Random nonlinear wave equations: Propagation of singularities.

*(English)*Zbl 0643.60045The paper is concerned with a random nonlinear wave equation of the form
\[
\partial^ 2X/\partial t^ 2(t,x)-\partial^ 2X/\partial^ 2x(t,x)=a(X(t,x))\xi (t,x)+b(X(t,x)),
\]
where x belongs to an interval I of R, a and b are continuously differentiable functions on R with bounded derivatives, and \(\xi\) is a space-time white noise. If I has finite endpoints, the Dirichlet boundary condition is imposed there. The solution is defined via an integral form. The paper is devoted to the investigation of the existence and the propagation as t varies of the singularities of X as a function of (t,x). Deterministic hyperbolic equations are known to preserve the singularities of the initial conditions, and to propagate them along characteristic curves. The paper shows that this fact remains true in the random case.

Since solutions of the random wave equation become semimartingales when restricted to some one-dimensional curves of the (t,x)-plane, one can use a law of the iterated logarithm to find the local modulus of continuity of solutions along these curves, and to define a singularity as a failure to have this modulus of continuity. A law of the iterated logarithm for general continuous semimartingales is proved in the paper. It is shown that there are many singularities, existing almost surely at nonanticipative random points, and propagating in the orthogonal direction.

Since solutions of the random wave equation become semimartingales when restricted to some one-dimensional curves of the (t,x)-plane, one can use a law of the iterated logarithm to find the local modulus of continuity of solutions along these curves, and to define a singularity as a failure to have this modulus of continuity. A law of the iterated logarithm for general continuous semimartingales is proved in the paper. It is shown that there are many singularities, existing almost surely at nonanticipative random points, and propagating in the orthogonal direction.

Reviewer: F.Flandoli

##### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

35L70 | Second-order nonlinear hyperbolic equations |

35R60 | PDEs with randomness, stochastic partial differential equations |

60G44 | Martingales with continuous parameter |