Electronic Communications in Probability

Moment bounds for some fractional stochastic heat equations on the ball

Eulalia Nualart

Full-text: Open access

Abstract

In this paper, we obtain upper and lower bounds for the moments of the solution to a class of fractional stochastic heat equations on the ball driven by a Gaussian noise which is white in time and has a spatial correlation in space of Riesz kernel type. We also consider the space-time white noise case on an interval.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 41, 12 pp.

Dates
Received: 15 July 2017
Accepted: 25 June 2018
First available in Project Euclid: 25 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1532505672

Digital Object Identifier
doi:10.1214/18-ECP147

Mathematical Reviews number (MathSciNet)
MR3841402

Zentralblatt MATH identifier
1394.60069

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]
Secondary: 60G52: Stable processes 35K08: Heat kernel

Keywords
Stochastic heat equation fractional Laplacian Dirichlet boundary conditions heat kernel estimates

Rights
Creative Commons Attribution 4.0 International License.

Citation

Nualart, Eulalia. Moment bounds for some fractional stochastic heat equations on the ball. Electron. Commun. Probab. 23 (2018), paper no. 41, 12 pp. doi:10.1214/18-ECP147. https://projecteuclid.org/euclid.ecp/1532505672


Export citation

References

  • [1] Balan, R.M. and Conus, D. (2016), Intermittency for the wave and heat equations with fractional noise in time, The Annals of Probability 44, 1488–1534.
  • [2] Balan, R.M., Jolis, M. and Quer-Sardanyons, L. (2017), Intermittency for the Hyperbolic Anderson Model with rough noise in space, Stoch. Proc. Appl. 127, 2316–2338.
  • [3] Blumenthal, R.M. and Getoor, R.K. (1959), Asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math. 9, 399–408.
  • [4] Bogdan, K., Grzymny, T. and Ryznar, M. (2010), Heat kernel estimates for the fractional laplacian with Dirichlet conditions, The Annals of Probability 38, 1901–1923.
  • [5] Chen, Z.-Q. and Song, R. (1997), Intrinsic ultracontractivity and conditional gauge for symmetric stable processes, Journal of functional analysis 150, 204–239.
  • [6] Chen, Z.-Q., Kim, P. and Song, R. (2010), Heat kernel estimates for the Dirichlet fractional Laplacian, J. Eur. Math. Soc. 12, 1307–1329.
  • [7] Dalang, R.C. (1996), Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s, Electron. J. Probab. 4, 1–29.
  • [8] Dalang, R., Khoshnevisan, D., Mueller, C., Nualart, D. and Xiao, Y. (2009), A minicourse on stochastic partial differential equations, Lecture Notes in Mathematics 1962, Springer, Berlin.
  • [9] Foondun, M. and Joseph, M. (2014), Remarks on non-linear noise excitability of some stochastic heat equations, Stochastic Processes and their Applications 124, 3429–3440.
  • [10] Foondun, M. and Khoshnevisan, D. (2009), Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14, 548–568.
  • [11] Foondun, M. and Khoshnevisan, D. (2013), On the stochastic heat equation with spatially-colored random forcing Trans. Amer. Math. Soc. 365, 409-458.
  • [12] Foondun, M. and Nualart, E. (2015), On the behaviour of stochastic heat equations on bounded domains, ALEA Lat. Am. J. Probab. Math. Stat. 12, 551–571.
  • [13] Foondun, M., Guerngar, N. and Nane, E. (2017), Some properties of non-linear fractional stochastic heat equations on bounded domains, Chaos, Solitons & Fractals 102, 86–93.
  • [14] Foondun, M., Liu, W. and Omaba, M. (2017), Moment bounds for a class of fractional stochastic heat equations, The Annals of Probability 45, 2131–2153.
  • [15] Henry, D. (1981), Geometric theory of semilinear parabolic equations (1981), Lecture Notes in Mathematics 840, Springer-Verlag, Berlin.
  • [16] Hu, Y., Huang, J., Nualart, D. and S. Tindel (2015), Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency, Electro. J. Probab. 20, 1–50.
  • [17] Khoshnevisan, D. and Kim, K. (2015), Non-linear noise excitation and intermittency under high disorder, Proc. Amer. Math. Soc. 143, 4073–4083.
  • [18] Kwiecinska, A.A. (1999), Stabilization of partial differential equations by noise, Stochastic Process. Appl. 79, 179–184.
  • [19] Liu, W., Tian, K. and Foondun, M. (2017), On Some Properties of a Class of Fractional Stochastic Heat Equations, Journal of Theoretical Probability 30, 1310–1333.
  • [20] Riahi, L. (2013), Estimates for Dirichlet heat kernels, instrinsic ultracontractivity and expected exit time on Lipschitz domains, Communications in Mathematical Analysis, 15, 115–130.
  • [21] Walsh, J.B. (1986), An Introduction to Stochastic Partial Differential Equations, École d’été de Probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math. 1180, Springer, Berlin, 265–439.
  • [22] Xie, B. (2016) Some effects of the noise intensity upon non-linear stochastic heat equations on $[0, 1]$, Stochastic Processes and their Applications 126, 1184–1205.