Stochastics and Dynamics

We establish the existence of random stable and unstable manifolds for ill-posed stochastic partial differential equations (SPDEs). Namely, we assume that the linear part does not generate a $C_0$-semigroup. Using the theory of integrated semigroups, we are able to analyze the long-time behavior of random dynamical systems generated by such SPDEs.

The interplay between bifurcations and random switching processes of vector fields is studied. More precisely, we provide a classification of piecewise-deterministic Markov processes arising from stochastic switching dynamics near fold, Hopf, transcritical and pitchfork bifurcations. We prove the existence of invariant measures for different switching rates. We also study when the invariant measures are unique, when multiple measures occur, when measures have smooth densities, and under which conditions finite-time blow-up occurs. We demonstrate the applicability of our results for three nonlinear models arising in applications.

We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e. the number of ergodic attractors and their corresponding mixing components do not change under small perturbations of the map. Our methods provide a topological description of the attractor and give an elementary proof of the density of periodic orbits.

We consider a least square-type method to estimate the drift parameters for the mean-reverting Ornstein–Uhlenbeck process of the second kind $\{X_t,t\ge 0\}$ defined as $d{X}_t=𝜃(\mu +X_t)dt+d{Y}_{t,G}^{(1)},t\ge 0$, with unknown parameters $𝜃>0$ and $\mu \in ℝ$, where $Y_{t,G}^{(1)}:={\int }_0^t{e}^{-s}d{G}_{a_s}$ with $a_t={\mathrm{\text{He}}}^{\frac{t}{H}}$, and $\{G_t,t\ge 0\}$ is a Gaussian process. In order to establish the consistency and the asymptotic distribution of least square-type estimators of $𝜃$ and $\mu$ based on the continuous-time observations $\{X_t,t\in [0,T]\}$ as $T\to \infty$, we impose some technical conditions on the process $G$, which are satisfied, for instance, if $G$ is a fractional Brownian motion with Hurst parameter $H\in (0,1)$, $G$ is a subfractional Brownian motion with Hurst parameter $H\in (0,1)$ or $G$ is a bifractional Brownian motion with Hurst parameters $(H,K)\in (0,1)\times (0,1]$. Our method is based on pathwise properties of $\{X_t,t\ge 0\}$ and $\{Y_{t,G}^{(1)},t\ge 0\}$ proved in the sequel.

We prove an $L^2$-regularity result for the solutions of Forward Backward doubly stochastic differential equations (F-BDSDEs) under globally Lipschitz continuous assumptions on the coefficients. As an application of our result, we derive the rate of convergence in time for the (Euler time discretization-based) numerical scheme for F-BDSDEs proposed by Bachouch et al. (2016) under only globally Lipschitz continuous assumptions.

We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erdős–Rényi graph with parameter $p_n\in (0,1]$, where $n$ is the size of the graph (i.e. the number of particles). If $p_n\equiv 1$, the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as $n\to \infty$ to the solution of a PDE: a McKean–Vlasov (or Fokker–Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erdős–Rényi graphs with ${lim}_n{p}_{n}n=\infty$, and properly rescaling the interaction to account for the dilution introduced by $p_n$. However, these results have been proven under strong assumptions on the initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results — Law of Large Numbers and Large Deviation Principle — assuming only the convergence of the empirical measure of the initial condition.

We study the local entropy of typical infinite Bowen balls in random dynamical systems, and show the random entropy expansiveness for $C^1$ partially hyperbolic diffeomorphisms with multi one-dimensional centers. Moreover, we consider $C^1$ diffeomorphism $f$ with dominated splitting $TM=E^{cu}\oplus E_1^c\oplus \cdots \oplus E_k^c\oplus E^{cs}$ such that $dim{E}_i^c=1$ for every $1\le i\le k$, and all the Lyapunov exponents are non-negative along $E^{cu}$ and non-positive along $E^{cs}$, we prove the asymptotically random entropy expansiveness for $f$.

Random attractors and their higher-order regularity properties are studied for stochastic reaction–diffusion equations on time-varying domains. Some new a priori estimates for the difference of solutions near the initial time and the continuous dependence in initial data in $H_0^1$ are proved. Then attraction of the random attractors in the higher integrability space $L^{2+\delta }$ for any $\delta \in [0,\infty )$ and the regular space $H_0^1$ is established.