Potential Analysis

, Volume 45, Issue 1, pp 157–166 | Cite as

Decorrelation of Total Mass Via Energy

Article

Abstract

The main result of this small note is a quantified version of the assertion that if u and v solve two nonlinear stochastic heat equations, and if the mutual energy between the initial states of the two stochastic PDEs is small, then the total masses of the two systems are nearly uncorrelated for a very long time. One of the consequences of this fact is that a stochastic heat equation with regular coefficients is a finite system if and only if the initial state is integrable.

Keywords

The stochastic heat equation Finite particle systems Total mass Mutual energy 

Mathematics Subject Classification (2010)

Primary 60H15 60H25; Secondary 35R60 60K37 60J30 60B15 

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References

  1. 1.
    Chen, L., Dalang, R.C.: Moments and growth indices for nonlinear stochastic heat equation with rough initial conditions. Ann. Probab. 43(6), 3006–3051 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chen, L., Kim, K.: On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations. to appear in Ann. Instit. H. Poincaré. Preprint available at arXiv:1410.0604 (2014)
  3. 3.
    Dalang, R.C.: Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4(6) (1999). 29 pp. (electronic)Google Scholar
  4. 4.
    Dalang, RC., Mueller, C.: Some non-linear S.P.D.E.’s that are second order in time. Electron. J. Probab. 8(1) (2003). 21 pp. (electronic)Google Scholar
  5. 5.
    Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Foondun, M., Khoshnevisan, D.: On the global maximum of the solution to a stochastic heat equation with compact-support initial data. Ann. Inst. Henri Poincaré Probab. Stat. 46(4), 895–907 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Fukushima, M., Ōshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter & Co., Berlin (1994)CrossRefMATHGoogle Scholar
  8. 8.
    Khoshnevisan, D.: A primer on stochastic partial differential equations. In: A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics 1962 1–38, Springer, Berlin (2009)Google Scholar
  9. 9.
    Liggett, T.M.: Interacting Particle Systems. Springer-Verlag, New York (1985)CrossRefMATHGoogle Scholar
  10. 10.
    Mueller, C.: Some Tools and Results for Parabolic Stochastic Partial Differential Equations (English summary). In: A Minicourse on Stochastic Partial Differential Equations, 111–144, Lecture Notes in Mathematics, vol. 1962. Springer, Berlin (2009)Google Scholar
  11. 11.
    Mueller, C.: On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37(4), 225–245 (1991)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Spitzer, F.: Infinite systems with locally interacting components. Ann. Probab. 9, 349–364 (1981)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations. In: Ècole d’èté de probabilités de Saint-Flour, XIV—1984, 265–439. Lecture Notes in Mathematics, vol. 1180. Springer, Berlin (1986)Google Scholar

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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KansasLawrenceUSA
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  3. 3.Department of Industrial Engineering and ManagementTechnion—Israel Institute of TechnologyHaifaIsrael

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