With the measurement of the electromagnetic (EM) counterpart, a gravitational wave (GW) event could be treated as a standard siren. As a novel cosmological probe, GW standard sirens will bring significant implications for cosmology. In this paper, by considering the coincident detections of GW and associated γ ray burst (GRB), we find that only about 400 GW bright standard sirens from binary neutron star mergers could be detected in a 10-year observation of the Einstein Telescope and the THESEUS satellite mission. Based on this mock sample, we investigate the implications of GW standard sirens on the interaction between dark energy and dark matter. In our analysis, four viable interacting dark energy (IDE) models, with interaction forms Q = 3βHρ _de and Q = Q = 3βHρ _c, are considered. Compared with the traditional EM observational data such as CMB, BAO, and SN Ia, the combination of both GW and EM observations could effectively break the degeneracies between different cosmological parameters and provide more stringent cosmological fits. We find that the GW data could play a more important role for determining the interaction in the models with Q = 3βHρ _c, compared with the models with Q = 3βHρ _de. We also show that constraining IDE models with mock GW data based on different fiducial H ₀ values yield different results, indicating that accurate determination of H ₀ is significant for exploring the interaction between dark energy and dark matter.

In this paper, we consider two kinds of stochastic models with asymptotic perturbation. One is the non-nearest neighbor random walk with asymptotic perturbation, for which we consider the transience and recurrence of the random walk.We give the criterion for positive recurrent of the random walk. The other is the branching process in random environment with asymptotic perturbation,for which by analyzing the limit behavior of the associated random walk, we give the criterion of extinction and non-extinction of the branching process.

With the measurement of the electromagnetic (EM) counterpart, a gravitational wave (GW) event could be treated as a standard siren. As a novel cosmological probe, GW standard sirens will bring significant implications for cosmology. In this paper, by considering the coincident detections of GW and associated $\gamma$ ray burst (GRB), we find that only about 400 GW bright standard sirens from binary neutron star mergers could be detected in a 10-year observation of the Einstein Telescope and the THESEUS satellite mission. Based on this mock sample, we investigate the implications of GW standard sirens on the interaction between dark energy and dark matter. In our analysis, four viable interacting dark energy (IDE) models, with interaction forms $Q=3\beta H \rho_{\mathrm{de}}$ and $Q=3 \beta H \rho_{\mathrm{c}}$, are considered. Compared with the traditional EM observational data such as CMB, BAO, and SN Ia, the combination of both GW and EM observations could effectively break the degeneracies between different cosmological parameters and provide more stringent cosmological fits. We find that the GW data could play a more important role for determining the interaction in the models with $Q=3 \beta H \rho_{\mathrm{c}}$, compared with the models with $Q=3\beta H \rho_{\mathrm{de}}$. We also show that constraining IDE models with mock GW data based on different fiducial $H_0$ values yield different results, indicating that accurate determination of $H_0$ is significant for exploring the interaction between dark energy and dark matter.