Communications in Mathematical Physics

, Volume 360, Issue 1, pp 307–346 | Cite as

A Macroscopic Multifractal Analysis of Parabolic Stochastic PDEs

  • Davar Khoshnevisan
  • Kunwoo KimEmail author
  • Yimin Xiao


It is generally argued that the solution to a stochastic PDE with multiplicative noise—such as \({\dot{u}=\frac12 u''+u\xi}\) , where \({\xi}\) denotes space-time white noise—routinely produces exceptionally-large peaks that are “macroscopically multifractal.” See, for example, Gibbon and Doering (Arch Ration Mech Anal 177:115–150, 2005), Gibbon and Titi (Proc R Soc A 461:3089–3097, 2005), and Zimmermann et al. (Phys Rev Lett 85(17):3612–3615, 2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (J Phys A 22(13):2621–2626, 1989; Proc Lond Math Soc (3) 64:125–152, 1992). The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as “stretch factors.” A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.


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  1. 1.
    Adler, R.J.: An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. Instit. Math. Statist., Hayward (1990)Google Scholar
  2. 2.
    Amir G., Corwin I., Quastel J.: Probability distribution of the free energy of the continuum directed random polymer in 1 + 1 dimensions. Commun. Pure Appl. Math. 64, 466–537 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barlow M.T., Taylor S.J.: Fractional dimension of sets in discrete spaces. J. Phys. A 22(13), 2621–2626 (1989)MathSciNetCrossRefzbMATHADSGoogle Scholar
  4. 4.
    Barlow M.T., Taylor S.J.: Defining fractal subsets of \({\mathbb{Z}^d}\) . Proc. Lond. Math. Soc. (3) 64, 125–152 (1992)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bertini L., Cancrini N.: The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78(5/6), 1377–1402 (1994)MathSciNetzbMATHADSGoogle Scholar
  6. 6.
    Carmona, R.A., Molchanov, S.A.: Parabolic Anderson Problem and Intermittency, Memoires of the Amer. Math. Soc., vol. 108. American Mathematical Society, Rhode Island (1994)Google Scholar
  7. 7.
    Chen X.: Precise intermittency for the parabolic Anderson equation with an (1 +  1)-dimensional time-space white noise. Ann. Inst. Henri Poinc. 51(4), 1486–1499 (2015)MathSciNetCrossRefzbMATHADSGoogle Scholar
  8. 8.
    Chen X.: Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise. Ann. Probab. 44(2), 1535–1598 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Conus D., Joseph M., Khoshnevisan D.: On the chaotic character of the stochastic heat equation, before the onset of intermittency. Ann. Probab. 41(3B), 2225–2260 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Corwin, I.: The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 01, 1130001 (2012) [76 pages] (2012)Google Scholar
  11. 11.
    Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)CrossRefzbMATHGoogle Scholar
  12. 12.
    Dalang, R.C.: Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.s, Electron. J. Probab. 4(6), 29 pp. (1999). (electronic). [Corrigendum: Electron. J. Probab. 6, no. 6, 5 pp. (2001)]Google Scholar
  13. 13.
    Dalang, R.C., Khoshnevisan, D., Mueller, C., Nualart, D., Xiao, Y.: In: Rassoul-Agha, F., Khoshnevisan, D. (eds.) A Minicourse on Stochastic Partial Differential Equations. Springer, Berlin (2009)Google Scholar
  14. 14.
    Edwards S.F., Wilkinson D.R.: The surface statistics of a granular aggregate. Proc. R Soc. Lond. Ser. A 381(1780), 17–31 (1982)CrossRefADSGoogle Scholar
  15. 15.
    Foondun, M., Khoshnevisan, D.: Intermittence and nonlinear stochastic partial differential equations. Electron. J. Probab. 14(21), 548–568 (2009)Google Scholar
  16. 16.
    Gibbon J.D., Doering C.R: Intermittency and regularity issues in 3D Navier–Stokes turbulence. Arch. Ration. Mech. Anal. 177, 115–150 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gibbon J.D., Titi E.S.: Cluster formation in complex multi-scale systems. Proc. R. Soc. A 461, 3089–3097 (2005)MathSciNetCrossRefzbMATHADSGoogle Scholar
  18. 18.
    Joseph M., Khoshnevisan D., Mueller C.: Strong invariance and noise comparison principles for some parabolic SPDEs. Ann. Probab. 45(1), 377–403 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kardar M.: Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nuclear Phys. B290, 582–602 (1987)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Kardar M., Parisi G., Zhang Y.-C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986)CrossRefzbMATHADSGoogle Scholar
  21. 21.
    Khoshnevisan, D.: Analysis of Stochastic Partial Differential Equations, Published by the AMS on behalf of CBMS Regional Conference Series in Mathematics 119, (116 pp), Providence RI (2014)Google Scholar
  22. 22.
    Khoshnevisan D., Kim K., Xiao Y.: Intermittency and multifractality: a case study via stochastic PDEs. Ann. Probab. 45(6A), 3697–3751 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mandelbrot B.B.: Fractal Geometry of Nature. W. H. Freeman and Co., San Francisco (1982)zbMATHGoogle Scholar
  24. 24.
    Molchanov S.A.: Ideas in the theory of random media. Acta Appl. Math. 22, 139–282 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mueller C.: On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37(4), 225–245 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Peszat S., Zabczyk J.: Nonlinear stochastic wave and heat equations. Probab. Theory Relat. Fields 116(3), 421–443 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Walsh, J.B.: An introduction to stochastic partial differential equations, In: École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, pp. 265–439. Springer, Berlin (1986)Google Scholar
  28. 28.
    Zeldovich Y.B., Ruzmaikin A.A., Sokoloff D.D.: The Almighty Chance. World Scientific, Singapore (1990)CrossRefzbMATHGoogle Scholar
  29. 29.
    Zimmermann M.G., Toral R., Piro O., San Miguel M.: Stochastic spatiotemporal intermittency and noise-induced transition to an absorbing phase. Phys. Rev. Lett. 85(17), 3612–3615 (2000)CrossRefADSGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Department of MathematicsPohang University of Science and Technology (POSTECH)PohangKorea
  3. 3.Department Statistics and ProbabilityMichigan State UniversityEast LansingUSA

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