The Annals of Probability

Intermittency and multifractality: A case study via parabolic stochastic PDEs

Davar Khoshnevisan, Kunwoo Kim, and Yimin Xiao

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Let $\xi $ denote space–time white noise, and consider the following stochastic partial differential equations on $\mathbb{R}_{+}\times \mathbb{R}$: (i) $\dot{u}=\frac{1}{2}u"+u\xi $, started identically at one; and (ii) $\dot{Z}=\frac{1}{2}Z"+\xi $, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes.

We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on $\mathbb{R}_{+}\times \mathbb{R}^{d}$ with $d\ge 2$. Gregory Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question.

As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein–Uhlenbeck process on $\mathbb{R}$ are multifractal.

Throughout, we make extensive use of the macroscopic fractal theory of Barlow and Taylor [J. Phys. A 22 (1989) 2621–2628; Proc. Lond. Math. Soc. (3) 64 (1992) 125–152]. We expand on aspects of the Barlow–Taylor theory, as well.

Article information

Ann. Probab., Volume 45, Number 6A (2017), 3697-3751.

Received: March 2015
Revised: July 2016
First available in Project Euclid: 27 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 60K37: Processes in random environments

Intermittency multifractality macroscopic/large-scale Hausdorff dimension stochastic partial differential equations


Khoshnevisan, Davar; Kim, Kunwoo; Xiao, Yimin. Intermittency and multifractality: A case study via parabolic stochastic PDEs. Ann. Probab. 45 (2017), no. 6A, 3697--3751. doi:10.1214/16-AOP1147.

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  • [1] Albeverio, S., Molchanov, S. A. and Surgailis, D. (1994). Stratified structure of the Universe and Burgers’ equation—A probabilistic approach. Probab. Theory Related Fields 100 457–484.
  • [2] Albin, J. M. P. and Choi, H. (2010). A new proof of an old result by Pickands. Electron. Commun. Probab. 15 339–345.
  • [3] Barlow, M. T. and Taylor, S. J. (1989). Fractional dimension of sets in discrete spaces. J. Phys. A 22 2621–2628.
  • [4] Barlow, M. T. and Taylor, S. J. (1992). Defining fractal subsets of $\mathbb{Z}^{d}$. Proc. Lond. Math. Soc. (3) 64 125–152.
  • [5] Barral, J. and Seuret, S. (2011). A localized Jarník–Besicovitch theorem. Adv. Math. 226 3191–3215.
  • [6] Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78 1377–1401.
  • [7] Borodin, A. and Corwin, I. (2014). Moments and Lyapunov exponents for the parabolic Anderson model. Ann. Appl. Probab. 24 1172–1198.
  • [8] Carmona, R. A. and Molchanov, S. A. (1994). Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 viii+125.
  • [9] Chen, X. (2010). Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs 157. Amer. Math. Soc., Providence, RI.
  • [10] Chen, X. (2015). Precise intermittency for the parabolic Anderson equation with an $(1+1)$-dimensional time–space white noise. Ann. Inst. Henri Poincaré B, Calc. Probab. Stat. 51 1486–1499.
  • [11] Chen, X. (2016). Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise. Ann. Probab. 44 1535–1598.
  • [12] Collela, P. and Lanford, O. E. (1973). Appendix: Sample field behavior for the free Markov random field. In Constructive Quantum Field Theory (G. Velo and A. S. Wightman, eds.). Lecture Notes in Physics 25 44–70.
  • [13] Conus, D. (2013). Moments for the parabolic Anderson model: On a result of Hu and Nualart. Commun. Stoch. Anal. 7 125–152.
  • [14] Conus, D., Joseph, M. and Khoshnevisan, D. (2013). On the chaotic character of the stochastic heat equation, before the onset of intermitttency. Ann. Probab. 41 2225–2260.
  • [15] Conus, D., Joseph, M. and Khoshnevisan, D. (2013). Correlation-length bounds, and estimates for intermittent islands in parabolic SPDEs. Electron. J. Probab. 17 1–15.
  • [16] Conus, D., Joseph, M., Khoshnevisan, D. and Shiu, S.-Y. (2013). On the chaotic character of the stochastic heat equation, II. Probab. Theory Related Fields 156 483–533.
  • [17] Dalang, R., Khoshnevisan, D., Mueller, C., Nualart, D. and Xiao, Y. (2006). A Minicourse in Stochastic Partial Differential Equations (D. Khoshnevisan and F. Rassoul-Agha, eds.). Lecture Notes in Math. 1962. Springer, Berlin.
  • [18] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous S.P.D.E.’s. Electron. J. Probab. 4. 1–29.
  • [19] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [20] Foondun, M. and Khoshnevisan, D. (2009). Intermittence and nonlinear stochastic partial differential equations. Electron. J. Probab. 14 548–568.
  • [21] Gel’fand, I. M. and Vilenkin, N. Ya. (1964). Applications of Harmonic Analysis. Generalized Functions 4. Academic Press, New York. Translated from the Russian by Amiel Feinstein.
  • [22] Gibbon, J. D. and Titi, E. S. (2005). Cluster formation in complex multi-scale systems. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 3089–3097.
  • [23] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559–664.
  • [24] Hairer, M. and Labbé, C. (2017). Multiplicative stochastic heat equations on the whole space. J. Eur. Math. Soc. To appear. Preprint available at
  • [25] Harper, A. J. (2017). Pickand’s constant $H_{\alpha }$ does not equal $1/\Gamma (1/\alpha)$, for small $\alpha $. Bernoulli 23 582–602.
  • [26] Hu, Y. and Nualart, D. (2009). Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 285–328.
  • [27] Jaffard, S. (1999). The multifractal nature of Lévy processes. Probab. Theory Related Fields 114 207–227.
  • [28] Joseph, M., Khoshnevisan, D. and Mueller, C. (2017). Strong invariance and noise comparison principles for some parabolic SPDE. Ann. Probab. To appear. Preprint available at
  • [29] Kardar, M. (1987). Replica Bethe ansatz studies of two-dimensional interfaces with quenched random impurities. Nuclear Phys. B 290 582–602.
  • [30] Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 889–892.
  • [31] Kardar, M. and Zhang, Y.-C. (1987). Scaling of directed polymers in random media. Phys. Rev. Lett. 58 2087–2090.
  • [32] Khoshnevisan, D. (2014). Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics 119. Amer. Math. Soc., Providence, RI.
  • [33] Motoo, M. (1958). Proof of the law of iterated logarithm through diffusion equation. Ann. Inst. Statist. Math. 10 21–28.
  • [34] Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37 225–245.
  • [35] Mueller, C. and Nualart, D. (2008). Regularity of the density for the stochastic heat equation. Electron. J. Probab. 13 2248–2258.
  • [36] Naudts, J. (1988). Dimension of discrete fractal spaces. J. Phys. A 21 447–452.
  • [37] Orey, S. and Pruitt, W. E. (1973). Sample functions of the $N$-parameter Wiener process. Ann. Probab. 1 138–163.
  • [38] Paladin, G., Peliti, L. and Vulpiani, A. (1986). Intermittency as multifractality in history space. J. Phys. A 19 L991–L996.
  • [39] Paley, R. E. A. C. and Zygmund, A. (1932). A note on analytic functions on the circle. Math. Proc. Cambridge Philos. Soc. 28 266–272.
  • [40] Pickands, J. III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51–73.
  • [41] Qualls, C. and Watanabe, H. (1971). An asymptotic 0–1 behavior of Gaussian processes. Ann. Math. Stat. 42 2029–2035.
  • [42] Strassen, V. (1964). An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 3 211–226.
  • [43] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.
  • [44] Weber, M. (2004). Some examples of application of the metric entropy method. Acta Math. Hungar. 105 39–83.