Nonlocal gradient operators are basic elements of nonlocal vector calculus that play important roles in nonlocal modeling and analysis. In this work, we extend earlier analysis on nonlocal gradient operators. In particular, we study a nonlocal Dirichlet integral that is given by a quadratic energy functional based on nonlocal gradients. Our main finding, which differs from claims made in previo...

This paper examines a particular class of continuous-time stochastic processes commonly known as af¿ne diffusions (AD) and af¿ne jump-diffusions (AJD) in which the drift, the diffusion and the jump coef¿cients are all af¿ne functions of the state variables. By deriving the joint characteristic function associated with a vector of observed state variables for such models, we are able to examine ...

In this paper, we study quasi-stationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to −∞ at the origin, and the diffusion to have an entrance boundary at +∞. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth–death processes. Generalized F...

Abstract. We propose a compact reaction rate formula that accounts for a non-equilibrium distribution of residence times of complex molecules, based on a detailed study of the coarse-grained phase space of a reaction coordinate. We take the structural transition dynamics of a six-atomMorse cluster between two isomers as a prototype of multi-dimensional molecular reactions. Residence time distri...

Diffusion processes are a promising instrument to realistically model the time-continuous evolution of natural phenomena in life sciences. However, approximation of a given system is often carried out heuristically, leading to diffusions that do not correctly reflect the true dynamics of the original process. Moreover, statistical inference for diffusions proves to be challenging in practice as...

The article is concerned with a new nonlocal model based on Eringen’s nonlocal elasticity and generalized thermoelasticity. A study is made of the magneto-thermoelastic waves in an isotropic conducting thermoelastic finite rod subjected to moving heat sources permeated by a primary uniform magnetic field and rotating with a uniform angular velocity. The Laplace transform technique with respect ...

An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theo...

In analogy to a perturbed harmonic oscillator, we calculate the fundamental and some other higher order soliton solutions of the nonlocal nonlinear Schödinger equation (NNLSE) in the second approximation in the generally nonlocal case. Comparing with numerical simulations we show that soliton solutions in the second approximation can describe the generally nonlocal soliton states of the NNLSE m...