Abstract
The solutions to certain stochastic partial differential equations with linear Gaussian noise constitute interesting examples of self-similar processes. In this chapter we analyze these classes of self-similar processes. We focus on the solution to the linear heat and wave equation driven by a Gaussian noise which behaves as a Brownian motion or fractional Brownian motion with respect to the time variable and is white or colored with respect to the space variable. We consider various aspects of these self-similar processes. In particular we present the conditions for the existence of the solution, the sharp regularity of their trajectories, we study the law of the solution to the linear heat equation and its connection with the bifractional Brownian motion.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R. Balan, C.A. Tudor, The stochastic heat equation with fractional-colored noise: existence of the solution. Latin Am. J. Probab. Math. Stat. 4, 57–87 (2008)
R.M. Balan, C.A. Tudor, The stochastic wave equation with fractional noise: a random field approach. Stoch. Process. Appl. 120, 2468–2494 (2010)
E. Bayraktar, V. Poor, R. Sircar, Estimating the fractal dimension of the SP 500 index using wavelet analysis. Int. J. Theor. Appl. Finance 7, 615–643 (2004)
T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems. Electron. Commun. Probab. 12, 161–172 (2007)
S. Bourguin, C.A. Tudor, On the law of the solution to a stochastic heat equation with fractional noise in time. Preprint (2011)
D. del Castillo-Negrete, B.A. Carreras, V.E. Lynch, Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach. Phys. Rev. Lett. 91 (2003)
J. Clarke De la Cerda, C.A. Tudor, Hitting times for the stochastic wave equation with fractional-colored noise. Rev. Mat. Iberoam. (2012, to appear)
R.C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDE’s. Electron. J. Probab. 4, 1–29 (1999). Erratum in Electr. J. Probab. 6 (2001) 5 pp.
R.C. Dalang, N. Frangos, The stochastic wave equation in two spatial dimensions. Ann. Probab. 26, 187–212 (1998)
R.C. Dalang, C. Mueller, Some non-linear SPDE’s that are second order in time. Electron. J. Probab. 8, 1–21 (2003)
R.C. Dalang, M. Sanz-Solé, Regularity of the sample paths of a class of second order SPDE’s. J. Funct. Anal. 227, 304–337 (2005)
R.C. Dalang, M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension three. Mem. Am. Math. Soc. 199(931), 1–70 (2009)
R.C. Dalang, M. Sanz-Solé, Criteria for hitting probabilities with applications to systems of stochastic wave equations. Bernoulli 16(4), 1343–1368 (2010)
G. Denk, D. Meintrup, S. Schaffer, Modeling, simulation and optimization of integrated circuits. Int. Ser. Numer. Math. 146, 251–267 (2004)
K. Dzhaparidze, H. van Zanten, A series expansion of fractional Brownian motion. Probab. Theory Relat. Fields 130, 39–55 (2004)
M. Gubinelli, A. Lejay, S. Tindel, Young integrals and SPDE’s. Potential Anal. 25, 307–326 (2006)
S.C. Kou, X. Sunney, Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule. Phys. Rev. Lett. 93(18) (2004)
J.A. Leon, S. Tindel, Ito’s formula for linear fractional PDEs. Stoch. Int. J. Probab. Stoch. Process. 80(5), 427–450 (2008)
M. Maejima, C.A. Tudor, Wiener integrals and a Non-central limit theorem for Hermite processes. Stoch. Anal. Appl. 25(5), 1043–1056 (2007)
B. Maslovski, D. Nualart, Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202, 277–305 (2003)
A. Millet, M. Sanz-Solé, A stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27, 803–844 (1999)
D. Nualart, Malliavin Calculus and Related Topics, 2nd edn. (Springer, New York, 2006)
D. Nualart, P.-A. Vuillermont, Variational solutions for partial differential equations driven by fractional a noise. J. Funct. Anal. 232, 390–454 (2006)
H. Ouahhabi, C.A. Tudor, Additive functionals of the solution to fractional stochastic heat equation. Preprint. J. Fourier Anal. Appl. (2012). doi:10.1007/s00041-013-9272-7
S. Peszat, J. Zabczyk, Nonlinear stochastic wave and heat equations. Probab. Theory Relat. Fields 116, 421–443 (2000)
L. Quer-Sardanyons, S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet. Stoch. Process. Appl. 117, 1448–1472 (2007)
M. Sanz-Solé, Malliavin Calculus with Applications to Stochastic Partial Differential Equations (EPFL Press, Lausanne, 2005)
J. Swanson, Variations of the solution to a stochastic heat equation. Ann. Probab. 35(6), 2122–2159 (2007)
S. Tindel, C.A. Tudor, F. Viens, Stochastic evolution equations with fractional Brownian motion. Probab. Theory Relat. Fields 127, 186–204 (2003)
F. Treves, Basic Linear Partial Differential Equations (Academic Press, New York, 1975)
J.B. Walsh, An introduction to stochastic partial differential equations, in Ecole d’Eté de Probabilités de Saint-Flour XIV. Lecture Notes in Math., vol. 1180 (Springer, Berlin, 1986), pp. 265–439
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Tudor, C.A. (2013). Solutions to the Linear Stochastic Heat and Wave Equation. In: Analysis of Variations for Self-similar Processes. Probability and Its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-00936-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-00936-0_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00935-3
Online ISBN: 978-3-319-00936-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)