Sticky-reflected stochastic heat equation driven by colored noise
Abstract
UDC 519.21
We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval $[0,1]$ driven by colored noise.
The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on the real line, but now the solution obeys the usual stochastic heat equation except for points where it reaches zero.
The solution has no noise at zero and a drift pushes it to stay positive.
The proof is based on a new approach that can also be applied to other types of SPDEs with discontinuous coefficients.
References
A. Alonso, F. Brambila-Paz, $L^p$ -continuity of conditional expectations, J. Math. Anal. and Appl., 221, No. 1, 161–176 (1998), https://doi.org/10.1006/jmaa.1998.5818 DOI: https://doi.org/10.1006/jmaa.1998.5818
D. Aldous, Stopping times and tightness, Ann. Probab., 6, No. 2, 335 – 340 (1978), https://doi.org/10.1214/aop/1176995579 DOI: https://doi.org/10.1214/aop/1176995579
S. A. Cherny, H.-J. Engelbert, Singular stochastic differential equations, Lect. Notes Math., vol. 1858, Springer-Verlag,Berlin (2005). viii+128 pp. ISBN: 3-540-24007-1, https://doi.org/10.1007/b104187 DOI: https://doi.org/10.1007/b104187
R. Chitashvili, On the nonexistence of a strong solution in the boundary problem for a sticky Brownian motion, Proc. A. Razmadze Math. Inst., 115, 17 – 31 (1997).
N. Dunford, J. T. Schwartz, Linear operators, Part I, General theory, John Wiley & Sons, Inc., New York (1988). xiv + 858 pp. ISBN: 0-471-60848-3
S. N. Ethier, T. G. Kurtz, Markov processes, Wiley Ser. Probab. and Math. Statist.: Probab. and Math. Statist., JohnWiley & Sons, Inc., New York (1986). x + 534 pp. ISBN: 0-471-08186-8 , https://doi.org/10.1002/9780470316658 DOI: https://doi.org/10.1002/9780470316658
H.-J. Engelbert, G. Peskir, Stochastic differential equations for sticky Brownian motion, Stochastics, 86, No. 6,993 – 1021 (2014), https://doi.org/10.1080/17442508.2014.899600 DOI: https://doi.org/10.1080/17442508.2014.899600
T. Fattler, M. Grothaus, R. Vobhall, Construction and analysis of a sticky reflected distorted Brownian motion, Ann. Inst. Henri Poincare Probab. Stat., ´52, No. 2, 735 – 762 (2016), https://doi.org/10.1214/14-AIHP650 DOI: https://doi.org/10.1214/14-AIHP650
T. Funaki, S. Olla, Fluctuations for $nablaphi$ interface model on a wall, Stoch. Process. and Appl., 94, No. 1, 1 – 27 (2001), https://doi.org/10.1016/S0304-4149(00)00104-6 DOI: https://doi.org/10.1016/S0304-4149(00)00104-6
M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and symmetric Markov processes, extended ed., De Gruyter Stud. Math., vol. 19, Walter de Gruyter & Co., Berlin (2011). x+489 pp. ISBN: 978-3-11-021808-4
T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J., 89, 129–193(1983), https://doi.org/10.1017/S0027763000020298 DOI: https://doi.org/10.1017/S0027763000020298
M. Grothaus, R. Vobhall, Stochastic differential equations with sticky reflection and boundary diffusion, Electron. J.Probab., 22, Paper No. 7 (2017), https://doi.org/10.1214/17-EJP27 DOI: https://doi.org/10.1214/17-EJP27
M. Grothaus, R. Vobhall, Strong Feller property of sticky reflected distorted Brownian motion, J. Theoret. Probab., 31, No. 2, 827 – 852 (2018), https://doi.org/10.1007/s10959-016-0735-z DOI: https://doi.org/10.1007/s10959-016-0735-z
U. G. Haussmann, E. Pardoux, ´ Stochastic variational inequalities of parabolic type, Appl. Math. and Optim., 20, No. 2, 163 – 192 (1989), https://doi.org/10.1007/BF01447653 DOI: https://doi.org/10.1007/BF01447653
N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, second ed., North-Holland Math. Library, vol. 24, North-Holland Publ. Co., Amsterdam; Kodansha, Ltd., Tokyo (1989), xvi+555 pp. ISBN: 0-444-87378-3
J. Jacod, A. N. Shiryaev, Limit theorems for stochastic processes, second ed., Grundlehren Math. Wiss., vol. 288, Springer-Verlag, Berlin (2003). xx+661 pp. ISBN: 3-540-43932-3, https://doi.org/10.1007/978-3-662-05265-5 DOI: https://doi.org/10.1007/978-3-662-05265-5
O. Kallenberg, Foundations of modern probability, second ed., Probab. and Appl. (N.Y.), Springer-Verlag, New York (2002). xx+638 pp. ISBN: 0-387-95313-2, https://doi.org/10.1007/978-1-4757-4015-8 DOI: https://doi.org/10.1007/978-1-4757-4015-8
V. Konarovskyi, Coalescing-fragmentating Wasserstein dynamics: particle approach, arXiv:1711.03011 (2017).
I. Karatzas, A. N. Shiryaev, M. Shkolnikov, On the one-sided Tanaka equation with drift, Electron. Commun. Probab.,16, 664 – 677 (2011), https://doi.org/10.1214/ECP.v16-1665 DOI: https://doi.org/10.1214/ECP.v16-1665
R. S. Liptser, A. N. Shiryaev, Statistics of random processes, I, General theory, Appl. Math. (N.Y.), vol. 5, SpringerVerlag, Berlin (2001). xvi+427 pp. ISBN: 3-540-63929-2
Zhi Ming Ma, M. Rockner, ¨ Introduction to the theory of (non-symmetric) Dirichlet forms, Springer-Verlag, Berlin (1992). {rm vi}+209 pp. ISBN: 3-540-55848-9, https://doi.org/10.1007/978-3-642-77739-4 DOI: https://doi.org/10.1007/978-3-642-77739-4
D. Nualart, E. Pardoux, ´ White noise driven quasilinear SPDEs with reflection, Probab. Theory and Related Fields, 93, No. 1, 77 – 89 (1992), https://doi.org/10.1007/BF01195389 DOI: https://doi.org/10.1007/BF01195389
N. N. Vakhania, V. I. Tarieladze, S. A. Chobanyan, Probability distributions on Banach spaces, Math. and Appl.(Sov. Ser.), vol. 14, D. Reidel Publ. Co., Dordrecht (1987). xxvi+482 pp. ISBN: 90-277-2496-2, https://doi.org/10.1007/978-94-009-3873-1 DOI: https://doi.org/10.1007/978-94-009-3873-1
L. Zambotti, A reflected stochastic heat equation as symmetric dynamics with respect to the 3-d Bessel bridge,J. Funct. Anal., 180, No. 1, 195 – 209 (2001), https://doi.org/10.1006/jfan.2000.3685 DOI: https://doi.org/10.1006/jfan.2000.3685
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