A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

In this paper we prove a necessary condition of the optimal control problem for a class of general mean-field forward-backward stochastic systems with jumps in the case where the diffusion coefficients depend on control, the control set does not need to be convex, the coefficients of jump terms are independent of control as well as the coefficients of mean-field backward stochastic differential equations depend on the joint law of $(X(t),Y(t))$. Two new adjoint equations are brought in as well as several new generic estimates of their solutions are investigated for analysing the higher terms, especially, those involving the expectation which come from the derivatives of the coefficients with respect to the measure. Utilizing these subtle estimates, the second-order expansion of the cost functional, which is the key point to analyse the necessary condition, is obtained, and whereafter the stochastic maximum principle.Comment: 32 page

Majorana Edge States in Interacting Two-chain Ladders of Fermions

In this work we study interacting spinless fermions on a two-chain ladder with inter-chain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a $\mathbb{Z}_2$ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground state degeneracy against local perturbations. Lattice fermion models realizing such effective field theory are discussed.Comment: 6 pages, 1 figur

Progenitor delay-time distribution of short gamma-ray bursts: Constraints from observations

Context. The progenitors of short gamma-ray bursts (SGRBs) have not yet been well identified. The most popular model is the merger of compact object binaries (NS-NS/NS-BH). However, other progenitor models cannot be ruled out. The delay-time distribution of SGRB progenitors, which is an important property to constrain progenitor models, is still poorly understood. Aims. We aim to better constrain the luminosity function of SGRBs and the delay-time distribution of their progenitors with newly discovered SGRBs. Methods. We present a low-contamination sample of 16 Swift SGRBs that is better defined by a duration shorter than 0.8 s. By using this robust sample and by combining a self-consistent star formation model with various models for the distribution of time delays, the redshift distribution of SGRBs is calculated and then compared to the observational data. Results. We find that the power-law delay distribution model is disfavored and that only the lognormal delay distribution model with the typical delay tau >= 3 Gyr is consistent with the data. Comparing Swift SGRBs with T90 > 0.8 s to our robust sample (T90 < 0.8 s), we find a significant difference in the time delays between these two samples. Conclusions. Our results show that the progenitors of SGRBs are dominated by relatively long-lived systems (tau >= 3 Gyr), which contrasts the results found for Type Ia supernovae. We therefore conclude that primordial NS-NS systems are not favored as the dominant SGRB progenitors. Alternatively, dynamically formed NS-NS/BH and primordial NS-BH systems with average delays longer than 5 Gyr may contribute a significant fraction to the overall SGRB progenitors.Comment: 8 pages, 6 figures, Astron. Astrophys. in pres