## Electronic Journal of Probability

### Wavelet analysis of the Besov regularity of Lévy white noise

#### Abstract

We characterize the local smoothness and the asymptotic growth rate of the Lévy white noise. We do so by characterizing the weighted Besov spaces in which it is located. We extend known results in two ways. First, we obtain new bounds for the local smoothness via the Blumenthal-Getoor indices of the Lévy white noise. We also deduce the critical local smoothness when the two indices coincide, which is true for symmetric-$\alpha$-stable, compound Poisson, and symmetric-gamma white noises to name a few. Second, we express the critical asymptotic growth rate in terms of the moment properties of the Lévy white noise. Previous analyses only provided lower bounds for both the local smoothness and the asymptotic growth rate. Showing the sharpness of these bounds requires us to determine in which Besov spaces a given Lévy white noise is (almost surely) not. Our methods are based on the wavelet-domain characterization of Besov spaces and precise moment estimates for the wavelet coefficients of the Lévy white noise.

#### Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 158, 38 pp.

Dates
Accepted: 11 November 2020
First available in Project Euclid: 24 December 2020

https://projecteuclid.org/euclid.ejp/1608779099

Digital Object Identifier
doi:10.1214/20-EJP554

#### Citation

Aziznejad, Shayan; Fageot, Julien. Wavelet analysis of the Besov regularity of Lévy white noise. Electron. J. Probab. 25 (2020), paper no. 158, 38 pp. doi:10.1214/20-EJP554. https://projecteuclid.org/euclid.ejp/1608779099

#### References

• [1] A. Abdesselam, A second-quantized kolmogorov–chentsov theorem via the operator product expansion, Communications in Mathematical Physics (2020), 1–54.
• [2] R.J. Adler, D. Monrad, R.H. Scissors, and R. Wilson, Representations, decompositions and sample function continuity of random fields with independent increments, Stochastic Process. Appl. 15 (1983), no. 1, 3–30.
• [3] D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2009.
• [4] O.E. Barndorff-Nielsen, Processes of normal inverse Gaussian type, Finance and Stochastics 2 (1997), no. 1, 41–68.
• [5] D. Berger, Lévy driven linear and semilinear stochastic partial differential equations, arXiv preprint arXiv:arXiv:1907.01926 (2019).
• [6] D. Berger, Lévy driven CARMA generalized processes and stochastic partial differential equations, Stochastic Processes and their Applications 130 (2020), no. 10, 5865–5887.
• [7] J. Bertoin, Lévy processes, vol. 121, Cambridge University Press, 1998.
• [8] H. Biermé, O. Durieu, and Y. Wang, Generalized random fields and Lévy’s continuity theorem on the space of tempered distributions, Commun. Stoch. Anal. 12 (2018), no. 4, Article 4, 427–445.
• [9] R.M. Blumenthal and R.K. Getoor, Sample functions of stochastic processes with stationary independent increments, Journal of Mathematics and Mechanics 10 (1961), 493–516.
• [10] E. Bostan, J. Fageot, U.S. Kamilov, and M. Unser, MAP estimators for self-similar sparse stochastic models, Proceedings of the Tenth International Workshop on Sampling Theory and Applications (SampTA13), Bremen, Germany, 2013, pp. 197–199.
• [11] B. Böttcher, R.L. Schilling, and J. Wang, Lévy Matters iii: Lévy-Type Processes: Construction, Approximation and Sample Path Properties, vol. 2099, Springer, 2014.
• [12] P.J. Brockwell and J. Hannig, CARMA $(p,q)$ generalized random processes, Journal of Statistical Planning and Inference 140 (2010), no. 12, 3613–3618.
• [13] P. Cartier, Processus aléatoires généralisés, Séminaire Bourbaki 8 (1963), 425–434.
• [14] C. Chong, R.C. Dalang, and T. Humeau, Path properties of the solution to the stochastic heat equation with Lévy noise, Stochastics and Partial Differential Equations: Analysis and Computations 7 (2019), no. 1, 123–168.
• [15] Z. Ciesielski, Orlicz spaces, spline systems, and brownian motion, Constructive Approximation 9 (1993), no. 2-3, 191–208.
• [16] Z. Ciesielski, G. Kerkyacharian, and B. Roynette, Quelques espaces fonctionnels associés à des processus gaussiens, Studia Mathematica 107 (1993), no. 2, 171–204.
• [17] P.A. Cioica and S. Dahlke, Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains, International Journal of Computer Mathematics 89 (2012), no. 18, 2443–2459.
• [18] P.A. Cioica, S. Dahlke, N. Döhring, S. Kinzel, F. Lindner, T. Raasch, K. Ritter, and R.L. Schilling, Adaptive wavelet methods for elliptic stochastic partial differential equations, BIT Numerical Mathematics 52 (2012), no. 3, 589–614.
• [19] E. Clarkson and H.H. Barrett, Characteristic functionals in imaging and image-quality assessment: tutorial, JOSA A 33 (2016), no. 8, 1464–1475.
• [20] R.C. Dalang and T. Humeau, Lévy processes and Lévy white noise as tempered distributions, The Annals of Probability 45 (2017), no. 6b, 4389–4418.
• [21] R.C. Dalang and T. Humeau, Random field solutions to linear SPDEs driven by symmetric pure jump Lévy space-time white noises, Electronic Journal of Probability 24 (2019).
• [22] R.C. Dalang and J.B. Walsh, The sharp Markov property of Lévy sheets, The Annals of Probability (1992), 591–626.
• [23] I. Daubechies, Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics 41 (1988), no. 7, 909–996.
• [24] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.
• [25] C.S. Deng and R.L. Schilling, On shift Harnack inequalities for subordinate semigroups and moment estimates for Lévy processes, Stochastic Processes and Their Applications 125 (2015), 3851–3878.
• [26] A. Durand and S. Jaffard, Multifractal analysis of Lévy fields, Probability Theory and Related Fields 153 (2012), no. 1-2, 45–96.
• [27] D.E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Tracts in Mathematics, vol. 120, Cambridge University Press, Cambridge, 2008.
• [28] J. Fageot, Gaussian versus sparse stochastic processes: Construction, regularity, compressibility, EPFL thesis no. 7657 (2017), 231 p., Swiss Federal Institute of Technology Lausanne (EPFL), 2017.
• [29] J. Fageot, A. Amini, and M. Unser, On the continuity of characteristic functionals and sparse stochastic modeling, Journal of Fourier Analysis and Applications 20 (2014), 1179–1211.
• [30] J. Fageot, E. Bostan, and M. Unser, Wavelet statistics of sparse and self-similar images, SIAM Journal on Imaging Sciences 8 (2015), no. 4, 2951–2975.
• [31] J. Fageot, A. Fallah, and M. Unser, Multidimensional Lévy white noise in weighted Besov spaces, Stochastic Processes and Their Applications 127 (2017), no. 5, 1599–1621.
• [32] J. Fageot and T. Humeau, Unified view on Lévy white noises: General integrability conditions and applications to linear SPDE, arXiv preprint arXiv:arXiv:1708.02500 (2017).
• [33] J. Fageot, V. Uhlmann, and M. Unser, Gaussian and sparse processes are limits of generalized Poisson processes, Applied and Computational Harmonic Analysis, arXiv preprint arXiv:arXiv:1702.05003 (in press).
• [34] J. Fageot and M. Unser, Scaling limits of solutions of linear stochastic differential equations driven by Lévy white noises, Journal of Theoretical Probability 32 (2019), no. 3, 1166–1189.
• [35] J. Fageot, M. Unser, and J.P. Ward, On the Besov regularity of periodic Lévy noises, Applied and Computational Harmonic Analysis 42 (2017), no. 1, 21–36.
• [36] J. Fageot, M. Unser, and J.P. Ward, The $n$-term approximation of periodic generalized Lévy processes, Journal of Theoretical Probability (in press).
• [37] W. Farkas, N. Jacob, and R.L. Schilling, Function spaces related to continuous negative definite functions: $\psi$-Bessel potential spaces, Dissertationes Math. (Rozprawy Mat.) 393 (2001), 1–62.
• [38] X. Fernique, Processus linéaires, processus généralisés, Annales de l’Institut Fourier 17 (1967), 1–92.
• [39] P. Flandrin, Wavelet analysis and synthesis of fractional Brownian motion, IEEE Transactions on Information Theory 38 (1992), no. 2, 910–917.
• [40] I.M. Gel’fand, Generalized random processes, Doklady Akademii Nauk SSSR 100 (1955), 853–856.
• [41] I.M. Gel’fand and N.Y. Vilenkin, Generalized Functions. Vol. 4: Applications of Harmonic Analysis, Academic Press, New York-London, 1964.
• [42] M. Griffiths and M. Riedle, Modelling Lévy space-time white noises, arXiv preprint arXiv:arXiv:1907.04193 (2019).
• [43] M. Hairer, A theory of regularity structures, Inventiones mathematicae 198 (2014), no. 2, 269–504.
• [44] M. Hairer and C. Labbé, The reconstruction theorem in Besov spaces, Journal of Functional Analysis 273 (2017), no. 8, 2578–2618.
• [45] V. Herren, Lévy-type processes and Besov spaces, Potential Analysis 7 (1997), no. 3, 689–704.
• [46] C. Houdré and R. Kawai, On layered stable processes, Bernoulli 13 (2007), no. 1, 252–278.
• [47] T. Hytönen and M.C. Veraar, On Besov regularity of Brownian motions in infinite dimensions, Probability and Mathematical Statistics 28 (2008), no. 1, 143–162.
• [48] K. Itô, Stationary random distributions, Kyoto Journal of Mathematics 28 (1954), no. 3, 209–223.
• [49] K. Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces, vol. 47, SIAM, 1984.
• [50] S. Jaffard, The multifractal nature of Lévy processes, Probability Theory and Related Fields 114 (1999), no. 2, 207–227.
• [51] M. Kabanava, Tempered Radon measures, Revista Matemática Complutense 21 (2008), no. 2, 553–564.
• [52] S. Koltz, T.J. Kozubowski, and K. Podgorski, The laplace distribution and generalizations, Boston, MA: Birkhauser, 2001.
• [53] F. Kühn, Existence and estimates of moments for Lévy-type processes, Stochastic Processes and Their Applications 127 (2017), no. 3, 1018–1041.
• [54] F. Kühn, Lévy Matters vi: Lévy-type processes: Moments, construction and heat kernel estimates, vol. 2187, Springer, 2017.
• [55] F. Kühn and R.L. Schilling, On the domain of fractional Laplacians and related generators of Feller processes, Journal of Functional Analysis 276 (2019), no. 8, 2397–2439.
• [56] G. Laue, Remarks on the relation between fractional moments and fractional derivatives of characteristic functions, Journal of Applied Probability (1980), 456–466.
• [57] H. Luschgy and G. Pagès, Moment estimates for Lévy processes, Electronic Communications in Probability 13 (2008), 422–434.
• [58] S. Mallat, A Wavelet Tour of Signal Processing, third ed., Elsevier/Academic Press, Amsterdam, 2009, The sparse way, With contributions from G. Peyré.
• [59] Y. Meyer, Wavelets and operators, Cambridge Studies in Advanced Mathematics, vol. 37, Cambridge University Press, Cambridge, 1992.
• [60] Y. Meyer, F. Sellan, and M.S. Taqqu, Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion, Journal of Fourier Analysis and Applications 5 (1999), no. 5, 465–494.
• [61] R.A. Minlos, Generalized random processes and their extension in measure, Trudy Moskovskogo Matematicheskogo Obshchestva 8 (1959), 497–518.
• [62] T. Mori, Representation of linearly additive random fields, Probability theory and related fields 92 (1992), no. 1, 91–115.
• [63] E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, arXiv preprint arXiv:arXiv:1104.4345 (2011).
• [64] P. Pad and M. Unser, Optimality of operator-like wavelets for representing sparse AR(1) processes, IEEE Transactions on Signal Processing 63 (2015), no. 18, 4827–4837.
• [65] W.E. Pruitt, The growth of random walks and Lévy processes, The Annals of Probability 9 (1981), no. 6, 948–956.
• [66] B.S. Rajput and J. Rosinski, Spectral representations of infinitely divisible processes, Probability Theory and Related Fields 82 (1989), no. 3, 451–487.
• [67] M. Rosenbaum, First order $p$-variations and Besov spaces, Statistics & Probability Letters 79 (2009), no. 1, 55–62.
• [68] B. Roynette, Mouvement brownien et espaces de Besov, Stochastics: An International Journal of Probability and Stochastic Processes 43 (1993), no. 3-4, 221–260.
• [69] G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance, Stochastic Modeling, Chapman & Hall, New York, 1994.
• [70] K. Sato, Lévy Processes and Infinitely Divisible Distributions, vol. 68, Cambridge University Press, Cambridge, 2013.
• [71] R.L. Schilling, On Feller processes with sample paths in Besov spaces, Mathematische Annalen 309 (1997), no. 4, 663–675.
• [72] R.L. Schilling, Growth and Hölder conditions for the sample paths of Feller processes, Probability Theory and Related Fields 112 (1998), no. 4, 565–611.
• [73] R.L. Schilling, Function spaces as path spaces of Feller processes, Mathematische Nachrichten 217 (2000), no. 1, 147–174.
• [74] H.-J. Schmeisser and H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley Chichester, 1987.
• [75] B. Simon, Functional integration and quantum physics, vol. 86, Academic press, 1979.
• [76] B. Simon, Distributions and their Hermite expansions, Journal of Mathematical Physics 12 (2003), no. 1, 140–148.
• [77] P. Sjögren, Riemann sums for stochastic integrals and ${L}_{p}$ moduli of continuity, Probability Theory and Related Fields 59 (1982), no. 3, 411–424.
• [78] L.N. Slobodeckii, Generalized Sobolev spaces and their application to boundary problems for partial differential equations, Leningrad. Gos. Ped. Inst. Ucen. Zap 197 (1958), 54–112.
• [79] F. Trèves, Topological Vector Spaces, Distributions and Kernels, Academic Press, New York-London, 1967.
• [80] H. Triebel, Function Spaces and Wavelets on Domains, EMS Tracts in Mathematics, vol. 7, European Mathematical Society (EMS), Zürich, 2008.
• [81] Hans Triebel, Theory of Function Spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2010.
• [82] M. Unser and P. D. Tafti, Stochastic models for sparse and piecewise-smooth signals, IEEE Transactions on Signal Processing 59 (2011), no. 3, 989–1006.
• [83] M. Unser and P. D. Tafti, An introduction to sparse stochastic processes, Cambridge University Press, 2014.
• [84] N.G. Ushakov, Selected topics in characteristic functions, Walter de Gruyter, 2011.
• [85] M.C. Veraar, Correlation inequalities and applications to vector-valued gaussian random variables and fractional brownian motion, Potential Analysis 30 (2009), no. 4, 341–370.
• [86] M.C. Veraar, Regularity of Gaussian white noise on the $d$-dimensional torus, Marcinkiewicz centenary volume 95 (2011), 385–398.
• [87] J.B. Walsh, An introduction to stochastic partial differential equations, École d’Été de Probabilités de Saint Flour XIV-1984, Springer, 1986, pp. 265–439.