Non elliptic SPDEs and ambit fields: existence of densities

Abstract

Relying on the method developed in [debusscheromito2014], we prove the existence of a density for two different examples of random fields indexed by $(t,x)\in(0,T]\times \Rd$. The first example consists of SPDEs with Lipschitz continuous coefficients driven by a Gaussian noise white in time and with a stationary spatial covariance, in the setting of [dalang1999]. The density exists on the set where the nonlinearity $σ$ of the noise does not vanish. This complements the results in [sanzsuess2015] where $σ$ is assumed to be bounded away from zero. The second example is an ambit field with a stochastic integral term having as integrator a Lévy basis of pure-jump, stable-like type.

23 pages