We consider nonparametric Bayesian inference in a reflected diffusion model dXt = b(Xt)dt+σ(Xt)dWt, with discretely sampled observations X0, X∆, . . . , Xn∆. We analyse the nonlinear inverse problem corresponding to the 'low frequency sampling' regime where ∆ > 0 is fixed and n → ∞. A general theorem is proved that gives conditions for prior distributions Π on the diffusion coefficient σ and the drift function b that ensure minimax optimal contraction rates of the posterior distribution over Hölder-Sobolev smoothness classes. These conditions are verified for natural examples of nonparametric random wavelet series priors. For the proofs we derive new concentration inequalities for empirical processes arising from discretely observed diffusions that are of independent interest. Primary 62G05; secondary 60J60 62F15 62G20

A method is proposed for calculating the shear viscosity of a liquid from finite-size effects of self-diffusion coefficients in Molecular Dynamics simulations. This method uses the difference in the self-diffusivities, computed from at least two system sizes, and an analytic equation to calculate the shear viscosity. To enable the efficient use of this method, a set of guidelines is developed. The most efficient number of system sizes is two and the large system is at least four times the small system. The number of independent simulations for each system size should be assigned in such a way that 50%–70% of the total available computational resources are allocated to the large system. We verified the method for 250 binary and 26 ternary Lennard-Jones systems, pure water, and an ionic liquid ([Bmim][Tf2N]). The computed shear viscosities are in good agreement with viscosities obtained from equilibrium Molecular Dynamics simulations for all liquid systems far from the critical point. Our results indicate that the proposed method is suitable for multicomponent mixtures and highly viscous liquids. This may enable the systematic screening of the viscosities of ionic liquids and deep eutectic solvents.

We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the formwhere N (t) is a standard Poisson process of intensity λ, and Z k are drawn i.i.d. from jump measure µ. A high-dimensional wavelet series prior for the Lévy measure ν = λµ is devised and the posterior distribution arises from observing discrete samples Y ∆ , Y 2∆ , . . . , Y n∆ at fixed observation distance ∆, giving rise to a nonlinear inverse inference problem. We derive contraction rates in uniform norm for the posterior distribution around the true Lévy density that are optimal up to logarithmic factors over Hölder classes, as sample size n increases. We prove a functional Bernstein-von Mises theorem for the distribution functions of both µ and ν, as well as for the intensity λ, establishing the fact that the posterior distribution is approximated by an infinite-dimensional Gaussian measure whose covariance structure is shown to attain the information lower bound for this inverse problem. As a consequence posterior based inferences, such as nonparametric credible sets, are asymptotically valid and optimal from a frequentist point of view.MSC 2000 subject classification: 62G20, 65N21, 60G51, 60J75

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a Gaussian process that can be obtained from the composition of a Brownian motion with a covariance operator determined by the Lévy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the Lévy jump measure. As an application we deduce Kolmogorov-Smirnov type tests and confidence bands.MSC 2000 subject classification: Primary: 60F05; Secondary: 60G51, 62G05

We estimate linear functionals in the classical deconvolution problem by kernel estimators. We obtain a uniform central limit theorem with √ n-rate on the assumption that the smoothness of the functionals is larger than the ill-posedness of the problem, which is given by the polynomial decay rate of the characteristic function of the error. The limit distribution is a generalized Brownian bridge with a covariance structure that depends on the characteristic function of the error and on the functionals. The proposed estimators are optimal in the sense of semiparametric efficiency. The class of linear functionals is wide enough to incorporate the estimation of distribution functions. The proofs are based on smoothed empirical processes and mapping properties of the deconvolution operator.