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We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influence of noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Itô-Tanaka type.

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Campi Flegrei is an active volcanic area situated in the Campanian Plain (Italy) and dominated by a resurgent caldera. The great majority of past eruptions have been explosive, variable in magnitude, intensity, and in their vent locations. In this hazard assessment study we present a probabilistic analysis using a variety of volcanological data sets to map the background spatial probability of vent opening conditional on the occurrence of an event in the foreseeable future. The analysis focuses on the reconstruction of the location of past eruptive vents in the last 15 ka, including the distribution of faults and surface fractures as being representative of areas of crustal weakness. One of our key objectives was to incorporate some of the main sources of epistemic uncertainty about the volcanic system through a structured expert elicitation, thereby quantifying uncertainties for certain important model parameters and allowing outcomes from different expert weighting models to be evaluated. Results indicate that past vent locations are the most informative factors governing the probabilities of vent opening, followed by the locations of faults and then fractures. Our vent opening probability maps highlight the presence of a sizeable region in the central eastern part of the caldera where the likelihood of new vent opening per kilometer squared is about 6 times higher than the baseline value for the whole caldera. While these probability values have substantial uncertainties associated with them, our findings provide a rational basis for hazard mapping of the next eruption at Campi Flegrei caldera.

A 2-dimensional Navier-Stokes equation perturbed by a sufficiently distributed white noise is considered. Existence of invariant measures is known from previous works. The aim is to prove uniqueness of the invariant measures, strong law of large numbers, and convergence to equilibrium.

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier-Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier-Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.