## The Annals of Probability

### Intermittency for the stochastic heat equation with Lévy noise

#### Abstract

We investigate the moment asymptotics of the solution to the stochastic heat equation driven by a $(d+1)$-dimensional Lévy space-time white noise. Unlike the case of Gaussian noise, the solution typically has no finite moments of order $1+2/d$ or higher. Intermittency of order $p$, that is, the exponential growth of the $p$th moment as time tends to infinity, is established in dimension $d=1$ for all values $p\in (1,3)$, and in higher dimensions for some $p\in (1,1+2/d)$. The proof relies on a new moment lower bound for stochastic integrals against compensated Poisson measures. The behavior of the intermittency exponents when $p\to 1+2/d$ further indicates that intermittency in the presence of jumps is much stronger than in equations with Gaussian noise. The effect of other parameters like the diffusion constant or the noise intensity on intermittency will also be analyzed in detail.

#### Article information

Source
Ann. Probab., Volume 47, Number 4 (2019), 1911-1948.

Dates
Revised: March 2018
First available in Project Euclid: 4 July 2019

https://projecteuclid.org/euclid.aop/1562205693

Digital Object Identifier
doi:10.1214/18-AOP1297

Mathematical Reviews number (MathSciNet)
MR3980911

#### Citation

Chong, Carsten; Kevei, Péter. Intermittency for the stochastic heat equation with Lévy noise. Ann. Probab. 47 (2019), no. 4, 1911--1948. doi:10.1214/18-AOP1297. https://projecteuclid.org/euclid.aop/1562205693

#### References

• [1] Ahn, H. S., Carmona, R. A. and Molchanov, S. A. (1992). Nonstationary Anderson model with Lévy potential. In Stochastic Partial Differential Equations and Their Applications (Charlotte, NC, 1991). Lect. Notes Control Inf. Sci. 176 1–11. Springer, Berlin.
• [2] Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Applications of Mathematics (New York) 51. Springer, New York.
• [3] Balan, R. M. and Ndongo, C. B. (2016). Intermittency for the wave equation with Lévy white noise. Statist. Probab. Lett. 109 214–223.
• [4] Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78 1377–1401.
• [5] Besala, P. (1963). On solutions of Fourier’s first problem for a system of non-linear parabolic equations in an unbounded domain. Ann. Polon. Math. 13 247–265.
• [6] Bichteler, K. and Jacod, J. (1983). Random measures and stochastic integration. In Theory and Application of Random Fields (Bangalore, 1982). Lect. Notes Control Inf. Sci. 49 1–18. Springer, Berlin.
• [7] Carmona, R. A. and Molchanov, S. A. (1994). Parabolic Anderson Model and Intermittency. Amer. Math. Soc., Providence, RI.
• [8] Chen, B., Chong, C. and Klüppelberg, C. (2016). Simulation of stochastic Volterra equations driven by space–time Lévy noise. In The Fascination of Probability, Statistics and Their Applications 209–229. Springer, Cham.
• [9] Chen, L. and Dalang, R. C. (2015). Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions. Ann. Probab. 43 3006–3051.
• [10] Chong, C. (2017). Lévy-driven Volterra equations in space and time. J. Theoret. Probab. 30 1014–1058.
• [11] Chong, C. (2017). Stochastic PDEs with heavy-tailed noise. Stochastic Process. Appl. 127 2262–2280.
• [12] Conus, D. and Khoshnevisan, D. (2012). On the existence and position of the farthest peaks of a family of stochastic heat and wave equations. Probab. Theory Related Fields 152 681–701.
• [13] Cranston, M., Mountford, T. S. and Shiga, T. (2005). Lyapunov exponent for the parabolic Anderson model with Lévy noise. Probab. Theory Related Fields 132 321–355.
• [14] Dalang, R. C. and Mueller, C. (2009). Intermittency properties in a hyperbolic Anderson problem. Ann. Inst. Henri Poincaré Probab. Stat. 45 1150–1164.
• [15] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential B. Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam. Translated from the French by J. P. Wilson.
• [16] Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
• [17] Foondun, M. and Khoshnevisan, D. (2009). Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 548–568.
• [18] Foondun, M. and Khoshnevisan, D. (2010). On the global maximum of the solution to a stochastic heat equation with compact-support initial data. Ann. Inst. Henri Poincaré Probab. Stat. 46 895–907.
• [19] Hu, Y., Huang, J. and Nualart, D. (2016). On the intermittency front of stochastic heat equation driven by colored noises. Electron. Commun. Probab. 21 Paper No. 21, 13.
• [20] Khoshnevisan, D. (2014). Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics 119. Amer. Math. Soc., Providence, RI.
• [21] Marinelli, C. and Röckner, M. (2014). On maximal inequalities for purely discontinuous martingales in infinite dimensions. In Séminaire de Probabilités XLVI. Lecture Notes in Math. 2123 293–315. Springer, Cham.
• [22] Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37 225–245.
• [23] Saint Loubert Bié, E. (1998). Étude d’une EDPS conduite par un bruit poissonnien. Probab. Theory Related Fields 111 287–321.
• [24] Shiryaev, A. N. (1996). Probability, 2nd ed. Graduate Texts in Mathematics 95. Springer, New York. Translated from the first (1980) Russian edition by R. P. Boas.
• [25] Veraar, M. C. (2006). Stochastic integration in Banach spaces and applications to parabolic evolution equations. Ph.D. thesis, Technical Univ. Delft.
• [26] 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.