy Institute of Mathematical Statistics and Applied Mathematics, Bucharest, Romania

    Ciprian A. Tudor

Authors
  1. Junfeng Liu
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  2. Ciprian A. Tudor
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Correspondence to Ciprian A. Tudor.

Additional information

Supported by NSFC (No. 11401313), NSFJS (No. BK20161579), 2014 QingLan Project and CPSF (Nos. 2014M560368, 2015T80475).

Partially supported by the MEC project PAI 80160047 Conycit, Chile and by the CNRS-FAPESP grant 267378.

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Liu, J., Tudor, C.A. Generalized Anderson Model with Time-Space Multiplicative Fractional Noise. Results Math 72, 1967–1989 (2017). https://doi.org/10.1007/s00025-017-0739-8

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  • DOI: https://doi.org/10.1007/s00025-017-0739-8

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