Investigation of sample paths properties of sub-Gaussian type random fields, with application to stochastic heat equations
Pub. online: 27 February 2025
Type: Research Article
Open Access
Open Access
Received
12 May 2024
12 May 2024
Revised
1 December 2024
1 December 2024
Accepted
30 January 2025
30 January 2025
Published
27 February 2025
27 February 2025
Abstract
The paper presents bounds for the distributions of suprema for a particular class of sub-Gaussian type random fields defined over spaces with anisotropic metrics. The results are applied to random fields related to stochastic heat equations with fractional noise: bounds for the tail distributions of suprema and estimates for the rate of growth are provided for such fields.
References
Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer, New York (2007). 472 p. MR2319516
Balan, R., Jolis, M., Quer-Sardanyons, L.: SPDEs with affine multiplicative fractional noise in space with index $\frac{1}{4}\lt H\lt \frac{1}{2}$. Electron. J. Probab., 1–36 20(54). 2015. MR3354614. https://doi.org/10.1214/EJP.v20-3719
Basse-O’ConnorA, Graversen, S.-E., Pedersen, J.: Multiparameter processes with stationary increments: Spectral representation and integration. Electron. J. Probab. 17, 1–21 (2012). MR2968681. https://doi.org/10.1214/EJP.v17-2287
Beghin, L., Kozachenko, Yu., Orsingher, E., Sakhno, L.: On the Solutions of Linear Odd-Order Heat-Type Equations with Random Initial Conditions. J. Stat. Phys. 127(4), 721–739 (2007). MR2319850. https://doi.org/10.1007/s10955-007-9309-x
Buldygin, V.V., Kozachenko, Yu.V.: Metric characterization of random variables and random processes. Translations of Mathematical Monographs, vol. 188. AMS, American Mathematical Society, Providence, RI (2000). 257 p. MR1743716. https://doi.org/10.1090/mmono/188
Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, vol. 152. Cambridge University Press, (2014). MR3236753. https://doi.org/10.1017/CBO9781107295513
D’Ovidio, M., Orsingher, E., Sakhno, L.: Spectral densities related to some fractional stochastic differential equations. Electron. Commun. Probab. 21(18), 1–15 (2016). MR3485387. https://doi.org/10.1214/16-ecp4411
Dozzi, M., Kozachenko, Yu., Mishura, Yu., Ralchenko, K.: Asymptotic growth of trajectories of multifractional Brownian motion with statistical applications to drift parameter estimation. Stat. Inference Stoch. Process. 21(1), 21–52 (2018). MR3769831. https://doi.org/10.1007/s11203-016-9147-z
Giuliano Antonini, R., Kozachenko, Yu.V., Nikitina, T.: Spaces of φ-subgaussian random variables. Rend. Accad. Naz. Sci. Detta Accad. XL, Parte I, Mem. Mat. 121. Vol. XXVII, 95–124 (2003). MR2056414
Hong, J., Liu, Z., Optimal, S.D.: Hölder continuity and hitting probabilities for SPDEs with rough fractional noises. J. Math. Anal. Appl. 512(1), 126125 (2022). MR4390469. https://doi.org/10.1016/j.jmaa.2022.126125
Hopkalo, O., Sakhno, L.: Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes. Mod. Stoch. Theory Appl. 8(1), 41–62 (2021). MR4235563. https://doi.org/10.15559/21-vmsta171
Hopkalo, O.M., Sakhno, L.M., Vasylyk, O.I.: Properties of solutions to linear KdV equations with φ-sub-Gaussian initial conditions. Bulletin of the Taras Shevchenko National University of Kyiv. Physics and Mathematics 2022(2), 11–19 (2022). https://doi.org/10.17721/1812-5409.2022/2.1
Hopkalo, O.M., Sakhno, L.M., Vasylyk, O.I.: Bounds for the Tail Distributions of Suprema of Sub-Gaussian Type Random Fields. Austrian J. Stat. 52(SI), 54–70 (2023). https://doi.org/10.17713/ajs.v52iSI.1753
Jolis, M.: The Wiener integral with respect to second order processes with stationary increments. J. Math. Anal. Appl. 366(2), 607–620 (2010). MR2600506. https://doi.org/10.1016/j.jmaa.2010.01.058
Kozachenko Yu, V., Koval’chuk, Y.A.: Boundary value problems with random initial conditions and series of functions of $Su{b_{\varphi }}(\Omega )$. Ukr. Math. J. 50(4), 572–585 (1998). MR1698149. https://doi.org/10.1007/BF02487389
Kozachenko, Yu., Olenko, A.: Whitaker-Kotelnikov-Shanon approximation of φ-sub-Gaussian random processes. J. Math. Anal. Appl. 442(2), 924–946 (2016). MR3514327. https://doi.org/10.1016/j.jmaa.2016.05.052
Kozachenko, Yu., Orsingher, E., Sakhno, L., Vasylyk, O.: Estimates for functionals of solutions to higher-order heat-type equations with random initial condition. J. Stat. Phys. 172(6), 1641–1662 (2018). MR3856958. https://doi.org/10.1007/s10955-018-2111-0
Kozachenko, Yu., Orsingher, E., Sakhno, L., Vasylyk, O.: Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions. Mod. Stoch. Theory Appl. 7(1), 79–96 (2020). MR4085677. https://doi.org/10.15559/19-vmsta146
Kozachenko, Yu.V., Ostrovskij, E.I.: Banach spaces of random variables of sub-Gaussian type. Theory Probab. Math. Stat. 32, 45–56 (1986). MR0882158
Meerschaert, M.M., Wang, W., Xiao, Y.: Fernique-type inequalities and moduli of continuity for anisotropic Gaussian random fields. Trans. Am. Math. Soc. 365(2), 1081–1107 (2013). MR2995384. https://doi.org/10.1090/S0002-9947-2012-05678-9
Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press, (2007). MR2356959. https://doi.org/10.1017/CBO9780511721373
Sakhno, L.: Estimates for distributions of suprema of spherical random fields. Stat. Optim. Inf. Comput. 11(2), 186–195 (2023). MR4574745. https://doi.org/10.19139/soic-2310-5070-1705
Sakhno, L.: Investigation of Airy equations with random initial conditions. Electron. Commun. Probab. 28, 1–14 (2023). MR4568937. https://doi.org/10.1214/23-ECP522
Sakhno, L.M., Vasylyk, O.I.: Investigation of solutions to higher-order dispersive equations with φ-sub-Gaussian initial conditions. Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics 2(2021), 78–84 (2021). https://doi.org/10.17721/1812-5409.2021/2.11
Walsh, J.B.: An introduction to stochastic partial differential equations. In: Hennequin, P.L. (ed.) École d’Été de Probabilités de Saint Flour XIV – 1984. Lecture Notes in Mathematics, vol. 1180. Springer, Berlin, Heidelberg (1986). MR0876085. https://doi.org/10.1007/BFb0074920
Xiao, Y.: Sample paths properties of anisotropic Gaussian random fields. A minicourse on stochastic partial differential equations. In: Khoshnevisan, D., Rassoul-Agha, F. (eds.) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math., vol. 1962, pp. 145–212. Springer, New York (2010). MR2508776. https://doi.org/10.1007/978-3-540-85994-9_5
