Optimal distributed control of a stochastic Cahn-Hilliard equation

We study an optimal distributed control problem associated to a stochastic Cahn-Hilliard equation with a classical double-well potential and Wiener multiplicative noise, where the control is represented by a source-term in the definition of the chemical potential. By means of probabilistic and analytical compactness arguments, existence of an optimal control is proved. Then the linearized system and the corresponding backward adjoint system are analysed through monotonicity and compactness arguments, and first-order necessary conditions for optimality are proved.Comment: Key words and phrases: stochastic Cahn-Hilliard equation, phase separation, optimal control, linearized state system, adjoint state system, first-order optimality condition

Ergodicity and Kolmogorov equations for dissipative SPDEs with singular drift: a variational approach

We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures $\mu$, solvability of the associated Kolmogorov equation in $L^1(\mu)$ is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variational setting.Comment: 32 page

Fr\'echet differentiability of mild solutions to SPDEs with respect to the initial datum

We establish n-th order Fr\'echet differentiability with respect to the initial datum of mild solutions to a class of jump-diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order G\^ateaux differentiability of their solutions with respect to the initial datum, extending previous results in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.Comment: 30 pages, no figure

The referendum incentive compatibility hypothesis: Some new results using information messages

We report results from a laboratory experiment that allows us to test the incentive compatibility hypothesis of hypothetical referenda used in CV studies through the public or private provision of information messages. One of the main methodological issues about hypothetical markets regards whether people behave differently when bidding for a public good through casting a ballot vote than when they are privately purchasing an equivalent good. This study tried to address the core of this issue by using a good that can be traded both as private and public: information messages. This allows the elimination of confounding effects associated with the specific good employed. In our case information dispels some of the uncertainty about a potential gain from a gamble. So, the approximate value of the message can be inferred once the individual measure of risk aversion is known. Decision tasks are then framed in a systematic manner according to the hypothetical vs real nature of the decision and the public vs private nature of the message. A sample of 536 university students across three countries (I, UK and NZ) participated into this lab experiment. The chosen countries reflect diversity in exposure to the practice of advisory (NZ) and abrogative (Italy) referenda, with the UK not having any exposure to it. Under private provision the results show that the fraction of participants unwilling to buy information is slightly higher in the real treatment than in the hypothetical one. Under public provision, instead, there is no statistical difference between real and hypothetical settings, confirming in part the finding of previous researchers. A verbal protocol analysis of the thought processes during choice highlights that public provision of information systematically triggers concerns and motivations different from those arising under the private provision setting. These findings suggest that the incentive compatibility of public referenda is likely to rely more on affective and psychological factors than on the strategic behaviour assumptions theorised by economists

Testing Newtonian gravity with distant globular clusters: NGC1851 and NGC1904

Globular clusters are useful to test the validity of Newtonian dynamics in the low acceleration regime typical of galaxies, without the complications of non-baryonic dark matter. Specifically, in absence of disturbing effects, e.g. tidal heating, their velocity dispersion is expected to vanish at large radii. If such behaviour is not observed, and in particular if, as observed in elliptical galaxies, the dispersion is found constant at large radii below a certain threshold acceleration, this might indicate a break down of Newtonian dynamics. To minimise the effects of tidal heating in this paper we study the velocity dispersion profile of two distant globular clusters, NGC 1851 and NGC 1904. The velocity dispersion profile is derived from accurate radial velocities measurements, obtained at the ESO 8m VLT telescope. Reliable data for 184 and 146 bona fide cluster star members, respectively for NGC 1851 and NGC 1904, were obtained. These data allow to trace the velocity dispersion profile up to ~2r0, where r0 is the radius at which the cluster internal acceleration of gravity is a0 = 10e-8 cm/s/s. It is found that in both clusters the velocity dispersion becomes constant beyond ~r0. These new results are fully in agreement with those found for other five globular clusters previously investigated as part of this project. Taken all together, these 7 clusters support the claim that the velocity dispersion is constant beyond r0, irrespectively of the specific physical properties of the clusters: mass, size, dynamical history, and distance from the Milky Way. The strong similarly with the constant velocity dispersion observed in elliptical galaxies beyond r0 is suggestive of a common origin for this phenomenon in the two class of objects, and might indicate a breakdown of Newtonian dynamics below a0.Comment: Accepted for publication by A&A main journal. 12 pages, 12 figure