Mixing it up: A general framework for Markovian statistics

Up to now, the nonparametric analysis of multidimensional continuous-time Markov processes has focussed strongly on specific model choices, mostly related to symmetry of the semigroup. While this approach allows to study the performance of estimators for the characteristics of the process in the minimax sense, it restricts the applicability of results to a rather constrained set of stochastic processes and in particular hardly allows incorporating jump structures. As a consequence, for many models of applied and theoretical interest, no statement can be made about the robustness of typical statistical procedures beyond the beautiful, but limited framework available in the literature. To close this gap, we identify $\beta$-mixing of the process and heat kernel bounds on the transition density as a suitable combination to obtain $\sup$-norm and $L^2$ kernel invariant density estimation rates matching the case of reversible multidimenisonal diffusion processes and outperforming density estimation based on discrete i.i.d. or weakly dependent data. Moreover, we demonstrate how up to $\log$-terms, optimal $\sup$-norm adaptive invariant density estimation can be achieved within our general framework based on tight uniform moment bounds and deviation inequalities for empirical processes associated to additive functionals of Markov processes. The underlying assumptions are verifiable with classical tools from stability theory of continuous time Markov processes and PDE techniques, which opens the door to evaluate statistical performance for a vast amount of Markov models. We highlight this point by showing how multidimensional jump SDEs with L\'evy driven jump part under different coefficient assumptions can be seamlessly integrated into our framework, thus establishing novel adaptive $\sup$-norm estimation rates for this class of processes

Universal spectral features of different classes of random diffusivity processes

Stochastic models based on random diffusivities, such as the diffusing- diffusivity approach, are popular concepts for the description of non-Gaussian diffusion in heterogeneous media. Studies of these models typically focus on the moments and the displacement probability density function. Here we develop the complementary power spectral description for a broad class of random diffusivity processes. In our approach we cater for typical single particle tracking data in which a small number of trajectories with finite duration are garnered. Apart from the diffusing-diffusivity model we study a range of previously unconsidered random diffusivity processes, for which we obtain exact forms of the probability density function. These new processes are different versions of jump processes as well as functionals of Brownian motion. The resulting behaviour subtly depends on the specific model details. Thus, the central part of the probability density function may be Gaussian or non-Gaussian, and the tails may assume Gaussian, exponential, log-normal or even power-law forms. For all these models we derive analytically the moment-generating function for the single-trajectory power spectral density. We establish the generic $1/f^2$-scaling of the power spectral density as function of frequency in all cases. Moreover, we establish the probability density for the amplitudes of the random power spectral density of individual trajectories. The latter functions reflect the very specific properties of the different random diffusivity models considered here. Our exact results are in excellent agreement with extensive numerical simulations.Severo Ochoa.SEV-2017-0718 BERC.2018-202

Cutoff Thermalization for Ornstein-Uhlenbeck Systems with Small Levy Noise in the Wasserstein Distance : Cutoff Thermalization for Ornstein–Uhlenbeck Systems...

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein-Uhlenbeck systems with small additive Lévy noise and any nonzero initial value.Peer reviewe