Mean field approach to stochastic control with partial information

In our present article, we follow our way of developing mean field type control theory in our earlier works [4], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as [3, 13], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gaussian case. Besides, our problem considered here can be reduced to the model in [2], which is fundamentally different from our present proposed framework.

Addressing indirect frequency coupling via partial generalized coherence

Distinguishing between direct and indirect frequency coupling is an important aspect of functional connectivity analyses because this distinction can determine if two brain regions are directly connected. Although partial coherence quantifies partial frequency coupling in the linear Gaussian case, we introduce a general framework that can address even the nonlinear and non-Gaussian case. Our technique, partial generalized coherence (PGC), expands prior work by allowing pairwise frequency coupling analyses to be conditioned on other processes, enabling model-free partial frequency coupling results. By taking advantage of recent advances in conditional mutual information estimation, we are able to implement our technique in a way that scales well with dimensionality, making it possible to condition on many processes and produce a partial frequency coupling graph. We analyzed both linear Gaussian and nonlinear simulated networks. We then performed PGC analysis of calcium recordings from mouse olfactory bulb glomeruli under anesthesia and quantified the dominant influence of breathing-related activity on the pairwise relationships between glomeruli for breathing-related frequencies. Overall, we introduce a technique capable of eliminating indirect frequency coupling in a model-free way, empowering future research to correct for potentially misleading frequency interactions in functional connectivity analyses.

Examining vehicular speed characteristics through divergences from prior distributions

A variety of approaches, within literature, has been conducted to interpret vehicular speed characteristics. This study turns the attention to the entropy-based approaches, and thus focuses on the maximum entropy method of statistical mechanics and the Kullback–Leibler (KL) divergence approach to examining the vehicular speeds. The vehicle speeds at the selected highway are analyzed in order to find out the disparities among them. However, it is turned out that the speed dynamics could not be distinguished over the speed distributions; hence the maximization of Shannon entropy seems insufficient to compare the speed distributions of each data set. For this reason, the KL divergence approach was performed. This approach displays the comparison, among the speed distributions, based on two prior distribution models, i.e., uniform and Gauss. The examination of the trends of KL divergences obtained from both distributions was made. It was concluded that the KL divergence values for the highway speed data sets ranged between about 0.53 and 0.70 for the uniform case, while for the Gaussian case the obtained values are between 0.16 and 0.33. The KL divergence trends for the real speeds were obtained analogous for both cases, but they differed significantly when the synthetic data sets were employed. As a result, the KL divergence approach proves suitable as an appropriate indicator to compare the speed distributions.

Dynamic probabilistic constraints under continuous random distributions

The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from $$L^2$$ L 2 is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented.

Towards spacetime entanglement entropy for interacting theories

Entanglement entropy of quantum fields in gravitational settings is a topic of growing importance. This entropy of entanglement is conventionally computed relative to Cauchy hypersurfaces where it is possible via a partial tracing to associate a reduced density matrix to the spacelike region of interest. In recent years Sorkin has proposed an alternative, manifestly covariant, formulation of entropy in terms of the spacetime two-point correlation function. This formulation, developed for a Gaussian scalar field theory, is explicitly spacetime in nature and evades some of the possible non-covariance issues faced by the conventional formulation. In this paper we take the first steps towards extending Sorkin’s entropy to non-Gaussian theories where Wick’s theorem no longer holds and one would expect higher correlators to contribute. We consider quartic perturbations away from the Gaussian case and find that to first order in perturbation theory, the entropy formula derived by Sorkin continues to hold but with the two-point correlators replaced by their perturbation-corrected counterparts. We then show that our results continue to hold for arbitrary perturbations (of both bosonic and fermionic theories). This is a non-trivial and, to our knowledge, novel result. Furthermore we also derive closed-form formulas of the entanglement entropy for arbitrary perturbations at first and second order. Our work also suggests avenues for further extensions to generic interacting theories.

Mean exit time and escape probability for the stochastic logistic growth model with multiplicative α-stable Lévy noise

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 [Formula: see text]. 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 [Formula: see text], carrying capacity [Formula: see text], intensity of Gaussian noise [Formula: see text], noise intensity [Formula: see text] and stability index [Formula: see text] vary. The MET from the interval [Formula: see text] at the right boundary is finite if [Formula: see text]. For [Formula: see text], the MET from [Formula: see text] at this boundary is infinite. A larger stability index [Formula: see text] is less likely leading to the extinction of the fish population.

On the Capacity Regions of Degraded Relay Broadcast Channels with and without Feedback

The four-node relay broadcast channel (RBC) is considered, in which a transmitter communicates with two receivers with the assistance of a relay node. We first investigate three types of physically degraded RBCs (PDRBCs) based on different degradation orders among the relay and the receivers’ observed signals. For the discrete memoryless (DM) case, only the capacity region of the second type of PDRBC is already known, while for the Gaussian case, only the capacity region of the first type of PDRBC is already known. In this paper, we step forward and make the following progress: (1) for the first type of DM-PDRBC, a new outer bound is established, which has the same rate expression as an existing inner bound, with only a slight difference on the input distributions; (2) for the second type of Gaussian PDRBC, the capacity region is established; (3) for the third type of PDRBC, the capacity regions are established both for DM and Gaussian cases. Besides, we also consider the RBC with relay feedback where the relay node can send the feedback signal to the transmitter. A new coding scheme based on a hybrid relay strategy and a layered Marton’s coding is proposed. It is shown that our scheme can strictly enlarge Behboodi and Piantanida’s rate region, which is tight for the second type of DM-PDRBC. Moreover, we show that capacity regions of the second and third types of PDRBCs are exactly the same as that without feedback, which means feedback cannot enlarge capacity regions for these types of RBCs.

Linear differential equations for the resolvents of the classical matrix ensembles

The spectral density for random matrix [Formula: see text] ensembles can be written in terms of the average of the absolute value of the characteristic polynomial raised to the power of [Formula: see text], which for even [Formula: see text] is a polynomial of degree [Formula: see text]. In the cases of the classical Gaussian, Laguerre, and Jacobi weights, we show that this polynomial, and moreover, the spectral density itself, can be characterized as the solution of a linear differential equation of degree [Formula: see text]. This equation, and its companion for the resolvent, are given explicitly for [Formula: see text] and [Formula: see text] for all three classical cases, and also for [Formula: see text] in the Gaussian case. Known dualities for the spectral moments relating [Formula: see text] to [Formula: see text] then imply corresponding differential equations in the case [Formula: see text], and for the Gaussian ensemble, the case [Formula: see text]. We apply the differential equations to give a systematic derivation of recurrences satisfied by the spectral moments and by the coefficients of their [Formula: see text] expansions, along with first-order differential equations for the coefficients of the [Formula: see text] expansions of the corresponding resolvents. We also present the form of the differential equations when scaled at the hard or soft edges.