Non elliptic SPDEs and ambit fields: existence of densities

TL;DR

This paper proves the existence of probability densities for certain stochastic partial differential equations and ambit fields driven by Gaussian and Lévy noise, extending previous results to cases where the nonlinearity may vanish.

Contribution

It introduces new existence results for densities of SPDEs and ambit fields with non-vanishing and vanishing nonlinearities, using advanced probabilistic methods.

Findings

Density exists where the nonlinearity does not vanish

Extends previous results to unbounded nonlinearities

Addresses ambit fields with Lévy basis integrators

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 $\sigma$ of the noise does not vanish. This complements the results in [sanzsuess2015] where $\sigma$ is assumed to be bounded away from zero. The second example is an ambit field with a stochastic integral term having as integrator a L\'evy basis of pure-jump, stable-like type.