Heat equation in a multidimensional domain with a general stochastic measure
Authors:
I. M. Bodnarchuk and G. M. Shevchenko
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 93 (2016), 1-17
MSC (2010):
Primary 60H15; Secondary 60G17, 60G57
DOI:
https://doi.org/10.1090/tpms/991
Published electronically:
February 7, 2017
MathSciNet review:
3553436
Full-text PDF Free Access
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Additional Information
Abstract: Stochastic heat equation on $[0,T]\times \mathbb {R}^d$, $d\ge 1$, driven by a general stochastic measure $\mu (t)$, $t\in [0,T]$, is studied in this paper. The existence, uniqueness, and Hölder regularity of a mild solution are proved.
References
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References
- V. Radchenko, Heat equation with general stochastic measure colored in time, Modern Stochastics: Theory and Applications 1 (2014), 129–138. MR 3316482
- V. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231–251. MR 2539554
- I. Bodnarchuk, Mild solution of a wave equation with a general random measure, Visnyk Kyiv University. Mathematics and Mechanics 24 (2010), 28–33. (Ukrainian)
- V. M. Radchenko, Cable equation with a general stochastic measure, Teor. Imovirnost. Matem. Statyst. 84 (2011), 123–130; English transl. in Theor. Probability and Math. Statist. 84 (2012) 131–138. MR 2857423
- V. Radchenko and M. Zähle, Heat equation with a general stochastic measure on nested fractals, Stat. Probab. Lett. 82 (2012), 699–704. MR 2887489
- R. M. Balan and C. A Tudor, Stochastic heat equation with multiplicative fractional-colored noise, J. Theor. Probab. 23 (2010), no. 3, 834–870. MR 2679959
- C. A. Tudor, Analysis of Variations for Self-similar Processes. A Stochastic Calculus Approach, Probability and Its Applications, Springer, Cham–Heidelberg, 2013. MR 3112799
- E. Nualart and L. Quer-Sardanyons, Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension, Stoch. Process. Appl. 122 (2012), 418–447. MR 2860455
- S. Kwapień and W. A. Woyczyński, 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, Kiev, 1999. (Russian)
- V. M. Radchenko, Integral equations with a general stochastic measure, Teor. Imovirnost. Matem. Statyst. 91 (2014), 154–163; English transl. in Theor. Probability and Math. Statist. 91 (2015), 169–179. MR 3364132
- V. N. Radchenko, Evolution equations driven by general stochastic measures in Hilbert space, Teor. Veroyatnost. Primenen. 59 (2014), no. 2, 375–386; English transl. in 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
- A. M. Il’in, A. S. Kalashnikov, and O. A. Oleinik, Linear equations of the second order of parabolic type, Uspekhi Mat. Nauk. 17 (1962), no. 3 (105), 3–146. (Russian) MR 0138888
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Additional Information
I. M. Bodnarchuk
Affiliation:
Department of Mathematical Analysis, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
robeiko_i@ukr.net
G. M. Shevchenko
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 6, Kyiv 03127, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Stochastic measure,
stochastic heat equation,
mild solution,
Hölder condition,
Besov space
Received by editor(s):
July 28, 2015
Published electronically:
February 7, 2017
Additional Notes:
The paper was prepared following the talk at the International 5conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015
The results of the paper have also been presented at the conference “Stochastic Processes in Abstract Spaces (SPAS 2015)”, October 14–16, 2015
Article copyright:
© Copyright 2017
American Mathematical Society
