Electronic Communications in Probability

On the multifractal local behavior of parabolic stochastic PDEs

Abstract

Consider the stochastic heat equation $\dot{u} =\frac 12 u''+\sigma (u)\xi$ on $(0\,,\infty )\times \mathbb{R}$ subject to $u(0)\equiv 1$, where $\sigma :\mathbb{R} \to \mathbb{R}$ is a Lipschitz (local) function that does not vanish at $1$, and $\xi$ denotes space-time white noise. It is well known that $u$ has continuous sample functions [22]; as a result, $\lim _{t\downarrow 0}u(t\,,x)= 1$ almost surely for every $x\in \mathbb{R}$.

The corresponding fluctuations are also known [14, 16, 20]: For every fixed $x\in \mathbb{R}$, $t\mapsto u(t\,,x)$ looks locally like a fixed multiple of fractional Brownian motion (fBm) with index $1/4$. In particular, an application of Fubini’s theorem implies that, on an $x$-set of full Lebesgue measure, the short-time behavior of the peaks of the random function $t\mapsto u(t\,,x)$ are governed by the law of the iterated logarithm for fBm, up to possibly a suitable normalization constant. By contrast, the main result of this paper claims that, on an $x$-set of full Hausdorff dimension, the short-time peaks of $t\mapsto u(t\,,x)$ follow a non-iterated logarithm law, and that those peaks contain a rich multifractal structure a.s.

Large-time variations of these results were predicted in the physics literature a number of years ago and proved very recently in [10, 11]. To the best of our knowledge, the short-time results of the present paper are observed here for the first time.

Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 49, 11 pp.

Dates
Accepted: 7 September 2017
First available in Project Euclid: 2 October 2017

https://projecteuclid.org/euclid.ecp/1506931449

Digital Object Identifier
doi:10.1214/17-ECP86

Mathematical Reviews number (MathSciNet)
MR3710805

Zentralblatt MATH identifier
1378.60091

Citation

Huang, Jingyu; Khoshnevisan, Davar. On the multifractal local behavior of parabolic stochastic PDEs. Electron. Commun. Probab. 22 (2017), paper no. 49, 11 pp. doi:10.1214/17-ECP86. https://projecteuclid.org/euclid.ecp/1506931449

References

• [1] Borell, Christer (1975). The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30 207–216.
• [2] Dalang, R. C., Davar Khoshnevisan, Carl Mueller, David Nualart, and Yimin Xiao (2009). A Minicourse on Stochastic Partial Differential Equations, Springer, Berlin.
• [3] Dudley, R. M. (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Anal. 1 290–330.
• [4] Hairer, Martin (2013). Solving the KPZ equation, Ann. Math. 178(2) 559–664.
• [5] Hairer, Martin (2015). A theory of regularity structures. Invent. Math. 198(2) 269–504.
• [6] Hairer, Martin and Cyril Labbé (2015). Multiplicative stochastic heat equations on the whole space. Preprint. Document available electronically at arXiv:1504.07162.
• [7] Hairer, Martin and Étienne Pardoux (2015). A Wong–Zakai theorem for stochastic PDEs, J. Math. Soc. Japan 67(4) 1551–1604.
• [8] Joyce, H. and D. Preiss (1995). On the existence of subsets of finite positive packing measure, Mathematika 42 15–24.
• [9] Khoshnevisan, Davar (2002). Multiparameter Processes, Springer, New York.
• [10] Khoshnevisan, Davar, Kunwoo Kim, and Yimin Xiao (2016). Intermittency and multifractality: A case study via stochastic PDEs, Ann. Probab. (to appear). Document available electronically at arXiv:1503.06249.
• [11] Khoshnevisan, Davar, Kunwoo Kim, and Yimin Xiao (2016). A macroscopic multifractal analysis of parabolic stochastic PDEs (in preparation).
• [12] Khoshnevisan, Davar, Yuval Peres, and Yimin Xiao (2000). Limsup random fractals, Electr. J. Probab. 5(4) 1–24.
• [13] Khoshnevisan, Davar and Zhan Shi: Fast sets and points for fractional Brownian motion. In Séminaire de Probabilités, XXXIV, volume 1729 of Lecture Notes in Math., pages 393-416. Springer, Berlin, 2000.
• [14] Khoshnevisan, Davar, Jason Swanson, Yimin Xiao, and Liang Zhang (2013). Weak existence of a solution to a differential equation driven by a very rough fBm. Preprint. Document available electronically at arXiv:1309.3613.
• [15] Ledoux, Michel (2001). The Concentration of Measure Phenomenon, Amer. Math. Soc., Rhose Island.
• [16] Lei, Pedro and David Nualart (2009). A decomposition of the bifractional Brownian motion and some applications, Stat. Probab. Lett. 79 619–624.
• [17] Marcus, Michael B. and Jay Rosen (2006). Markov Processes, Gaussian Processes, and Local Times, Cambridge University Press, Cambridge.
• [18] Matilla, Pertti (1995). Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge.
• [19] Peres, Yuval (1996). Intersection-equivalence of Brownian paths and certain branching processes, Comm. Math. Phys. 177 417–434.
• [20] Pospíšil, Jan and Rogert Tribe (2007). Parameter estimates and exact variations for stochastic heat equations driven by space-time white noise, Stoch. Anal. Appl. 25(3) 593–611.
• [21] Sudakov, V. N. and B. S. T’sirelson (1978). Extremal properties of half-spaces for spherically-invariant measures. J. Soviet Math. 9 9–18; translated from Zap. Nauch. Sem. L.O.M.I. 41 14–24 (1974).
• [22] Walsh, John B. (1986). An Introduction to Stochastic Partial Differential Equations, in: École d’été de probabilités de Saint-Flour, XIV—1984, 265–439, Lecture Notes in Math., vol. 1180, Springer, Berlin.