Stochastics and Dynamics

Based on our deterministic models for cholera epidemics, we propose a stochastic model for cholera epidemics to incorporate environmental fluctuations which is a nonlinear system of Itô stochastic differential equations. We conduct an asymptotical analysis of dynamical behaviors for the model. The basic stochastic reproduction value ${\mathcal{ℛ}}_s$ is defined in terms of the basic reproduction number $R_0$ for the corresponding deterministic model and noise intensities. The basic stochastic reproduction value determines the dynamical patterns of the stochastic model. When ${\mathcal{ℛ}}_s<1$, the cholera infection will extinct within finite periods of time almost surely. When ${\mathcal{ℛ}}_s>1$, the cholera infection will persist most of time, and there exists a unique stationary ergodic distribution to which all solutions of the stochastic model will approach almost surely as noise intensities are bounded. When the basic reproduction number $R_0$ for the corresponding deterministic model is greater than 1, and the noise intensities are large enough such that ${\mathcal{ℛ}}_s<1$, the cholera infection is suppressed by environmental noises. We carry out numerical simulations to illustrate our analysis, and to compare with the corresponding deterministic model. Biological implications are pointed out.

In this work, we prove a version of Hörmander’s theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent $\frac{1}{2}<H<1$ and an analytic semigroup on a given separable Hilbert space. In contrast to the classical finite-dimensional case, the Jacobian operator in typical solutions of parabolic stochastic PDEs is not invertible which causes a severe difficulty in expressing the Malliavin matrix in terms of an adapted process. Under a Hörmander’s bracket condition and some algebraic constraints on the vector fields combined with the range of the semigroup, we prove that the law of finite-dimensional projections of such solutions has a density with respect to Lebesgue measure. The argument is based on rough path techniques and a suitable analysis on the Gaussian space of the fractional Brownian motion.

We consider the sequence of spatial quadratic variations of the solution to the stochastic heat equation with space-time white noise. This sequence satisfies a Central Limit Theorem. By using Malliavin calculus, we refine this result by proving the convergence of the sequence of densities and by finding the second-order term in the asymptotic expansion of the densities. In particular, our proofs are based on sharp estimates of the correlation structure of the solution, which may have their own interest.

In this paper, we shall prove the homeomorphism property of solutions for multidimensional stochastic differential equations (SDEs) driven by $G$-Brownian motion with non-Lipschitz coefficients with respect to the initial values.

We consider a Piecewise Deterministic Markov Process given by random switching between finitely many vector fields vanishing at $0$. It has been shown recently that the behavior of this process is mainly determined by the signs of Lyapunov exponents. However, results have only been given when all these exponents have the same sign. In this paper, we consider the degenerate case where the process leaves invariant some face and results are stated when the Lyapunov exponents are of opposite signs. Our results enable in particular to close a gap in a discussion on random switching between two Lorenz vector fields by Bakhtin and Hurth, Invariant densities for dynamical systems with random switching, Nonlinearity 25 (2012) 2937–2952.

In this paper, we generalize the recently introduced concept of fair measure [M. Misiurewicz and A. Rodrigues, Counting preimages, Ergod. Theor. Dyn. Syst. 38 (2018) 1837–1856]. We study fair measures for Markov and mixing interval maps with countably many branches. We investigate them in terms of the recurrence properties of some underlying countable Markov shifts, both from the stochastic viewpoint and from the viewpoint of thermodynamical formalism. Finally, we move beyond the interval and look for fair measures for graph maps.

This paper aims to investigate the controllability for impulsive neutral stochastic delay partial differential equations (PDEs) driven by fractional Brownian motion (fBm) with Hurst index $H\in (\frac{1}{2},1)$ and Lévy noise in Hilbert spaces. By using a fixed point approach without imposing a severe compactness condition on the semigroup, a new set of sufficient conditions is derived. The results in this paper are generalization and continuation of the recent results on this issue. At the end, an application to the stochastic nonlinear heat equation with delays driven by a fBm and Lévy noise is given.

For the Ornstein–Uhlenbeck process in stationary and explosive cases, this paper studies Cramér-type moderate deviations for the log-likelihood ratio process. As an application, we give the negative regions of drift testing problem, and also obtain the decay rates of the error probabilities. The main methods of this paper consist of mod-$\varphi$ convergence approach, deviation inequalities for multiple Wiener–Itô integrals and asymptotic analysis techniques.