We consider simple autoregressive processes of type AR(1), whose stationary distribution is supported on a subset of the unit interval and is an affine transformation of a Bernoulli convolution. A new structural representation of the stationary distribution as a product of a power function with a log-periodic function near the origin is given, which gives structural insight to the stationary distribution on the whole unit interval by using a characteristic functional equation. This enables to prove that the stationary distribution of the AR(1)-process belongs to the domain of geometric partial attraction of a max-semistable law. We further prove uniform convergence of the distribution function of normalized maxima of the AR(1)-process to a certain power of the max-semistable law in the spirit of a merge theorem and point out connections to deterministic and random dynamical systems.
We study linear and symmetric diffusion processes on $\mathbb{R}^{\mathbb{Z}^d}$ that can be seen as the Langevin dynamics of the (inhomogeneous) harmonic crystal for given conductances, with a particular focus on their stationary distributions. Despite the linearity of the interaction, we exhibit a rich variety of beh...
J. Deuschel, Jonas Köppl, Yannic Steenbeck et al.· 0 citations
Stationarity of a stochastic process seems connected to the idea of constancy. But ergodicity is needed for the property that almost surely the observation of a trajectory from time -∞ to 0 makes possible the identification of the law of the whole process, including the future. When the stationary process is a Markov c...
M. Valadier· Journal of Convex Analysis· 0 citations
We prove unique ergodicity of the projective process associated with the two dimensional Navier--Stokes equation in vorticity form, with additive diagonal noise acting on every nonzero real Fourier phase and satisfying two sided power law bounds. Consequently, the exact Furstenberg--Khasminskii formula for the top Lyap...
We study the vanishing noise limit of the rate function for the Donsker-Varadhan large deviation principle for one-dimensional diffusion processes on a circle. As is well known, the rate function can be represented either as the Legendre transform of principal eigenvalue of the perturbed infinitesimal generator or by a...
Milan Koresski· Electronic Journal of Differ...· 0 citations
We study the iteration of (T(n)=\psi(n)-n), where (\psi) is the Dedekind psi function. We show that, for almost all integers, successive iterates satisfy strong asymptotic growth relations governed by (T(n)/n). As an application, we prove that for every fixed (\ell\ge1), the set of integers satisfying (T^{(\ell)}(n)=n)...
We study high-dimensional conditioned dynamics obtained from an arbitrary number of independent copies of a mixing Markov chain. The dynamics is conditioned to avoid a family of holes whose stationary measure vanishes as the dimension grows, and we ask whether the resulting process admits a quasi-stationary measure clo...
Matteo Tanzi, Giuseppe Tenaglia· 0 citations
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