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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
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