The Annals of Probability

Stochastic heat equation with rough dependence in space

Yaozhong Hu, Jingyu Huang, Khoa Lê, David Nualart, and Samy Tindel

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Abstract

This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter $H\in (\frac{1}{4},\frac{1}{2})$ in the space variable. The existence and uniqueness of the solution $u$ are proved assuming the nonlinear coefficient $\sigma(u)$ is differentiable with a Lipschitz derivative and $\sigma(0)=0$.

Article information

Source
Ann. Probab., Volume 45, Number 6B (2017), 4561-4616.

Dates
Received: May 2015
Revised: December 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1513069267

Digital Object Identifier
doi:10.1214/16-AOP1172

Mathematical Reviews number (MathSciNet)
MR3737918

Zentralblatt MATH identifier
06838127

Subjects
Primary: 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 65C30: Stochastic differential and integral equations

Keywords
Stochastic heat equation fractional Brownian motion Feynman–Kac formula Wiener chaos expansion intermittency

Citation

Hu, Yaozhong; Huang, Jingyu; Lê, Khoa; Nualart, David; Tindel, Samy. Stochastic heat equation with rough dependence in space. Ann. Probab. 45 (2017), no. 6B, 4561--4616. doi:10.1214/16-AOP1172. https://projecteuclid.org/euclid.aop/1513069267


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References

  • [1] Bahouri, H., Chemin, J.-Y. and Danchin, R. (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften 343. Springer, Heidelberg.
  • [2] Balan, R. M., Jolis, M. and Quer-Sardanyons, L. (2015). SPDEs with affine multiplicative fractional noise in space with index $H<1/2$. Electron. J. Probab. 20 36 pp.
  • [3] Brzeźniak, Z. and Peszat, S. (1999). Space–time continuous solutions to SPDE’s driven by a homogeneous Wiener process. Studia Math. 137 261–299.
  • [4] Dalang, R. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 29 pp.
  • [5] Dalang, R. C. (2001). Corrections to: “Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s” [Electron J. Probab. 4 (1999) 29 pp. (electronic); MR1684157 (2000b:60132)]. Electron. J. Probab. 6 5 pp.
  • [6] Dalang, R. C. and Quer-Sardanyons, L. (2011). Stochastic integrals for spde’s: A comparison. Expo. Math. 29 67–109.
  • [7] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [8] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1981). Higher Transcendental Functions. Vol. III. Based on Notes Left by Harry Bateman. Robert E. Krieger Publishing, Melbourne, FL.
  • [9] Gyöngy, I. (1998). Existence and uniqueness results for semilinear stochastic partial differential equations. Stochastic Process. Appl. 73 271–299.
  • [10] Gyöngy, I. and Krylov, N. (1996). Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Related Fields 105 143–158.
  • [11] Gyöngy, I. and Nualart, D. (1999). On the stochastic Burgers’ equation in the real line. Ann. Probab. 27 782–802.
  • [12] Hanche-Olsen, H. and Holden, H. (2010). The Kolmogorov–Riesz compactness theorem. Expo. Math. 28 385–394.
  • [13] Peszat, S. and Zabczyk, J. (1997). Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process. Appl. 72 187–204.
  • [14] Pipiras, V. and Taqqu, M. S. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 251–291.
  • [15] Samko, S., Kilbas, A. and Marichev, O. (1993). Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Science Publishers, Yverdon.
  • [16] Zorko, C. T. (1986). Morrey space. Proc. Amer. Math. Soc. 98 586–592.