Jump processes as generalized gradient flows

We have created a functional framework for a class of non-metric gradient systems. The state space is a space of nonnegative measures, and the class of systems includes the Forward Kolmogorov equations for the laws of Markov jump processes on Polish spaces. This framework comprises a definition of a notion of solutions, a method to prove existence, and an archetype uniqueness result. We do this by using only the structure that is provided directly by the dissipation functional, which need not be homogeneous, and we do not appeal to any metric structure.

Diffusion in Metallic Glasses and in Oxide Glasses - An Overview

We remind the reader to some common features of metallic and oxide glasses. We then introduce the radiotracer method for diffusion studies, which can be applied for both types of glasses. We provide an overview on diffusion in metallic glasses in which we consider both types of metallic glasses – conventional and bulk metallic glasses. In the last part we discuss diffusion and ionic conduction in oxide glasses. For ionic glasses, conductivity measurements are an important complement to tracer diffusion studies. We remind the reader to the method of impedance spectroscopy. We discuss results for soda-lime silicate glasses, single alkali borate glasses and mixed alkali borate glasses and present evidence for collective jump processes in glasses.

Resource System with Losses in a Random Environment

The article deals with queueing systems with random resource requirements modeled as bivariate Markov jump processes. One of the process components describes the service system with limited resources. Another component represents a random environment that submits multi-class requests for resources to the service system. If the resource request is lost, then the state of the service system does not change. The change in the state of the environment interacting with the service system depends on whether the resource request has been lost. Thus, unlike in known models, the service system provides feedback to the environment in response to resource requests. By analyzing the properties of the system of integral equations for the stationary distribution of the corresponding random process, we obtain the conditions for the stationary distribution to have a product form. These conditions are expressed in the form of three systems of nonlinear equations. Several special cases are explained in detail.

Fuzzy logic: about the origins of fast ion dynamics in crystalline solids

Nuclear magnetic resonance offers a wide range of tools to analyse ionic jump processes in crystalline and amorphous solids. Both high-resolution and time-domain 1 , 2 H , 6 , 7 Li , 19 F , 23 Na NMR helps throw light on the origins of rapid self-diffusion in materials being relevant for energy storage. It is well accepted that Li + ions are subjected to extremely slow exchange processes in compounds with strong site preferences. The loss of this site preference may lead to rapid cation diffusion, as is also well known for glassy materials. Further examples that benefit from this effect include, e.g. cation-mixed, high-entropy fluorides ( Ba, Ca) F 2 , Li-bearing garnets ( Li 7 La 3 Zr 2 O 12 ) and thiophosphates such as LiTi 2 ( PS 4 ) 3 . In non-equilibrium phases site disorder, polyhedra distortions, strain and the various types of defects will affect both the activation energy and the corresponding attempt frequencies. Whereas in ( Me, Ca ) F 2 ( Me = Ba , Pb ) cation mixing influences F anion dynamics, in Li 6 PS 5 X ( X = Br , Cl , I ) the potential landscape can be manipulated by anion site disorder. On the other hand, in the mixed conductor Li 4 + x Ti 5 O 12 cation-cation repulsions immediately lead to a boost in Li + diffusivity at the early stages of chemical lithiation. Finally, rapid diffusion is also expected for materials that are able to guide the ions along (macroscopic) pathways with confined (or low-dimensional) dimensions, as is the case in layer-structured RbSn 2 F 5 or MeSnF 4 . Diffusion on fractal systems complements this type of diffusion. This article is part of the Theo Murphy meeting issue ‘Understanding fast-ion conduction in solid electrolytes’.

Gradient flows in metric random walk spaces

Recently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each $$x\in X$$ x ∈ X , which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.

Variational inference for Markovian queueing networks

Queueing networks are stochastic systems formed by interconnected resources routing and serving jobs. They induce jump processes with distinctive properties, and find widespread use in inferential tasks. Here, service rates for jobs and potential bottlenecks in the routing mechanism must be estimated from a reduced set of observations. However, this calls for the derivation of complex conditional density representations, over both the stochastic network trajectories and the rates, which is considered an intractable problem. Numerical simulation procedures designed for this purpose do not scale, because of high computational costs; furthermore, variational approaches relying on approximating measures and full independence assumptions are unsuitable. In this paper, we offer a probabilistic interpretation of variational methods applied to inference tasks with queueing networks, and show that approximating measure choices routinely used with jump processes yield ill-defined optimization problems. Yet we demonstrate that it is still possible to enable a variational inferential task, by considering a novel space expansion treatment over an analogous counting process for job transitions. We present and compare exemplary use cases with practical queueing networks, showing that our framework offers an efficient and improved alternative where existing variational or numerically intensive solutions fail.

On Itô formulas for jump processes

A well-known Itô formula for finite-dimensional processes, given in terms of stochastic integrals with respect to Wiener processes and Poisson random measures, is revisited and is revised. The revised formula, which corresponds to the classical Itô formula for semimartingales with jumps, is then used to obtain a generalisation of an important infinite-dimensional Itô formula for continuous semimartingales from Krylov (Probab Theory Relat Fields 147:583–605, 2010) to a class of $$L_p$$ L p -valued jump processes. This generalisation is motivated by applications in the theory of stochastic PDEs.