Narrow Results

This paper is devoted to studying parameter estimation for a class of stochastic dynamical systems with oscillating coefficients. We show that the homogenized systems faithfully capture the dynamical quantities such as mean exit time and escape probability. Exacting data from observations on the mean exit time (or escape probability) of the original systems, we try to fit the mean exit time (or escape probability) of the homogenized systems by least square method. In this way, we can accurately estimate the unknown parameter in the drift under appropriate assumptions. Furthermore, we conduct some numerical experiments to illustrate our method.

In this paper, we formulate a stochastic logistic fish growth model driven by both white noise and non-Gaussian noise. We focus our study on the mean time to extinction, escape probability to measure the noise-induced extinction probability and the Fokker–Planck equation for fish population $X(t)$. In the Gaussian case, these quantities satisfy local partial differential equations while in the non-Gaussian case, they satisfy nonlocal partial differential equations. Following a discussion of existence, uniqueness and stability, we calculate numerical approximations of the solutions of those equations. For each noise model we then compare the behaviors of the mean time to extinction and the solution of the Fokker–Planck equation as growth rate $r$, carrying capacity $K$, intensity of Gaussian noise $\lambda$, noise intensity $\sigma$ and stability index $\alpha$ vary. The MET from the interval $(0,1)$ at the right boundary is finite if $\lambda <\sqrt{2}$. For $\lambda >\sqrt{2}$, the MET from $(0,1)$ at this boundary is infinite. A larger stability index $\alpha$ is less likely leading to the extinction of the fish population.

In this paper, we consider some properties of switching Brownian motion. Combining the analytic method and probabilistic method, some explicit expressions of density functions, the mean exit time and Laplace transform of exit time are given. This paper reveals how drift coefficients impact the first passage probabilities, scale functions and the mean exit time for switching Brownian motion.

We investigate the functional $\mathrm{\Omega }\mapsto {ℰ}(\mathrm{\Omega })$ where $\mathrm{\Omega }$ runs through the set of compact domains of fixed volume $v$ in any Riemannian manifold $(M,g)$ and where ${ℰ}(\mathrm{\Omega })$ is the mean exit time from$\mathrm{\Omega }$of the Brownian motion. We give an alternative analytical proof of a well-known fact on its critical points proved by McDonald: the critical points of ${ℰ}(\mathrm{\Omega })$ are harmonic domains.