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Hitting probabilities for non-linear systems of stochastic waves

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We consider a d-dimensional random eld u = fu(t; x)g thatsolves a non-linear system of stochastic wave equations in spatial dimensions k 2 f1; 2; 3g, driven by a spatially homogeneous Gaussian noise that is white in time. We mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent . Using Malliavin calculus, we establish upper and lower bounds on the probabilities that the random eld visits a deterministic subset of Rd, in terms, respectively, of Hausdor measure and Newtonian capacity of this set. The dimension that appears in the Hausdor measure is close to optimal, and shows that when d(2.. ) > 2(k+1), points are polar for u. Conversely, in low dimensions d, points are not polar. There is however an interval in which the question of polarity of points remains open. This is joint work with Marta Sanz-Sole.

This talk is part of the Isaac Newton Institute Seminar Series series.

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