Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)
Mild solution of a parabolic equation driven by a sigma-finite stochastic measure
O. O. Vertsimakha, V. M. Radchenko
Abstract: The stochastic parabolic equation on [0,T]xR driven by sigma-finite stochastic measure is investigated. For the integrator we assume sigma-additivity in probability on bounded Borel sets only. Existence and uniqueness of the mild solution is established. Holder continuity of the solution is proved. Thus, we get a generalisation of results obtained for usual stochastic measures in previous papers.
Keywords: Stochastic measure, sigma-finite stochastic measure, stochastic parabolic equation, mild solution, Holder continuity
1.D. Khoshnevisan. Analysis of stochastic partial differential equations. American Mathematical Soc., Providence, 2014.
2. P. L. Chow. Stochastic partial differential equations. CRC Press, Boca Raton, 2014.
3. P. A. Cioica, K. H. Kim, K. Lee, 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.
4. J.Dettweiler, L.Weis, J.van Neerven. Space-time regularity of solutions of the parabolic stochastic Cauchy problem. Stoch. Anal. Appl.,24 (2006), no.4, 843-869.
5. J.B.Walsh. An introduction to stochastic partial differential equations. Lecture Notes in Math., vol.1180, Springer, Berlin, 1986, 265-439.
6. L.Pryhara, G.Shevchenko. Approximations for a solution to stochastic heat equation with stable noise. Mod. Stoch. Theory Appl., 3(2016), no.2, 133-144.
7. I. M. Bodnarchuk, G. M. Shevchenko. Heat equation in a multidimensional domain with a general stochastic measure. Theory Probab. Math. Statist., 93(2016), 1-17.
8. 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.
9. I. M. Bodnarchuk. Regularity of the mild solution of a parabolic equation with stochastic measure. Ukrain. Mat. Zh., 69(2017), no.1, 3-16. (Ukrainian)
10. V. M. Radchenko. Mild solution of the heat equation with a general stochastic measure, Studia Math., 194 (2009), no. 3, 231-251.
11. S. Kwapien, W. A. Woyczinski. Random series and stochastic integrals: single and multiple, Birkhauser, Boston, 1992.
12. V. N. Radchenko. Integrals with respect to general stochastic measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine, Kyiv, 1999. (Russian)
13. A. I. Ilyin, A. S. Kalashnikov, I. A. Oleynik. Linear second-order partial dierential equations of the parabolic type, J. Math. Sci., 108 (2002), no. 4, 435-542.
14. V. N. Radchenko. Evolution equations with general stochastic measures in Hilbert space, Theory Probab. Appl., 59 (2015), no. 2, 328-339.
15. A. Kamont. A discrete characterization of Besov spaces, Approx. Theory Appl., 13 (1997), no. 2, 63-77.