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Fractional Brownian motion (fBm) is a canonical model for long-memory phenomena. In the presence of large amounts of potentially memory-bearing data, the data are often averaged, which can change the structure of the underlying…
A simple derivation of the effective spin-wave Hamiltonian for a double-exchange system with infinitely large Hund's-rule coupling is demonstrated. The formalism can be applied to models with arbitrary range of hopping as well as those with…
A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is…
We introduce the concept of a "transitory" dynamical system---one whose time-dependence is confined to a compact interval---and show how to quantify transport between two-dimensional Lagrangian coherent structures for the Hamiltonian case.…
In Part V of this study, we presented an original Lagrangian approach for computing the dynamic characteristics along stationary rays, by solving the linear, second-order Jacobi differential equation, considering four sets of initial…
Hamilton's equations with noise and friction possess a hidden supersymmetry, valid for time-independent as well as periodically time-dependent systems. It is used to derive topological properties of critical points and periodic trajectories…
We follow up on our previous works which presented a possible approach for deriving symplectic schemes for a certain class of highly oscillatory Hamiltonian systems. The approach considers the Hamilton-Jacobi form of the equations of…
This paper motivates the use of random-bridges -- stochastic processes conditioned to take target distributions at fixed timepoints -- in the realm of generative modelling. Herein, random-bridges can act as stochastic transports between two…
We present a systematic method for dealing with time dependent quantum dynamics, based on the quantum brachistochrone and matrix mechanics. We derive the explicit time dependence of the Hamiltonian operator for a number of constrained…
The geometric framework for the Hamilton-Jacobi theory developed in previous works is extended for multisymplectic first-order classical field theories. The Hamilton-Jacobi problem is stated for the Lagrangian and the Hamiltonian formalisms…
The equivalence of the chain method and Hamilton-Jacobi formalism is demonstrated. The stabilization algorithm of Hamilton-Jacobi formalism is clariffied and two examples are presented in details.
The incidence of rare events in fast-slow systems is investigated via analysis of the large deviation principle (LDP) that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior --…
We show that, in 't Hooft's large N limit, matrix models can be formulated as a classical theory whose equations of motion are the factorized Schwinger--Dyson equations. We discover an action principle for this classical theory. This action…
In this article, we study high-dimensional behavior of empirical spectral distributions $\{L_N(t), t\in[0,T]\}$ for a class of $N\times N$ symmetric/Hermitian random matrices, whose entries are generated from the solution of stochastic…
For continuous systems modeled by dynamical equations such as ODEs and SDEs, Bellman's Principle of Optimality takes the form of the Hamilton-Jacobi-Bellman (HJB) equation, which provides the theoretical target of reinforcement learning…
A general formalism is developed for constructing modified Hamiltonian dynamical systems which preserve a canonical equilibrium distribution by adding a time evolution equation for a single additional thermostat variable. When such systems…
In this work we study the theory of linearized gravity via the Hamilton-Jacobi formalism. We make a brief review of this theory and its Lagrangian description, as well as a review of the Hamilton-Jacobi approach for singular systems. Then…
In this paper, we study the limiting distribution of the eigenvalues for random tridiagonal matrix models. The limiting distribution is well described by its moments. Here, an analytical approach allows us, as in the case of Wigner…
We prove pathwise large-deviation principles of switching Markov processes by exploiting the connection to associated Hamilton-Jacobi equations, following Jin Feng's and Thomas Kurtz's method. In the limit that we consider, we show how the…
Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties…