Points of Positive Density for the Solution to a Hyperbolic SPDE

Using a slightly modified version of Aida–Kusuoka–Stroock's characterization of the points of strictly positive density for an arbitrary Wiener functional, we extend the theorem of Ben Arous–Léandre to solutions of hyperbolic SPDE's. Thus we show that the density f of the law of X_z is positive at y if and only if y can be achieved as S_z(h), where S(h) is the controlled equation corresponding to an element h of the Cameron–Martin space, and S(.)_z is a submersion at h. The proof depends on a convergence result for a sequence X^n, of perturbed processes (defined in terms of a non homogeneous linear interpolation of the Brownian sheet) to the solution X of the corresponding perturbed SPDE.