scholarly journals The martingale problem for a class of nonlocal operators of diagonal type

2019 ◽  
Vol 292 (11) ◽  
pp. 2316-2337 ◽  
Author(s):  
Jamil Chaker
Keyword(s):  
Martingale Problem ◽  
Nonlocal Operators

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

scholarly journals A path integral formulation for particle detectors: the Unruh-DeWitt model as a line defect

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Ivan M. Burbano ◽  
T. Rick Perche ◽  
Bruno de S. L. Torres
Keyword(s):  
Path Integral ◽  
Degrees Of Freedom ◽  
Wilson Line ◽  
Transition Probability ◽  
Relativistic Quantum ◽  
Quantum Fields ◽  
Line Defect ◽  
Particle Detectors ◽  
Nonlocal Operators ◽  
Detector Model

Abstract Particle detectors are an ubiquitous tool for probing quantum fields in the context of relativistic quantum information (RQI). We formulate the Unruh-DeWitt (UDW) particle detector model in terms of the path integral formalism. The formulation is able to recover the results of the model in general globally hyperbolic spacetimes and for arbitrary detector trajectories. Integrating out the detector’s degrees of freedom yields a line defect that allows one to express the transition probability in terms of Feynman diagrams. Inspired by the light-matter interaction, we propose a gauge invariant detector model whose associated line defect is related to the derivative of a Wilson line. This is another instance where nonlocal operators in gauge theories can be interpreted as physical probes for quantum fields.

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

The martingale comparison method for Markov processes

2021 ◽  
Vol 58 (1) ◽  
pp. 164-176
Author(s):  
Benedikt Köpfer ◽  
Ludger Rüschendorf
Keyword(s):  
Markov Processes ◽  
Martingale Problem ◽  
Function Class ◽  
Comparison Method ◽  
Stochastic Orders ◽  
Regularity Conditions ◽  
Evolution System ◽  
Basic Step ◽  
Theory Of Evolution ◽  
Comparison Results

AbstractComparison results for Markov processes with respect to function-class-induced (integral) stochastic orders have a long history. The most general results so far for this problem have been obtained based on the theory of evolution systems on Banach spaces. In this paper we transfer the martingale comparison method, known for the comparison of semimartingales to Markovian semimartingales, to general Markov processes. The basic step of this martingale approach is the derivation of the supermartingale property of the linking process, giving a link between the processes to be compared. This property is achieved using the characterization of Markov processes by the associated martingale problem in an essential way. As a result, the martingale comparison method gives a comparison result for Markov processes under a general alternative but related set of regularity conditions compared to the evolution system approach.

scholarly journals A note on symmetries of diffusions within a martingale problem approach

2019 ◽  
Vol 19 (02) ◽  
pp. 1950011 ◽  
Author(s):  
Francesco C. De Vecchi ◽  
Paola Morando ◽  
Stefania Ugolini
Keyword(s):  
Diffusion Processes ◽  
Martingale Problem ◽  
Second Order ◽  
Lie Symmetries ◽  
Definition Of

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second-order geometry and Itô integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.

scholarly journals The martingale problem for pseudo-differential operators on infinite-dimensional spaces

1999 ◽  
Vol 153 ◽  
pp. 101-118 ◽  
Author(s):  
V. Bogachev ◽  
P. Lescot ◽  
M. Röckner
Keyword(s):  
Differential Operators ◽  
Martingale Problem ◽  
Existence Of A Solution ◽  
Infinite Dimensional ◽  
Pseudo Differential Operators

AbstractA martingale problem for pseudo-differential operators on infinite dimensional spaces is formulated and the existence of a solution is proved. Applications to infinite dimensional “stable-like” processes are presented.

scholarly journals Regularity and rigidity theorems for a class of anisotropic nonlocal operators

2016 ◽  
Vol 153 (1-2) ◽  
pp. 53-70 ◽  
Author(s):  
Alberto Farina ◽  
Enrico Valdinoci
Keyword(s):  
Nonlocal Operators ◽  
Rigidity Theorems
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