We focus on the dynamics and Wong-Zakai approximation for a class of retarded partial differential equations subjected to multiplicative white noise. We show that when restricted to a local region and under certain conditions, there exists a unique global solution for the truncated system driven by either the white noise or the approximation noise. Such solution generates a random dynamical system, and the solutions of Wong-Zakai approximations are convergent to solutions of the stochastic retarded differential equation. We also show that there exist invariant manifolds for the truncated system driven by either the white noise or the approximation noise, which are then the local manifolds for the untruncated systems, and prove that such invariant manifolds of the Wong-Zakai approximations converge to those of the stochastic retarded differential equation.
REFERENCES
1.
2.
Boutet De Monvel
, L.
, Chueshov
, I. D.
, and Rezounenko
, A. V.
, “Inertial manifolds for retarded semilinear parabolic equations
,” Nonlinear Anal.
34
, 907
–925
(1998
).3.
Chueshov
, I. D.
and Scheutzow
, M.
, “Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations
,” J. Dyn. Differ. Equations
13
, 355
–380
(2001
).4.
Chueshov
, I. D.
, Scheutzow
, M.
, and Schmalfuss
, B.
, “Continuity properties of inertial manifolds for stochastic retarded semilinear parabolic equations
,” in Interacting Stochastic Systems
(Springer
, Berlin, Heidelberg
, 2005
), pp. 355
–377
.5.
Castaing
, C.
and Valadier
, M.
, Convex Analysis and Measurable Multifunctions
, Lecture Notes in Mathematics Vol. 580
(Springer
, Berlin, New York
, 1977
).6.
Duan
, J.
, Lu
, K.
, and Schmalfuss
, B.
, “Invariant manifolds for stochastic partial differential equations
,” Ann. Probab.
31
, 2109
–2135
(2003
).7.
Hadamard
, J.
, “Sur l’iteration et les solutions asymptotiques des equations differentielles
,” Bull. Soc. Math. Fr.
29
, 224
–228
(1901
).8.
Henry
, D.
, Geometric Theory of Semilinear Parabolic Equations
(Springer
, 2006
), Vol. 840
.9.
Lu
, K.
and Wang
, Q.
, “Chaotic behavior in differential equations driven by a Brownian motion
,” J. Differ. Equations
251
, 2853
–2895
(2011
).10.
Lu
, K.
and Wang
, B.
, “Wong-Zakai approximations and long term behavior of stochastic partial differential equations
,” J. Dyn. Differ. Equations
31
, 1341
–1371
(2017
).11.
Lyapunov
, A. M.
, Problème Général de la Stabilité du Mouvement
(Princeton University Press
, 1948
).12.
Pazy
, A.
, Semigroups of Linear Operators and Applications to Partial Differential Equations
, Applied Mathematical Sciences Vol. 44
(Springer-Verlag
, New York
, 1983
).13.
Perron
, O.
, “Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen
,” Math. Z.
29
, 129
–160
(1929
).14.
Shen
, J.
and Lu
, K.
, “Wong-Zakai approximations and center manifolds of stochastic differential equations
,” J. Differ. Equations
263
, 4929
–4977
(2017
).15.
Shen
, J.
, Lu
, K.
, and Zhang
, W.
, “Heteroclinic chaotic behavior driven by a Brownian motion
,” J. Differ. Equations
255
, 4185
–4225
(2013
).16.
Shen
, J.
, Zhao
, J.
, Lu
, K.
, and Wang
, B.
, “The Wong–Zakai approximations of invariant manifolds and foliations for stochastic evolution equations
,” J. Differ. Equations
266
, 4568
–4623
(2019
).17.
Travis
, C. C.
and Webb
, G. F.
, “Partial differential equations with deviating arguments in the time variable
,” J. Math. Anal. Appl.
56
, 397
–409
(1976
).18.
Wang
, X.
, Lu
, K.
, and Wang
, B.
, “Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise
,” J. Dyn. Differ. Equations
28
, 1309
–1335
(2016
).19.
Wang
, X.
, Lu
, K.
, and Wang
, B.
, “Wong–Zakai approximations and attractors for stochastic reaction-diffusion equations on unbounded domains
,” J. Differ. Equations
264
, 378
–424
(2018
).20.
Wong
, E.
and Zakai
, M.
, “On the convergence of ordinary integrals to stochastic integrals
,” Ann. Math. Stat.
36
, 1560
–1564
(1965
).21.
Wong
, E.
and Zakai
, M.
, “On the relation between ordinary and stochastic differential equations
,” Int. J. Eng. Sci.
3
, 213
–229
(1965
).22.
Zhao
, J.
, Shen
, J.
, and Lu
, K.
, “Persistence of C1 inertial manifolds under small random perturbations
,” J. Dyn. Differ. Equations
36
, 333
–385
(2024
).23.
Zang
, J.
, Shen
, J.
, and Wang
, X.
, “Stationary approximations of inertial manifolds for stochastic retarded semiliear parabolic equations
,” J. Math. Anal. Appl.
506
, 125668
(2022
).© 2024 Author(s). Published under an exclusive license by AIP Publishing.
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