Decompositions of stochastic convolution driven by a white-fractional Gaussian noise

Abstract

Let u=\{u(t,x);(t,x)\in {ℝ}_{+}\times ℝ\}be the solution to a linear stochastic heat equation driven by a Gaussian noise, which is a Brownian motion in time and a fractional Brownian motion in space with Hurst parameterH\in (\mathrm{0},\mathrm{1}): For any givenx\in ℝ(\text{resp}\text{.},t\in {ℝ}_{+}), we show a decomposition of the stochastic processt\to u(t,x)(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.,x\to u(t,x))as the sum of a fractional Brownian motion with Hurst parameter H/2 (resp., H) and a stochastic process with C^∞-continuous trajectories. Some applications of those decompositions are discussed.

Keywords

Stochastic heat equation / fractional Brownian motion (fBm) / path regularity / law of the iterated logarithm

References

[1] Balan R, Jolis M, Quer-Sardanyons L. SPDEs with affine multiplicative fractional noise in space with index 14<H<12. Electron J Probab, 2015, 20(54): 1–36

[2] Balan R, Jolis M, Quer-Sardanyons L. SPDEs with rough noise in space: Hölder continuity of the solution. Statist Probab Lett, 2016, 119: 310–316

[3] Dalang R C. Extending martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e's. Electron J Probab, 1999, 4(6): 1–29

[4] Foondun M, Khoshnevisan D, Mahboubi P. Analysis of the gradient of the solution to a stochastic heat equation via fractional Brownian motion. Stoch Partial Differ Equ Anal Comput, 2015, 3: 133–158

[5] Harnett D, Nualart D. Decomposition and limit theorems for a class of self-similar Gaussian processes. In: Baudoin F, Peterson J, eds. Stochastic Analysis and Related Topics—A Festschrift in Honor of Rodrigo Bañuelos. Progr Probab, Vol 72. Basel: Birkhñuser/Springer, 2017, 99–116

[6] Herrell R, Song R M, Wu D S, Xiao Y M. Sharp space time regularity of the solution to stochastic heat equation driven by fractional colored noise. Stoch Anal Appl, 2020, 38(4): 747–768

[7] Hu Y Z. Some recent progress on stochastic heat equations. Acta Math Sci Ser B Engl Ed, 2019, 39(3): 874–914

[8] Hu Y Z, Huang J Y, Lê K, Nualart D, Tindel S. Stochastic heat equation with rough dependence in space. Ann Probab, 2017, 45(6): 4561–4616

[9] Hu Y Z, Wang X. Stochastic heat equation with general noise. arXiv: 1912.05624

[10] Khalil M, Tudor C. On the distribution and q-variation of the solution to the heat equation with fractional Laplacian. Probab Math Statist, 2019, 39(2): 315–335

[11] Khoshnevisan D. Analysis of Stochastic Partial Differential Equations. CBMS Reg Conf Ser Math, No 119. Providence: Amer Math Soc, 2014

[12] Lei P, Nualart D. A decomposition of the bifractional Brownian motion and some applications. Statist Probab Lett, 2009, 79(5): 619–624

[13] Marcus M B, Rosen J. Markov Processes, Gaussian Processes, and Local Times. Cambridge Stud Adv Math, Vol 100. Cambridge: Cambridge Univ Press, 2006

[14] Mishura Y. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Math, Vol 1929. Berlin: Springer-Verlag, 2008

[15] Mishura Y, Ralchenko K, Shevchenko G. Existence and uniqueness of a mild solution to the stochastic heat equation with white and fractional noises. Theory Probab Math Statist, 2019, 98: 149–170

[16] Mishura Y, Ralchenko K, Zili M. On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity. Statist Probab Lett, 2020, 159: 108682

[17] Monrad D, Rootzén H. Small values of Gaussian processes and functional laws of the iterated logarithm. Probab Theory Related Fields, 1995, 192(91): 173–192

[18] Mueller C, Wu Z. Erratum: A connection between the stochastic heat equation and fractional Brownian motion and a simple proof of a result of Talagrand. Electron Commun Probab, 2012, 17(8): 1–10

[19] Peszat S, Zabczyk J. Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process Appl, 1997, 72(2): 187–204

[20] Pipiras V, Taqqu M S. Integration questions related to fractional Brownian motion. Probab Theory Related Fields, 2000, 291: 251–291

[21] Prato G D, Zabczyk J. Stochastic Equations in Infinite Dimensions. 2nd ed. Encyclopedia Math Appl, Vol 45. Cambridge: Cambridge Univ Press, 2014

[22] Song J, Song X M, Xu F. Fractional stochastic wave equation driven by a Gaussian noise rough in space. Bernoulli, 2020, 26(4): 2699–2726

[23] Tindel S, Tudor C, Viens F. Stochastic evolution equations with fractional Brownian motion. Probab Theory Related Fields, 2003, 127: 186–204

[24] Tudor C A, Xiao Y M. Sample paths of the solution to the fractional-colored stochastic heat equation. Stoch Dyn, 2017, 17(1): 1750004

[25] Walsh J B. An introduction to stochastic partial differential equations. In: Carmona R ,Kesten H, Walsh J B, eds. École d'Été de Probabilités de Saint Flour XIV-1984. Lecture Notes in Math, Vol 1180. Berlin: Springer-Verlag, 1986, 265–439

[26] Xiao Y M. Strong local nondeterminism and sample path properties of Gaussian random fields. In: Qian L F, Lai T L, Shao Q M, eds. Asymptotic Theory in Probability and Statistics with Applications. Adv Lect Math (ALM). Beijing: Higher Education Press, 2008, 136–176

RIGHTS & PERMISSIONS

2021 Higher Education Press