Stochastics and Dynamics

We study a rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against Euclidean space, where the rolling is coupled with random slipping and twisting. The system is modeled by a stochastic differential equation of Stratonovich-type driven by semimartingales on the orthonormal frame bundle. The stability of the system is examined via large deviations. We prove the large deviation principles for the projection curves on the base manifold and their horizontal lifts, respectively, provided that the large deviation holds for the random Euclidean curves as semimartingales. The large deviation results for the case of compact manifolds and two special cases of non-compact manifolds are established.

A new weak existence result for degenerate multi-dimensional stochastic McKean–Vlasov equation is established under relaxed regularity conditions.

We study the law of random self-similar series defined above an irrational rotation on the Circle. This provides a natural class of continuous singular non-Rajchman measures.

Feldman–Katok metric is the weakest metric in some sense that makes entropy formulae valid, which can also be considered as the topological counterpart of the edit distance. In this paper, we study Feldman–Katok metric of random dynamical systems and establish topological entropy formula as well as Brin–Katok local entropy formula and Katok entropy formula on Feldman–Katok metric.

We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-concave only outside some compact sets. The time discretization and the minibatch in SGLD introduce several difficulties when applying the reflection coupling, which are addressed by a series of careful estimates of conditional expectations. As a direct corollary, the SGLD with constant step size has an invariant distribution and we are able to obtain its geometric ergodicity in terms of $W_1$ distance. The generalization to non-gradient drifts is also included.

In this paper, we investigate the asymptotic behavior for the non-autonomous stochastic $p$-Laplacian equations with dynamic boundary conditions via the stationary Wong–Zakai approximations given by the Wiener shift. We first establish the existence and uniqueness of pullback random attractor for the Wong–Zakai approximation equations in $L^2({𝒪})\times L^2(\mathrm{\Gamma })$. Then we prove the upper semi-continuity of random attractors in $L^{q_1}({𝒪})\times L^{q_1}(\mathrm{\Gamma })$ and $L^l({𝒪})\times L^l(\mathrm{\Gamma })(\forall l>q_1)$ when the step size of approximations approaches zero.

We prove the Hölder–Dini regularity for a class of fractional Fokker–Planck–Kolmogorov equations. Then, by applying the Itô–Tanaka trick, we obtain the unique strong solvability of time inhomogeneous stochastic differential equations with drift coefficients in $L_t^{\infty }({{𝒞}}_{b,d}^{\beta ,\rho })$ for $\beta =1-\alpha /2$ and driven by $\alpha$-stable processes.

We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean–Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada–Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.