Mild solution of the parabolic equation driven by a $\sigma$-finite stochastic measure
Authors:
O. O. Vertsimakha and V. M. Radchenko
Translated by:
N. N. Semenov
Journal:
Theor. Probability and Math. Statist. 97 (2018), 17-32
MSC (2010):
Primary 60H15, 60G57, 60G17
DOI:
https://doi.org/10.1090/tpms/1045
Published electronically:
February 21, 2019
MathSciNet review:
3745996
Full-text PDF
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Abstract: Stochastic parabolic equation driven by a $\sigma$-finite stochastic measure in the interval $[0,T]\times \mathbb {R}$ is studied. The only condition imposed on the stochastic integrator is its $\sigma$-additivity in probability on bounded Borel sets. The existence, uniqueness, and Hölder continuity of a mild solution are proved. These results generalize those known earlier for usual stochastic measures.
References
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References
- D. Khoshnevisan, Analysis of Stochastic Partial Differential Equations, American Mathematical Soc., Providence, 2014. MR 3222416
- P. L. Chow, Stochastic Partial Differential Equations, CRC Press, Boca Raton, 2014. MR 3288853
- P. A. Cioica, K. H. Kim, K. Lee, and F. Lindner, On the $L_q$($L_p$)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains, Electron. J. Probab. 18 (2013), no. 82, 1–41. MR 3109621
- J. Dettweiler, L. Weis, and J. van Neerven, Space-time regularity of solutions of the parabolic stochastic Cauchy problem, Stoch. Anal. Appl. 24 (2006), no. 4, 843–869. MR 2241096
- J. B. Walsh, An Introduction to Stochastic Partial Differential Equations, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986. MR 876085
- L. Pryhara and G. Shevchenko, Approximations for a solution to stochastic heat equation with stable noise, Mod. Stoch. Theory Appl. 3 (2016), no. 2, 133–144. MR 3519720
- I. M. Bodnarchuk and G. M. Shevchenko, Heat equation in a multidimensional domain with a general stochastic measure, Theory Probab. Math. Statist. 93 (2016), 1–17. MR 3553436
- V. Radchenko, Riemann integral of a random function and the parabolic equation with a general stochastic measure, Theory Probab. Math. Statist. 87 (2013), 185–198. MR 3241455
- I. M. Bodnarchuk, Regularity of the mild solution of a parabolic equation with stochastic measure, Ukrain. Mat. Zh. 69 (2017), no. 1, 3–16; English transl. in Ukrain. Math. J. 69 (2017), no. 1, 1–18. MR 3631616
- V. M. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554
- S. Kwapień and W. A. Woycziński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992. MR 1167198
- V. N. Radchenko, Integrals with respect to general stochastic measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 1999. (Russian)
- A. M. Ilyin, A. S. Kalashnikov, and O. A. Oleynik, Linear second-order partial differential equations of the parabolic type, J. Math. Sci. 108 (2002), no. 4, 435–542.
- V. N. Radchenko, Evolution equations with general stochastic measures in Hilbert space, Theory Probab. Appl. 59 (2015), no. 2, 328–339. MR 3416054
- A. Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. 13 (1997), no. 2, 63–77. MR 1750304
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Additional Information
O. O. Vertsimakha
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
oksana.vertsim@gmail.com
V. M. Radchenko
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
vradchenko@univ.kiev.ua
Keywords:
Stochastic measure,
$\sigma$-finite stochastic measure,
stochastic parabolic equation,
mild solution,
Hölder continuity
Received by editor(s):
May 15, 2017
Published electronically:
February 21, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
