Existence and uniqueness of a mild solution to the stochastic heat equation with white and fractional noises
Authors:
Yu. Mishura, K. Ralchenko and G. Shevchenko
Journal:
Theor. Probability and Math. Statist. 98 (2019), 149-170
MSC (2010):
Primary 60H15, 35R60, 35K55, 60G22
DOI:
https://doi.org/10.1090/tpms/1068
Published electronically:
August 19, 2019
MathSciNet review:
3824684
Full-text PDF
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We prove the existence and uniqueness of a mild solution for a class of nonautonomous parabolic mixed stochastic partial differential equations defined on a bounded open subset $D \subset \mathbb {R}^d$ and involving standard and fractional $L^2(D)$-valued Brownian motions. We assume that the coefficients are homogeneous, Lipschitz continuous, and the coefficient at the fractional Brownian motion is an affine function.
References
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014. MR 3236753
- Jinqiao Duan and Wei Wang, Effective dynamics of stochastic partial differential equations, Elsevier Insights, Elsevier, Amsterdam, 2014. MR 3289240
- S. D. Èĭdel′man and S. D. Ivasišen, Investigation of the Green’s matrix of a homogeneous parabolic boundary value problem, Trudy Moskov. Mat. Obšč. 23 (1970), 179–234 (Russian). MR 0367455
- Samuil D. Eidelman and Nicolae V. Zhitarashu, Parabolic boundary value problems, Operator Theory: Advances and Applications, vol. 101, Birkhäuser Verlag, Basel, 1998. Translated from the Russian original by Gennady Pasechnik and Andrei Iacob. MR 1632789
- Peter Kotelenez, A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations, Stochastic Anal. Appl. 2 (1984), no. 3, 245–265. MR 757338, DOI https://doi.org/10.1080/07362998408809036
- Bohdan Maslowski and David Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal. 202 (2003), no. 1, 277–305. MR 1994773, DOI https://doi.org/10.1016/S0022-1236%2802%2900065-4
- Yuliya S. Mishura, Stochastic calculus for fractional Brownian motion and related processes, Lecture Notes in Mathematics, vol. 1929, Springer-Verlag, Berlin, 2008. MR 2378138
- Yuliya Mishura and Georgiy Shevchenko, Mixed stochastic differential equations with long-range dependence: Existence, uniqueness and convergence of solutions, Comput. Math. Appl. 64 (2012), no. 10, 3217–3227. MR 2989350, DOI https://doi.org/10.1016/j.camwa.2012.03.061
- David Nualart and Aurel Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- David Nualart and Pierre-A. Vuillermot, Variational solutions for partial differential equations driven by a fractional noise, J. Funct. Anal. 232 (2006), no. 2, 390–454. MR 2200741, DOI https://doi.org/10.1016/j.jfa.2005.06.015
- Marta Sanz-Solé and Pierre-A. Vuillermot, Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 4, 703–742 (English, with English and French summaries). MR 1983176, DOI https://doi.org/10.1016/S0246-0203%2803%2900015-3
- Marta Sanz-Solé and Pierre A. Vuillermot, Mild solutions for a class of fractional SPDEs and their sample paths, J. Evol. Equ. 9 (2009), no. 2, 235–265. MR 2511552, DOI https://doi.org/10.1007/s00028-009-0014-x
- G. Shevchenko, Mixed stochastic delay differential equations, Teor. Ĭmovīr. Mat. Stat. 89 (2013), 167–180 (English, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 89 (2014), 181–195. MR 3235184, DOI https://doi.org/10.1090/S0094-9000-2015-00944-3
- L. Tubaro, An estimate of Burkholder type for stochastic processes defined by the stochastic integral, Stochastic Anal. Appl. 2 (1984), no. 2, 187–192. MR 746435, DOI https://doi.org/10.1080/07362998408809032
- Mark C. Veraar, Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ. 10 (2010), no. 1, 85–127. MR 2602928, DOI https://doi.org/10.1007/s00028-009-0041-7
- Jerzy Zabczyk, A mini course on stochastic partial differential equations, Stochastic climate models (Chorin, 1999) Progr. Probab., vol. 49, Birkhäuser, Basel, 2001, pp. 257–284. MR 1948300
References
- G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, second ed., Encyclopedia of Mathematics and its Applications, vol. 152, Cambridge University Press, Cambridge, 2014. MR 3236753
- J. Duan and W. Wang, Effective dynamics of stochastic partial differential equations, Elsevier Insights, Elsevier, Amsterdam, 2014. MR 3289240
- S. D. Eidel’man and S. D. Ivasišen, Investigation of the Green’s matrix of a homogeneous parabolic boundary value problem, Trudy Moskov. Mat. Obšč. 23 (1970), 179–234; English transl. in Trans. Moscow Math. Soc. 23 (1970), 179–242. MR 0367455
- S. D. Eidelman and N. V. Zhitarashu, Parabolic boundary value problems, Operator Theory: Advances and Applications, vol. 101, Birkhäuser Verlag, Basel, 1998. MR 1632789
- P. Kotelenez, A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations, Stochastic Anal. Appl. 2 (1984), no. 3, 245–265. MR 757338
- B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal. 202 (2003), no. 1, 277–305. MR 1994773
- Y. Mishura, Stochastic calculus for fractional Brownian motion and related processes, vol. 1929, Springer Science & Business Media, 2008. MR 2378138
- Y. Mishura and G. Shevchenko, Mixed stochastic differential equations with long-range dependence: Existence, uniqueness and convergence of solutions, Comput. Math. Appl. 64 (2012), no. 10, 3217–3227. MR 2989350
- D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- D. Nualart and P.-A. Vuillermot, Variational solutions for partial differential equations driven by a fractional noise, J. Funct. Anal. 232 (2006), no. 2, 390–454. MR 2200741
- M. Sanz-Solé and P.-A. Vuillermot, Equivalence and Hölder–Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations, Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 4, 703–742. MR 1983176
- M. Sanz-Solé and P. A. Vuillermot, Mild solutions for a class of fractional SPDEs and their sample paths, J. Evol. Equ. 9 (2009), no. 2, 235–265. MR 2511552
- G. Shevchenko, Mixed stochastic delay differential equations, Theory Probab. Math. Statist. 89 (2014), 181–195. MR 3235184
- L. Tubaro, An estimate of Burkholder type for stochastic processes defined by the stochastic integral, Stochastic Anal. Appl. 2 (1984), no. 2, 187–192. MR 746435
- M. C. Veraar, Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations, J. Evol. Equ. 10 (2010), no. 1, 85–127. MR 2602928
- J. Zabczyk, A mini course on stochastic partial differential equations, Stochastic climate models (Chorin, 1999), Progr. Probab., vol. 49, Birkhäuser, Basel, 2001, 257–284. MR 1948300
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60H15,
35R60,
35K55,
60G22
Retrieve articles in all journals
with MSC (2010):
60H15,
35R60,
35K55,
60G22
Additional Information
Yu. Mishura
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine
Email:
myus@univ.kiev.ua
K. Ralchenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine
Email:
k.ralchenko@gmail.com
G. Shevchenko
Affiliation:
Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska, 01601 Kyiv, Ukraine
Email:
zhora@univ.kiev.ua
Keywords:
Fractional Brownian motion,
stochastic partial differential equation,
Green’s function
Received by editor(s):
March 5, 2018
Published electronically:
August 19, 2019
Article copyright:
© Copyright 2019
American Mathematical Society