×

zbMATH — the first resource for mathematics

Strong solutions to stochastic wave equations with values in Riemannian manifolds. (English) Zbl 1141.58019
The authors study the stochastic wave equation with values in Riemannian manifolds (opposite to Euclidean spaces). On a \(d\)-dimensional compact manifold \(M\), they consider the following one-dimensional stochastic wave equation \[ \mathbf{D}_t \partial_t u = \mathbf{D}_x \partial_x u + Y_u\, (\partial_t u, \partial_x u) \dot{W}, \] where \(\mathbf{D}\) is the connection on the pull-back bundle \(u^{-1} \text{TM}\) induced by the Riemannian connection on \(M\). They prove, after giving a rigorous formulation of the stochastic problem, existence and uniqueness of solutions of the above equation which are strong both in the PDE sense as well as in the probabilistic sense.

MSC:
58J65 Diffusion processes and stochastic analysis on manifolds
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Z. Brzeźniak, A. Carroll, The stochastic nonlinear heat equation, in preparation
[2] Brzeźniak, Z.; Elworthy, K.D., Stochastic differential equations on Banach manifolds, Methods funct. anal. topology, 6, 1, 43-84, (2000) · Zbl 0965.58028
[3] Cabaña, E., On barrier problems for the vibrating string, Z. wahrsch. verw. geb., 22, 13-24, (1972) · Zbl 0214.16801
[4] Carmona, R.; Nualart, D., Random nonlinear wave equations: propagation of singularities, Ann. probab., 16, 2, 730-751, (1988) · Zbl 0643.60045
[5] Carmona, R.; Nualart, D., Random nonlinear wave equations: smoothness of the solutions, Probab. theory related fields, 79, 4, 469-508, (1988) · Zbl 0635.60073
[6] A. Carroll, The stochastic nonlinear heat equation, PhD thesis, University of Hull, 1999
[7] Cazenave, T.; Shatah, J.; Tahvildar-Zadeh, A.S., Harmonic maps of the hyperbolic space and development of singularities in wave maps and yang – mills fields, Ann. inst. H. Poincaré phys. théor., 68, 3, 315-349, (1998) · Zbl 0918.58074
[8] Chojnowska-Michalik, A., Stochastic differential equations in Hilbert spaces, () · Zbl 0547.60067
[9] Chow, P.-L., Stochastic wave equations with polynomial nonlinearity, Ann. appl. probab., 12, 1, 361-381, (2002) · Zbl 1017.60071
[10] Christodoulou, D.; Tahvildar-Zadeh, A.S., On the regularity of spherically symmetric wave maps, Comm. pure appl. math., 46, 7, 1041-1091, (1993) · Zbl 0744.58071
[11] Da Prato, G.; Zabczyk, J., Stochastic equations in infinite dimensions, (1992), Cambridge Univ. Press Cambridge · Zbl 0761.60052
[12] Dalang, R.C., Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s, Electron. J. probab., 4, 6, 1-29, (1999) · Zbl 0922.60056
[13] Dalang, R.C.; Frangos, N.E., The stochastic wave equation in two spatial dimensions, Ann. probab., 26, 1, 187-212, (1998) · Zbl 0938.60046
[14] Dalang, R.C.; Lévêque, O., Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere, Ann. probab., 32, 1B, 1068-1099, (2004) · Zbl 1046.60058
[15] R.C. Dalang, M. Sanz-Solé, Hölder-Sobolev regularity of the solution to the stochastic wave equation in dimension 3, Mem. Amer. Math. Soc., in press
[16] Elworthy, K.D., Stochastic differential equations on manifolds, London math. soc. lecture note ser., vol. 70, (1982), Cambridge Univ. Press Cambridge · Zbl 0514.58001
[17] Evans, L.C., Partial differential equations, Grad. stud. math., vol. 19, (1998), Amer. Math. Soc. Providence, RI, xviii+662 pp
[18] Freire, A., Global weak solutions of the wave map system to compact homogeneous spaces, Manuscripta math., 91, 4, 525-533, (1996) · Zbl 0867.58019
[19] Friedman, A., Partial differential equations, (1969), Holt, Rinehart & Winston New York
[20] Funaki, T., A stochastic partial differential equation with values in a manifold, J. funct. anal., 109, 2, 257-288, (1992) · Zbl 0768.60055
[21] Ginibre, J.; Velo, G., The Cauchy problem for the \(\operatorname{O}(N), C \operatorname{P}(N - 1)\), and \(G_C(N, p)\) models, Ann. phys., 142, 2, 393-415, (1982) · Zbl 0512.58018
[22] Gu, C.H., On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space, Comm. pure appl. math., 33, 6, 727-737, (1980) · Zbl 0475.58005
[23] Hamilton, R.S., Harmonic maps of manifolds with boundary, Lecture notes in math., vol. 471, (1975), Springer-Verlag Berlin · Zbl 0308.35003
[24] Hermann, R., Differential geometry and the calculus of variations, Math. sci. eng., vol. 49, (1968), Academic Press New York · Zbl 0219.49023
[25] Karczewska, A.; Zabczyk, J., Stochastic PDE’s with function-valued solutions, (), 197-216 · Zbl 0990.60065
[26] Karczewska, A.; Zabczyk, J., A note on stochastic wave equations, (), 501-511 · Zbl 0978.60066
[27] Lions, J.L.; Magenes, E., Non-homogeneous boundary value problems and applications, I, (1972), Springer-Verlag Berlin · Zbl 0223.35039
[28] Malliavin, P., The canonic diffusion above the diffeomorphism group of the circle, C. R. acad. sci. Paris Sér. I math., 329, 4, 325-329, (1999) · Zbl 1006.60073
[29] Marcus, M.; Mizel, V.J., Stochastic hyperbolic systems and the wave equation, Stoch. stoch. rep., 36, 225-244, (1991) · Zbl 0739.60059
[30] Maslowski, B.; Seidler, J.; Vrkoč, I., Integral continuity and stability for stochastic hyperbolic equations, Differential integral equations, 6, 2, 355-382, (1993) · Zbl 0777.35096
[31] Millet, A.; Morien, P.L., On a nonlinear stochastic wave equation in the plane: existence and uniqueness of the solution, Ann. appl. probab., 11, 3, 922-951, (2001) · Zbl 1017.60072
[32] Millet, A.; Sanz-Solé, M., A stochastic wave equation in two space dimension: smoothness of the law, Ann. probab., 27, 2, 803-844, (1999) · Zbl 0944.60067
[33] Müller, S.; Struwe, M., Global existence of wave maps in \(1 + 2\) dimensions with finite energy data, Topol. methods nonlinear anal., 7, 2, 245-259, (1996) · Zbl 0896.35086
[34] Nash, J., The imbedding problem for Riemannian manifolds, Ann. of math. (2), 63, 20-63, (1956) · Zbl 0070.38603
[35] Ondreját, M., Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes math. (rozprawy mat.), 426, (2004), 63 pp · Zbl 1053.60071
[36] Ondreját, M., Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process, J. evol. equ., 4, 2, 169-191, (2004) · Zbl 1054.60068
[37] Ondreját, M., Existence of global martingale solutions to stochastic hyperbolic equations driven by a spatially homogeneous Wiener process, Stoch. dyn., 6, 1, 23-52, (2006) · Zbl 1092.60024
[38] O’Neill, B., Semi-Riemannian geometry. with applications to relativity, Pure appl. math., vol. 103, (1983), Academic Press New York · Zbl 0531.53051
[39] Pazy, A., Semigroups of linear operators and applications to partial differential equations, Appl. math. sci., vol. 44, (1983), Springer-Verlag New York, viii+279 pp · Zbl 0516.47023
[40] Peszat, S., The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. evol. equ., 2, 3, 383-394, (2002) · Zbl 1375.60109
[41] Peszat, S.; Zabczyk, J., Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic process. appl., 72, 187-204, (1997) · Zbl 0943.60048
[42] Peszat, S.; Zabczyk, J., Nonlinear stochastic wave and heat equations, Probab. theory related fields, 116, 3, 421-443, (2000) · Zbl 0959.60044
[43] Seidler, J.; Sobukawa, T., Exponential integrability of stochastic convolutions, J. London math. soc. (2), 67, 1, 245-258, (2003) · Zbl 1045.60070
[44] Shatah, J., Weak solutions and development of singularities of the \(\operatorname{SU}(2)\)σ-model, Comm. pure appl. math., 41, 4, 459-469, (1988) · Zbl 0686.35081
[45] Shatah, J.; Struwe, M., Geometric wave equations, Courant lect. notes math., vol. 2, (1998), New York Univ., Courant Inst. Math. Sci. New York · Zbl 0993.35001
[46] Tataru, D., The wave maps equation, Bull. amer. math. soc. (N.S.), 41, 2, 185-204, (2004) · Zbl 1065.35199
[47] Zhou, Y., Uniqueness of weak solutions of \(1 + 1\) dimensional wave maps, Math. Z., 232, 4, 707-719, (1999) · Zbl 0940.35141
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.