Theory of Probability and Mathematical Statistics
Heat equation in a multidimensional domain with a general stochastic measure
I. M. Bodnarchuk, G. M. Shevchenko
Download PDF
Abstract: Stochastic heat equation on [0,T]×R^d, d≥1, driven by a general stochastic measure µ(t), t∈[0,T], is studied in this paper. The existence, uniqueness, and Hölder regularity of a mild solution are proved.
Keywords: Stochastic measure, stochastic heat equation, mild solution, Hölder condition, Besov space
Bibliography: 1. V. Radchenko, Heat equation with general stochastic measure colored in time, Modern Stochastics: Theory and Applications 1 (2014), 129-138.
2. V. Radchenko, Mild solution of the heat equation with a general stochastic measure, Studia Math. 194 (2009), no. 3, 231-251.
3. I. Bodnarchuk, Mild solution of a wave equation with a general random measure, Visnyk Kyiv University. Mathematics and Mechanics 24 (2010), 28-33. (Ukrainian)
4. 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.
5. V. Radchenko and M. Zähle, Heat equation with a general stochastic measure on nested fractals, Stat. Probab. Lett. 82 (2012), 699-704.
6. R. M. Balan and C. A Tudor, Stochastic heat equation with multiplicative fractional-colored noise, J. Theor. Probab. 23 (2010), no. 3, 834-870.
7. C. A. Tudor, Analysis of Variations for Self-similar Processes. A Stochastic Calculus Approach, Probability and Its Applications, Springer, Cham-Heidelberg, 2013.
8. 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.
9. S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, 1992.
10. V. N. Radchenko, Integrals with respect to general stochastic measures, Proceedings of Institute of Mathematics, National Academy of Science of Ukraine, Kiev, 1999. (Russian)
11. 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.
12. 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.
13. A. Kamont, A discrete characterization of Besov spaces, Approx. Theory Appl. 13 (1997), no. 2, 63-77.
14. 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)