2,391 research outputs found
Markov selections for the 3D stochastic Navier-Stokes equations
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.Comment: 59 pages; corrected several errors and typos, added reference
Detecting adversarial manipulation using inductive Venn-ABERS predictors
Inductive Venn-ABERS predictors (IVAPs) are a type of probabilistic predictors with the theoretical guarantee that their predictions are perfectly calibrated. In this paper, we propose to exploit this calibration property for the detection of adversarial examples in binary classification tasks. By rejecting predictions if the uncertainty of the IVAP is too high, we obtain an algorithm that is both accurate on the original test set and resistant to adversarial examples. This robustness is observed on adversarials for the underlying model as well as adversarials that were generated by taking the IVAP into account. The method appears to offer competitive robustness compared to the state-of-the-art in adversarial defense yet it is computationally much more tractable
Communication and equilibrium in discontinuous games of incomplete information
This paper offers a new approach to the study of economic problems usually modeled as games of incomplete information with discontinuous payoffs. Typically, the discontinuities arise from indeterminacies (ties) in the underlying problem. The point of view taken here is that the tie-breaking rules that resolve these indeterminacies should be viewed as part of the solution rather than part of the description of the model. A solution is therefore a tie-breaking rule together with strategies satisfying the usual best-response criterion. When information is incomplete, solutions need not exist; that is, there may be no tie-breaking rule that is compatible with the existence of strategy profiles satisfying the usual best-response criteria. It is shown that the introduction of incentive compatible communication (cheap talk) restores existence
Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise
We prove that every Markov solution to the three dimensional Navier-Stokes
equation with periodic boundary conditions driven by additive Gaussian noise is
uniquely ergodic. The convergence to the (unique) invariant measure is
exponentially fast.
Moreover, we give a well-posedness criterion for the equations in terms of
invariant measures. We also analyse the energy balance and identify the term
which ensures equality in the balance.Comment: 32 page
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