Non-elliptic SPDEs and Ambit Fields: Existence of Densities

Abstract

Relying on the method developed in [11], we prove the existence of a density for two different examples of random fields indexed by (t, x) ∈ (0, T ] × R d . 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 [9]. The density exists on the set where the nonlinearity σ of the noise does not vanish. This complements the results in [20] 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.

Hölder regularity of the densities for the Navier–Stokes equations with noise

“…Time regularity of the density has been proved in [26]. The method introduced in [8] is simple but effective and has been already used in other problems (see for instance [3,10,28,29]). …”

“…Moreover, there is an inherent limitation in the Fokker-Planck approach, in that it is less flexible than the method developed in [8] and thus cannot be used, in general, to evaluate the density of quantities that do not have an associated evolution equation, as for instance in [28,29], as well as for nonlinear functionals of the solution of a diffusion process. On the other hand the method is by no means limited to the NavierStokes equations and can be, in principle, applied to other stochastic PDEs with similar features.…”

Existence and Besov regularity of the density for a class of SDEs with Volterra noise

“…These new techniques avoid the use of the Malliavin calculus, which requires strong regularity of the coefficients of the SDE. We refer, among others, to [1], [2], [3], [8], [9], [10] for several applications of the fractional integration by parts methodology to concrete examples.…”