Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces. (English) Zbl 1286.60058

This paper treats the existence problem of a global weak solution to the stochastic wave equations with values in compact Riemannian homogeneous spaces. Let \(M\) be a compact Riemannian homogeneous space. The authors consider the initial value problem for the stochastic wave equation: \[ \begin{aligned} {\mathbb D}_t \partial_t u = \sum_{k=1}^d {\mathbb D}_{x_k} \partial_{x_k} u &+ f(u, \partial_t u, \partial_{x_1} u, \dots, \partial_{ x_d} u ) \tag{1} \\ &+ g( u, \partial_t u, \partial_{x_1} u, \dots, \partial_{x_d} u ) \dot{W} \end{aligned} \] with a random initial data \(( u_0, v_0 ) \in TM\). Here, \({\mathbb D}\) is the connection on the pull-back bundle \(u^{-1} TM\) induced by the Riemannian connection on \(M\). For a nonlinear term \(f\) (and analogously \(g\)), assume that \[ \begin{aligned} f: \,\, T^{d+1} M &\ni (p, v_0, \dots, v_d) \mapsto \tag{2} \\ &f_0(p) v_0 + \sum_{k=1}^d f_k(p) v_k + f_{d+1}(p) \in TM, \end{aligned} \] where \(T^k M\) denotes the vector bundle over \(M\), whose fibre at \(p \in M\) is equal to the \(k\)-fold Cartesian product \(( T_p M)^k\) of \(T_p M\), \(f_{d+1}\) and \(g_{d+1}\) are continuous vector fields on \(M\), \(f_0, g_0\) : \(M \to {\mathbb R}\) are continuous functions and \(f_k, g_k\) : \(TM \to TM\) (for \(k=1,2 \dots, d\)) are continuous vector bundle homomorphisms. Further, suppose that \(W\) is a spatially homogeneous Wiener process with a spectral measure \(\mu\).
By virtue of the Nash isometric embedding theorem, \(M\) is assumed to be isometrically embedded into a certain Euclidean space \({\mathbb R}^n\). On this account, \(M\) is assumed to be a submanifold in \({\mathbb R}^n\), and the authors take advantage of this equivalent definition of a solution to (1), and study, instead of (1), the following classical second-order stochastic partial differential equation (SPDE): \[ \partial_{tt} u = \varDelta u + S_u( \partial_t u, \partial_t u) - \sum_{k=1}^d S_u ( u_{ x_k}, u_{x_k}) + f_u( D u) + g_u(D u) \dot{W}, \tag{3} \] where \(S\) is the second fundamental form of the submanifold \(M \subset {\mathbb R}^n\). Hence the authors define the notion of a weak solution to (1) as follows. Suppose that \(\Theta\) is a Borel probability measure on \[ {\mathcal H}_{\mathrm{loc}}(M) := \{ (u,v) \in {\mathcal H}_{\mathrm{loc}} \equiv H_{\mathrm{loc}}^1( {\mathbb R}^d) \oplus L_{\mathrm{loc}}^2( {\mathbb R}^d; {\mathbb R}^n); v(x) \in T_{u(x)}M, \,\, \text{a.e.} \,\, x \in M \}. \] A system \({\mathcal U} = ( \Omega, {\mathcal F}, {\mathbb F}, {\mathbb P}, W, z)\) consisting of (i) a stochastic basis \(( \Omega, {\mathcal F}, {\mathbb F}, {\mathbb P})\), (ii) a spatially homogeneous Wiener process \(W\), and (iii) an adapted weakly continuous \({\mathcal H}_{loc}(M)\)-valued process \(z=(u,v)\), is called a weak solution to (1) if and only if for all \(\varphi \in {\mathcal D}( {\mathbb R}^d)\), \[ \langle v(t), \varphi \rangle = \langle u(0), \varphi \rangle + \int_0^t \langle v(s), \varphi \rangle ds, \tag{4} \]
\[ \begin{aligned} \langle v(t), \varphi \rangle &= \langle v(0), \varphi \rangle + \int_0^t \langle S_{ u(s)}( v(s), v(s)), \varphi \rangle + \int_0^t \langle f( z(s), \nabla u(s)), \varphi \rangle ds \tag{5} \\ &+ \int_0^t \langle u(s), \varDelta \varphi \rangle ds - \sum_{k=1}^d \int_0^t \langle S_{u(s)} ( \partial_{x_k} u(s), \partial_{x_k} u(s)), \varphi \rangle \\ &+ \int_0^t \langle g( z(s), \nabla u(s)) dW, \varphi \rangle \end{aligned} \] hold \({\mathbb P}\)-a.e. for all \(t \geq 0\). Here is the main result.
{ Theorem.} Assume that \(\mu\) is a positive symmetric Borel measure on \({\mathbb R}^d\) such that \(\mu( {\mathbb R}^d) < \infty\). Then there exists a weak solution to (1) with the initial data \(\Theta\), where the law of \(z(0)\) is equal to \(\Theta\).
The proof is divided into several steps: (a) introducing an approximation of problem (3) via penalization; (b) finding a sufficiently large space in which the laws of the approximated sequence is tight; (c) optimizing the above space as small as possible, so that, after using the Skorokhod embedding theorem, the convergence in that space may be strong enough for the sequence of approximated solutions to be convergent; (d) and finally, using the symmetry of the target manifold to identify the limit with a solution to (3). Note that this method of constructing weak solutions of SPDEs does not rely on any kind of martingale representation theorems. For other related works, see [Z. Brzeźniak and M. Ondreját, J.Funct.Anal.253, No. 2, 449–481 (2007; Zbl 1141.58019); S. Peszat and J. Zabczyk Probab.Theory Related Fields 116, No. 3, 421–443 (2000; Zbl 0959.60044); D. Tataru Bull.Am.Math.Soc., New Ser.41, No. 2, 185–204 (2004; Zbl 1065.35199)].


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
58J65 Diffusion processes and stochastic analysis on manifolds
35L05 Wave equation
Full Text: DOI arXiv Euclid


[1] Adams, R. A. and Fournier, J. J. F. (2003). Sobolev Spaces , 2nd ed. Pure and Applied Mathematics ( Amsterdam ) 140 . Elsevier, Amsterdam. · Zbl 1098.46001
[2] Brzeźniak, Z. (1997). On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep. 61 245-295. · Zbl 0891.60056
[3] Brzeźniak, Z. and Carroll, A. The stochastic nonlinear heat equation. Unpublished manuscript. · Zbl 1040.60047
[4] Brzeźniak, Z. and Elworthy, K. D. (2000). Stochastic differential equations on Banach manifolds. Methods Funct. Anal. Topology 6 43-84. · Zbl 0965.58028
[5] Brzeźniak, Z. and Ondreját, M. Itô formula in \(L^{2}_\mathrm{loc}\) spaces with applications for stochastic wave equations. Unpublished manuscript. · Zbl 1238.60073
[6] Brzeźniak, Z. and Ondreját, M. (2007). Strong solutions to stochastic wave equations with values in Riemannian manifolds. J. Funct. Anal. 253 449-481. · Zbl 1141.58019
[7] Brzeźniak, Z. and Ondreját, M. (2012). Stochastic wave equations with values in Riemannian manifolds. In Stochastic Partial Differential Equations and Applications VIII . Quaderni di Matematica. · Zbl 1238.60073
[8] Brzeźniak, Z. and Peszat, S. (1999). Space-time continuous solutions to SPDE’s driven by a homogeneous Wiener process. Studia Math. 137 261-299. · Zbl 0944.60075
[9] Cabaña, E. M. (1972). On barrier problems for the vibrating string. Z. Wahrsch. Verw. Gebiete 22 13-24. · Zbl 0214.16801
[10] Carmona, R. and Nualart, D. (1988). Random nonlinear wave equations: Propagation of singularities. Ann. Probab. 16 730-751. · Zbl 0643.60045
[11] Carmona, R. and Nualart, D. (1988). Random nonlinear wave equations: Smoothness of the solutions. Probab. Theory Related Fields 79 469-508. · Zbl 0635.60073
[12] Carroll, A. (1999). The stochastic nonlinear heat equation. Ph.D. thesis, Univ. Hull. · Zbl 1016.46009
[13] Cazenave, T., Shatah, J. and Tahvildar-Zadeh, A. S. (1998). Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields. Ann. Inst. H. Poincaré Phys. Théor. 68 315-349. · Zbl 0918.58074
[14] Chojnowska-Michalik, A. (1979). Stochastic differential equations in Hilbert spaces. In Probability Theory ( Papers , VIIth Semester , Stefan Banach Internat. Math. Center , Warsaw , 1976). Banach Center Publications 5 53-74. Polish Acad. Sci., Warsaw. · Zbl 0414.60064
[15] Chow, P.-L. (2002). Stochastic wave equations with polynomial nonlinearity. Ann. Appl. Probab. 12 361-381. · Zbl 1017.60071
[16] Christodoulou, D. and Tahvildar-Zadeh, A. S. (1993). On the regularity of spherically symmetric wave maps. Comm. Pure Appl. Math. 46 1041-1091. · Zbl 0744.58071
[17] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44 . Cambridge Univ. Press, Cambridge. · Zbl 0761.60052
[18] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 29 pp. (electronic). · Zbl 0986.60053
[19] Dalang, R. C. and Frangos, N. E. (1998). The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 187-212. · Zbl 0938.60046
[20] Dalang, R. C. and Lévêque, O. (2004). Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere. Ann. Probab. 32 1068-1099. · Zbl 1046.60058
[21] Elworthy, K. D. (1982). Stochastic Differential Equations on Manifolds. London Mathematical Society Lecture Note Series 70 . Cambridge Univ. Press, Cambridge. · Zbl 0514.58001
[22] Flandoli, F. and Gatarek, D. (1995). Martingale and stationary solutions for stochastic Navier-Stokes equations. Probab. Theory Related Fields 102 367-391. · Zbl 0831.60072
[23] Freire, A. (1996). Global weak solutions of the wave map system to compact homogeneous spaces. Manuscripta Math. 91 525-533. · Zbl 0867.58019
[24] Friedman, A. (1969). Partial Differential Equations . Holt, Rinehart and Winston, New York. · Zbl 0224.35002
[25] Garsia, A. M., Rodemich, E. and Rumsey, H. Jr. (1970). A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 565-578. · Zbl 0252.60020
[26] Ginibre, J. and Velo, G. (1982). The Cauchy problem for the \(\mathrm{O}(N),C\mathrm{P}(N-1)\), and \(G_{C}(N,p)\) models. Ann. Physics 142 393-415. · Zbl 0512.58018
[27] Gu, C. H. (1980). On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space. Comm. Pure Appl. Math. 33 727-737. · Zbl 0475.58005
[28] Hamilton, R. S. (1975). Harmonic Maps of Manifolds with Boundary. Lecture Notes in Math. 471 . Springer, Berlin. · Zbl 0308.35003
[29] Hausenblas, E. and Seidler, J. (2001). A note on maximal inequality for stochastic convolutions. Czechoslovak Math. J. 51 785-790. · Zbl 1001.60065
[30] Hélein, F. (1991). Regularity of weakly harmonic maps from a surface into a manifold with symmetries. Manuscripta Math. 70 203-218. · Zbl 0718.58019
[31] Jakubowski, A. (1997). The almost sure Skorokhod representation for subsequences in nonmetric spaces. Theory Probab. Appl. 42 167-174. · Zbl 0923.60001
[32] Kelley, J. L. (1975). General Topology . Springer, New York. · Zbl 0306.54002
[33] Kirillov, A. Jr. (2008). An Introduction to Lie Groups and Lie Algebras. Cambridge Studies in Advanced Mathematics 113 . Cambridge Univ. Press, Cambridge. · Zbl 1153.17001
[34] Kneis, G. (1977). Zum Satz von Arzela-Ascoli in pseudouniformen Räumen. Math. Nachr. 79 49-54. · Zbl 0375.54018
[35] Ladyzhenskaya, O. A. and Shubov, V. I. (1981). On the unique solvability of the Cauchy problem for equations of two-dimensional relativistic chiral fields with values in complete Riemannian manifolds. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. ( LOMI ) 110 81-94, 242-243. · Zbl 0482.58011
[36] Marcus, M. and Mizel, V. J. (1991). Stochastic hyperbolic systems and the wave equation. Stochastics Stochastics Rep. 36 225-244. · Zbl 0739.60059
[37] Maslowski, B., Seidler, J. and Vrkoč, I. (1993). Integral continuity and stability for stochastic hyperbolic equations. Differential Integral Equations 6 355-382. · Zbl 0777.35096
[38] Millet, A. and Morien, P.-L. (2001). On a nonlinear stochastic wave equation in the plane: Existence and uniqueness of the solution. Ann. Appl. Probab. 11 922-951. · Zbl 1017.60072
[39] Millet, A. and Sanz-Solé, M. (1999). A stochastic wave equation in two space dimension: Smoothness of the law. Ann. Probab. 27 803-844. · Zbl 0944.60067
[40] Moore, J. D. and Schlafly, R. (1980). On equivariant isometric embeddings. Math. Z. 173 119-133. · Zbl 0421.53038
[41] Müller, S. and Struwe, M. (1996). Global existence of wave maps in \(1+2\) dimensions with finite energy data. Topol. Methods Nonlinear Anal. 7 245-259. · Zbl 0896.35086
[42] Nash, J. (1956). The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63 20-63. · Zbl 0070.38603
[43] Ondreját, M. (2004). Existence of global mild and strong solutions to stochastic hyperbolic evolution equations driven by a spatially homogeneous Wiener process. J. Evol. Equ. 4 169-191. · Zbl 1054.60068
[44] Ondreját, M. (2004). Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Math. ( Rozprawy Mat. ) 426 63. · Zbl 1053.60071
[45] Ondreját, M. (2005). Brownian representations of cylindrical local martingales, martingale problem and strong Markov property of weak solutions of SPDEs in Banach spaces. Czechoslovak Math. J. 55 1003-1039. · Zbl 1081.60049
[46] Ondreját, M. (2006). Existence of global martingale solutions to stochastic hyperbolic equations driven by a spatially homogeneous Wiener process. Stoch. Dyn. 6 23-52. · Zbl 1092.60024
[47] Ondreját, M. (2010). Stochastic nonlinear wave equations in local Sobolev spaces. Electron. J. Probab. 15 1041-1091. · Zbl 1225.60109
[48] O’Neill, B. (1983). Semi-Riemannian Geometry. With Applications to Relativity. Pure and Applied Mathematics 103 . Academic Press, New York. · Zbl 0531.53051
[49] Onishchik, A. L. and Vinberg, È. B. (1993). Foundations of Lie theory [see MR0950861 (89m:22010)]. In Lie Groups and Lie Algebras , I. Encyclopaedia of Mathematical Sciences 20 1-94, 231-235. Springer, Berlin.
[50] Peszat, S. (2002). The Cauchy problem for a nonlinear stochastic wave equation in any dimension. J. Evol. Equ. 2 383-394. · Zbl 1375.60109
[51] Peszat, S. and Zabczyk, J. (1997). Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process. Appl. 72 187-204. · Zbl 0943.60048
[52] Peszat, S. and Zabczyk, J. (2000). Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116 421-443. · Zbl 0959.60044
[53] Rudin, W. (1987). Real and Complex Analysis , 3rd ed. McGraw-Hill, New York. · Zbl 0925.00005
[54] Rudin, W. (1991). Functional Analysis , 2nd ed. McGraw-Hill, New York. · Zbl 0867.46001
[55] Seidler, J. (1993). Da Prato-Zabczyk’s maximal inequality revisited. I. Math. Bohem. 118 67-106. · Zbl 0785.35115
[56] Shatah, J. (1988). Weak solutions and development of singularities of the \(\mathrm{SU}(2)\) \(\sigma\)-model. Comm. Pure Appl. Math. 41 459-469. · Zbl 0686.35081
[57] Shatah, J. and Struwe, M. (1998). Geometric Wave Equations. Courant Lecture Notes in Mathematics 2 . New York Univ. Courant Institute of Mathematical Sciences, New York. · Zbl 0993.35001
[58] Tataru, D. (2004). The wave maps equation. Bull. Amer. Math. Soc. ( N.S. ) 41 185-204 (electronic). · Zbl 1065.35199
[59] Triebel, H. (1978). Interpolation Theory , Function Spaces , Differential Operators. North-Holland Mathematical Library 18 . North-Holland, Amsterdam. · Zbl 0387.46032
[60] Zhou, Y. (1999). Uniqueness of weak solutions of \(1+1\) dimensional wave maps. Math. Z. 232 707-719. · Zbl 0940.35141
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