Heat equation with a general stochastic measure in a bounded domain
Pub. online: 12 July 2024
Type: Research Article
Open Access
Open Access
Received
21 February 2024
21 February 2024
Revised
2 July 2024
2 July 2024
Accepted
3 July 2024
3 July 2024
Published
12 July 2024
12 July 2024
Abstract
A stochastic heat equation on $[0,T]\times B$, where B is a bounded domain, is considered. The equation is driven by a general stochastic measure, for which only σ-additivity in probability is assumed. The existence, uniqueness and Hölder regularity of the solution are proved.
References
Alois, E., Leon, J.A., Nualart, D.: Stochastic heat equation with random coefficients. Probab. Theory Relat. Fields 115, 41–94 (1999) MR1715545. https://doi.org/10.1007/s004400050236
Bodnarchuk, I.: Averaging principle for a stochastic cable equation. Modern Stoch. Theory Appl. 7(4), 449–467 (2020) MR4195646. https://doi.org/10.15559/20-vmsta168
Bodnarchuk, I.M., Shevchenko, G.M.: Heat equation in a multidimensional domain with a general stochastic measure. Theor. Probability ans Math. Statist. 93, 1–17 (2016) MR3553436. https://doi.org/10.1090/tpms/991
Cioica-Licht, P.A., Kim, K.-H., Lee, K.: On the regularity of the stochastic heat equation on polygonal domains in ${\mathbb{R}^{2}}$. J. Differential Equations 267, 6447–6479 (2019) MR4001061. https://doi.org/10.1016/j.jde.2019.06.027
Flandoli, F.: Dirichlet boundary value problem for stochastic parabolic equations: compatibility relations and regularity of the solutions. Stochastics 29, 331–357 (1990) MR1042066. https://doi.org/10.1080/17442509008833620
Friedman, A.: On quasi-linear parabolic equations of the second order II. Journ. Math. and Mech. 9, 539–556 (1960) MR0116134. https://doi.org/10.1512/iumj.1960.9.59030
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, New York (1964) MR0181836
Kamont, A.: A discrete characterization of Besov spaces. Approx. Theory Appl. (N.S.) 13, 63–77 (1997) MR1750304
Kim, K.-H.: A Sobolev space theory for parabolic stochastic PDEs driven by Lévy processes on ${C^{1}}$-domains. Stochastic Processes and their Applications 124, 440–474 (2014) MR3131301. https://doi.org/10.1016/j.spa.2013.08.008
Kwapień, S., Woyczyński, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Birkhäuser, Boston (1992) MR1167198. https://doi.org/10.1007/978-1-4612-0425-1
Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Soc. (1968) MR0244627
Lindner, F.: Singular behavior of the solution to the stochastic heat equation on a polygonal domain. Stoch. PDE: Anal. Comp. 2, 146–195 (2014) MR3249583. https://doi.org/10.1007/s40072-014-0030-x
Manikin, B.I.: Asymptotic properties of the parabolic equation driven by stochastic measure. Modern Stoch. Theory Appl. 9, 483–498 (2022) MR4510384
Memin, T., Mishura, Y., Valkeila, E.: Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51, 197–206 (2001) MR1822771. https://doi.org/10.1016/S0167-7152(00)00157-7
Radchenko, V.: Mild solution of the heat equation with a general stochastic measure. Studia Math. 194, 231–251 (2009) MR2539554. https://doi.org/10.4064/sm194-3-2
Radchenko, V.: Evolution equations driven by general stochastic measures in Hilbert space. Theory Probab. Appl. 59, 328–339 (2015) MR3416054. https://doi.org/10.1137/S0040585X97T987119
Radchenko, V.: General Stochastic Measures: Integration, Path Properties, and Equations. Wiley – ISTE, London (2022) MR4687103
Radchenko, V.: The Burgers equation driven by a stochastic measure. Modern Stoch. Theory Appl. 10, 229–246 (2023) MR4608186
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Chapman and Hall, London (1994) MR1280932
Shen, G., Wu, J.-L., Yin, X.: Averaging principle for fractional heat equations driven by stochastic measures. Appl. Math. Lett. 106, 106404 (2020) MR4090373. https://doi.org/10.1016/j.aml.2020.106404
Vertsimakha, O.O., Radchenko, V.M.: Mild solution of the parabolic equation driven by a σ-finite stochastic measure. Theor. Probab. Math. Statist. 97, 17–32 (2018) MR3745996. https://doi.org/10.1090/tpms/1045
