Existence and uniqueness of mild solution to fractional stochastic heat equation
Volume 6, Issue 1 (2019), pp. 57–79
Pub. online: 12 December 2018
Type: Research Article
Open Access
Open Access
Received
4 August 2018
4 August 2018
Revised
23 October 2018
23 October 2018
Accepted
23 October 2018
23 October 2018
Published
12 December 2018
12 December 2018
Abstract
For a class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset $D\subset {\mathbb{R}^{d}}$ and driven by an ${L^{2}}(D)$-valued fractional Brownian motion with the Hurst index $H>1/2$, a new result on existence and uniqueness of a mild solution is established. Compared to the existing results, the uniqueness in a fully nonlinear case is shown, not assuming the coefficient in front of the noise to be affine. Additionally, the existence of moments for the solution is established.
References
Chen, Y., Gao, H., Garrido-Atienza, M.J., Schmalfuss, B.: Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than 1/2 and random dynamical systems. Discrete Contin. Dyn. Syst. 34(1), 79–98 (2014). MR3072986. https://doi.org/10.3934/dcds.2014.34.79
Eidel’man, S.D., Ivasišen, S.D.: Investigation of the Green’s matrix of a homogeneous parabolic boundary value problem. Tr. Mosk. Mat. Obš. 23, 179–234 (1970). English transl. in Trans. Moscow Math. Soc. 23 (1970), 179–242 (1972). MR0367455
Eidelman, S.D., Zhitarashu, N.V.: 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. MR1632789. https://doi.org/10.1007/978-3-0348-8767-0
Friedman, A.: Partial Differential Equations. Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London (1969). MR0445088
Grecksch, W., Anh, V.V.: A parabolic stochastic differential equation with fractional Brownian motion input. Stat. Probab. Lett. 41(4), 337–346 (1999). MR1666072. https://doi.org/10.1016/S0167-7152(98)00147-3
Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. 2nd edn. Cambridge University Press, Cambridge (1952). MR0046395
Hu, Y.: Heat equations with fractional white noise potentials. Appl. Math. Optim. 43(3), 221–243 (2001). MR1885698. https://doi.org/10.1007/s00245-001-0001-2
Hu, Y.: Itô type stochastic differential equations driven by fractional Brownian motions of Hurst parameter $H>1/2$. Stochastics 90(5), 720–761 (2018). MR3810579. https://doi.org/10.1080/17442508.2017.1415342
Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York-London (1968). MR0244627
Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202(1), 277–305 (2003). MR1994773. https://doi.org/10.1016/S0022-1236(02)00065-4
Mishura, Y., Ralchenko, K., Shevchenko, G.: Existence and uniqueness of mild solution to stochastic heat equation with white and fractional noises. Teor. Imovir. Mat. Stat. 98, 142–162 (2018) MR3824684
Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics, vol. 1929. Springer, Berlin (2008). MR2378138. https://doi.org/10.1007/978-3-540-75873-0
Nualart, D., Răşcanu, A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53(1), 55–81 (2002). MR1893308
Nualart, D., Vuillermot, P.-A.: Variational solutions for partial differential equations driven by a fractional noise. J. Funct. Anal. 232(2), 390–454 (2006). MR2200741. https://doi.org/10.1016/j.jfa.2005.06.015
Sanz-Solé, M., Vuillermot, P.-A.: Equivalence and Hölder-Sobolev regularity of solutions for a class of non-autonomous stochastic partial differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 39(4), 703–742 (2003) MR1983176. https://doi.org/10.1016/S0246-0203(03)00015-3
Sanz-Solé, M., Vuillermot, P.A.: Mild solutions for a class of fractional SPDEs and their sample paths. J. Evol. Equ. 9(2), 235–265 (2009). MR2511552. https://doi.org/10.1007/s00028-009-0014-x
Shevchenko, G.: Mixed fractional stochastic differential equations with jumps. Stochastics 86(2), 203–217 (2014). MR3180033. https://doi.org/10.1080/17442508.2013.774404
Veraar, M.C.: Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations. J. Evol. Equ. 10(1), 85–127 (2010). MR2602928. https://doi.org/10.1007/s00028-009-0041-7
