Abstract
We study a class of second-order linear hyperbolic partial differential equations in spatial dimension d, driven by spatially correlated Gaussian noise that is white in time and concentrated in space on a hypersurface. For the case of isotropic Gaussian noise concentrated on a sphere, we give an explicit necessary and sufficient condition on the spatial covariance of the noise which guarantees that the solution of the equation is a function-valued process indexed by time. In the case of spatially homogeneous noise concentrated on a hyperplane H, we provide a necessary and sufficient condition for existence of a function-valued solution, defined everywhere outside of H, as well as a (different) necessary and sufficient condition for existence of a real-valued process solution, defined for all times and in all of space (including on H). A sufficient condition for Hölder continuity is provided, and existence and uniqueness for a non-linear form of the equation is established.
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Dalang, R.C., Lévêque, O. (2004). Second-Order Hyperbolic S.P.D.E.’s Driven by Boundary Noises. In: Dalang, R.C., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications IV. Progress in Probability, vol 58. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7943-9_6
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DOI: https://doi.org/10.1007/978-3-0348-7943-9_6
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