Existence of global martingale solutions to stochastic hyperbolic equations driven by a spatially homogeneous Wiener process. (English) Zbl 1092.60024

The author considers a stochastic partial differential equation \[ u_{tt}={\mathcal A}u+f(u)+g(u)\dot W,\quad u(0)=u_0,\;u_t(0)=v_0, \] in a domain \(G\subseteq {\mathbb R}^d\), where \(\mathcal A\) is a uniformly elliptic second-order differential operator, \(f,g\) are real continuous non-Lipschitz functions and \(W\) is a spatially homogeneous Wiener process in the tempered distribution space \({\mathcal S}'({\mathbb R}^d)\) with finite spectral measure. \(D\) can be either whole \({\mathbb R}^d\) or a bounded set with \(C^2\)-boundary, and then either Dirichlet or Neumann conditions are imposed. A solution of this equation is understood as a weak (i.e., defined via a suitable duality) solution of a martingale problem, appropriately (i.e., in a rather non-standard way) defined. It is proved that under some, quite intricate, growth conditions on \(f\) and \(g\), a global solution exists, whereas uniqueness is not discussed.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] DOI: 10.1016/S0304-4149(99)00034-4 · Zbl 0996.60074
[2] Brzeźniak Z., Probab. Th. Rel. Fields
[3] A. Chojnowska-Michalik, Probability Theory (Banach Center Publications, 1979) pp. 53–74.
[4] DOI: 10.1016/0022-247X(82)90110-X · Zbl 0496.60059
[5] Chow P.-L., Ann. Appl. Probab. 12 pp 361–
[6] DOI: 10.1007/978-1-4899-0399-0
[7] DOI: 10.1017/CBO9780511666223
[8] DOI: 10.1002/1521-4001(200211)82:11/12<745::AID-ZAMM745>3.0.CO;2-1 · Zbl 1009.60054
[9] DOI: 10.1512/iumj.1971.20.20046
[10] DOI: 10.1080/17442509408833868 · Zbl 0824.60052
[11] DOI: 10.1007/BFb0089647
[12] DOI: 10.1007/978-1-4684-0302-2
[13] DOI: 10.2969/jmsj/01420242 · Zbl 0108.11203
[14] Khrychev D. A., Math. Sb. 116 pp 398–
[15] DOI: 10.1007/978-3-0348-9234-6
[16] Millet A., Ann. Appl. Probab. 11 pp 922–
[17] Mizohata S., The Theory of Partial Differential Equations (1973) · Zbl 0263.35001
[18] Nomizu K., Fundamentals of Linear Algebra (1966) · Zbl 0148.01601
[19] Ondreját M., Czech. Math. J.
[20] Ondreját M., J. Evol. Eqns. 4 pp 169–
[21] Ondreját M., Diss. Math. 426 pp 1–
[22] Opic B., Pitman Research Notes in Mathematics 219, in: Hardy-Type Inequalities (1990)
[23] DOI: 10.1007/978-1-4612-5561-1 · Zbl 0516.47023
[24] S. Peszat, Stochastic Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math. 227, eds. G. Da Prato and L. Tubaro (Marcel Dekker, 2001) pp. 417–427.
[25] DOI: 10.1007/PL00013197 · Zbl 1375.60109
[26] DOI: 10.1016/S0304-4149(97)00089-6 · Zbl 0943.60048
[27] Reed M., Lecture Notes in Mathematics 507, in: Abstract Non Linear Wave Equations (1976) · Zbl 0317.35002
[28] Strauss W. A., Ana. Acad. Brasil. Ci. 42 pp 645–
[29] Strauss W. A., Nonlinear Wave Equations (1989)
[30] Triebel H., Interpolation Theory, Function Spaces, Differential Operators (1978) · Zbl 0387.46033
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