Stochastics and Dynamics

We prove that, for semi-invertible continuous cocycles, continuity of Lyapunov exponents is equivalent to continuity, in measure, of Oseledets subspaces.

D. Herzog and J. Mattingly have shown that a $ℂ$-valued polynomial ODE with finite-time blow-up solutions may be stabilized by the addition of $ℂ$-valued Brownian noise. In this paper, we extend their results to a ${ℂ}^2$-valued system of coupled ODEs with finite-time blow-up solutions. We show analytically and numerically that stabilization can be achieved in our setting by adding a suitable Brownian noise, and that the resulting system of SDEs is ergodic. The proof uses the Girsanov theorem to induce a time change from our ${ℂ}^2$-system to a quasi-$ℂ$-system similar to the one studied by Herzog and Mattingly.

We study the existence and regularity of the density for the solution $u(t,x)$ (with fixed $t>0$ and $x\in D$) of the heat equation in a bounded domain $D\subset {ℝ}^d$ driven by a stochastic inhomogeneous Neumann boundary condition with stochastic term. The stochastic perturbation is given by a fractional Brownian motion process. Under suitable regularity assumptions on the coefficients, by means of tools from the Malliavin calculus, we prove that the law of the solution has a smooth density with respect to the Lebesgue measure in $ℝ$.

We study the well-posedness for multi-dimensional and coupled systems of forward–backward SDEs when the generator can be separated into a quadratic and a subquadratic part. We obtain the existence and uniqueness of the solution on a small time interval. Moreover, the continuity and differentiability with respect to the initial value are presented.

Branching random walks on multidimensional lattice with heavy tails and a constant branching rate are considered. It is shown that under these conditions (heavy tails and constant rate), the front propagates exponentially fast, but the particles inside of the front are distributed very non-uniformly. The particles exhibit intermittent behavior in a large part of the region behind the front (i.e. the particles are concentrated only in very sparse spots there). The zone of non-intermittency (were particles are distributed relatively uniformly) extends with a power rate. This rate is found.

In this paper, we study one-dimensional Markov processes with spatial delay. Since the seminal work of Feller, we know that virtually any one-dimensional, strong, homogeneous, continuous Markov process can be uniquely characterized via its infinitesimal generator and the generator’s domain of definition. Unlike standard diffusions like Brownian motion, processes with spatial delay spend positive time at a single point of space. Interestingly, the set of times that a delay process spends at its delay point is nowhere dense and forms a positive measure Cantor set. The domain of definition of the generator has restrictions involving second derivatives. In this paper we provide a pathwise characterization for processes with delay in terms of an SDE and an occupation time formula involving the symmetric local time. This characterization provides an explicit Doob–Meyer decomposition, demonstrating that such processes are semi-martingales and that all of stochastic calculus including Itô formula and Girsanov formula applies. We also establish an occupation time formula linking the time that the process spends at a delay point with its symmetric local time there. A physical example of a stochastic dynamical system with delay is lastly presented and analyzed.

We propose and analyze a numerical scheme for the approximation of the solution for the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise. We prove the convergence of the scheme which is a linear evolution equation with additive noise.

In this paper, the probability properties are investigated for a stochastic SIS epidemic model. Transition probabilities of the susceptible process are obtained by using Laplace transform and perturbation variables. According to two cases: basic reproduction number $R_0\le 1$ and $R_0>1$, the dynamical behaviors in probability of the process are analyzed. It is shown that when $R_0\le 1$ the disease-free equilibrium is globally asymptotically stable with probability one, and when $R_0>1$ and $\frac{\mu +\gamma }{\beta }$ is a positive integer, the endemic equilibrium is globally asymptotically stable with probability one. These results coincide with the corresponding deterministic SIS epidemic model. However, when $R_0>1$ and $\frac{\mu +\gamma }{\beta }$ is not a positive integer, there are different properties between the deterministic and stochastic models. Numerical simulations are also performed to validate these results.