×

zbMATH — the first resource for mathematics

SPDEs with polynomial growth coefficients and the Malliavin calculus method. (English) Zbl 1295.60078
Summary: In this paper we study the existence and uniqueness of the \(L_\rho^{2p}(\mathbb R^d;\mathbb R^1)\times L_\rho^2(\mathbb R^d;\mathbb R^d)\) valued solutions of backward doubly stochastic differential equations (BDSDEs) with polynomial growth coefficients using weak convergence, equivalence of norm principle and Wiener-Sobolev compactness arguments. Then we establish a new probabilistic representation of the weak solutions of SPDEs with polynomial growth coefficients through the solutions of the corresponding BDSDEs. This probabilistic representation is then used to prove the existence of stationary solutions of SPDEs on \(\mathbb R^d\) via infinite horizon BDSDEs. The convergence of the solution of a finite horizon BDSDE, when its terminal time tends to infinity, to the solution of the infinite horizon BDSDE is shown to be equivalent to the convergence of the pull-back of the solution of corresponding SPDE to its stationary solution. This way we obtain the stability of the stationary solution naturally.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Arnold, L., Random dynamical systems, (1998), Springer-Verlag Berlin, Heidelberg
[2] Ash, R. B.; Doléans-Dade, C. A., Probability and measure theory, (1998), Academic Press · Zbl 0944.60004
[3] Bally, V.; Matoussi, A., Weak solutions for SPDEs and backward doubly stochastic differential equations, J. Theoret. Probab., 14, 125-164, (2001) · Zbl 0982.60057
[4] Bally, V.; Saussereau, B., A relative compactness criterion in Wiener-Sobolev spaces and application to semi-linear stochastic pdes, J. Funct. Anal., 210, 465-515, (2004) · Zbl 1055.60051
[5] Barles, G.; Lesigne, E., SDE, BSDE and PDE, (Backward Stochastic Differential Equations, Pitman Res. Notes Math., Ser., Vol. 364, (1997), Longman Harlow), 47-80 · Zbl 0886.60049
[6] Caraballo, T.; Kloeden, P. E.; Schmalfuss, B., Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50, 183-207, (2004) · Zbl 1066.60058
[7] Caselles, V.; Morel, J.-M.; Sbert, C., An axiomatic approach to image interpolation, IEEE Trans. Image Process., 7, 376-386, (1998) · Zbl 0993.94504
[8] Da Prato, G.; Malliavin, P.; Nualart, D., Compact families of Wiener functionals, C. R. Acad. Sci. Paris Ser. I. Math., 315, 1287-1291, (1992) · Zbl 0782.60002
[9] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge University Press · Zbl 0761.60052
[10] Duan, J.; Lu, K.; Schmalfuss, B., Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31, 2109-2135, (2003) · Zbl 1052.60048
[11] Feng, C. R.; Zhao, H. Z., Random periodic solutions of SPDEs via integral equations and Wiener-Sobolev compact embedding, J. Funct. Anal., 262, 4377-4422, (2012) · Zbl 1242.60065
[12] Feng, C. R.; Zhao, H. Z.; Zhou, B., Pathwise random periodic solutions of stochastic differential equations, J. Differential Equations, 251, 119-149, (2011) · Zbl 1227.34058
[13] Freidlin, M. I., (Functional Integration and Partial Differential Equations, Annals of Mathematics Studies, vol. 109, (1985), Princeton University Press Princeton)
[14] Kobylanski, M., Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28, 558-602, (2000) · Zbl 1044.60045
[15] Kunita, H., Stochastic flow acting on Schwartz distributions, J. Theoret. Probab., 7, 247-278, (1994) · Zbl 0818.60044
[16] Lepeltier, J. P.; San Martin, J., Backward stochastic differential equations with continuous coefficient, Statist. Probab. Lett., 32, 425-430, (1997) · Zbl 0904.60042
[17] Lian, Z.; Lu, K., Lyapunov exponents and invariant manifolds for random dynamical systems in a Banach space, Mem. Amer. Math. Soc., 206, 106, (2010) · Zbl 1200.37047
[18] Mattingly, J. C., Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity, Comm. Math. Phys., 206, 273-288, (1999) · Zbl 0953.37023
[19] Mohammed, S.-E. A.; Zhang, T.; Zhao, H. Z., The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equations, Mem. Amer. Math. Soc., 196, 105, (2008) · Zbl 1169.60014
[20] Nualart, D., The Malliavin calculus and related topics, (1996), Springer Berlin
[21] Pardoux, E., BSDE’s weak convergence and homogenization of semilinear PDE’s, (Clarke, F.; Stern, R., Nonlin. Analy., Diff. Equa. and Control, (1999), Kluwer Acad. Publi Dordrecht), 503-549 · Zbl 0959.60049
[22] Pardoux, E.; Peng, S., Backward stochastic differential equations and quasilinear parabolic partial differential equations, (Rozuvskii, B. L.; Sowers, R. B., Stochastic Partial Differential Equations and their Applications, Lect. Notes Control Inf. Sci., vol. 176, (1992), Springer Berlin, Heidelberg New York), 200-217 · Zbl 0766.60079
[23] Pardoux, E.; Peng, S., Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Related Fields, 98, 209-227, (1994) · Zbl 0792.60050
[24] Peszat, S., On a Sobolev space of functions of infinite number of variables, Bull. Pol. Acad. Sci. Math., 41, 55-60, (1993) · Zbl 0801.46040
[25] Sinai, Ya., Two results concerning asymptotic behaviour of solutions of Burgers equation with force, J. Stat. Phys., 64, 1-12, (1991) · Zbl 0978.35500
[26] Sinai, Ya., Burgers system dsiven by a periodic stochastic flows, (Itô’s Stochastic Calculus and Probability Theory, (1996), Springer Tokyo), 347-353 · Zbl 0866.35148
[27] E, W.; Khanin, K.; Mazel, A.; Sinai, Ya., Invariant measures for Burgers equation with stochastic forcing, Ann. of Math., 151, 877-960, (2000) · Zbl 0972.35196
[28] F. Wei, H.Z. Zhao, Stationary solutions of the stochastic parabolic infinity Laplacian equations in image processing, Preprint.
[29] Zhang, Q.; Zhao, H. Z., Stationary solutions of SPDEs and infinite horizon bdsdes, J. Funct. Anal., 252, 171-219, (2007) · Zbl 1127.60059
[30] Zhang, Q.; Zhao, H. Z., Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients, J. Differential Equations, 248, 953-991, (2010) · Zbl 1196.60120
[31] Zhang, Q.; Zhao, H. Z., Probabilistic representation of weak solutions of partial differential equations with polynomial growth coefficients, J. Theoret. Probab., 25, 396-423, (2012) · Zbl 1255.60103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.