# To Do List

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== Things that need to be done == | == Things that need to be done == | ||

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The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki. | The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki. | ||

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* Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]]. | * Some discussion on [[Dirichlet form|Dirichlet forms]], and maybe some models from [[nonlocal image processing]]. | ||

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* Fractional curvatures in conformal geometry. | * Fractional curvatures in conformal geometry. | ||

- | * Create a page on [[Non-local Lagrangian | Non-local Lagrangians]] | + | * Create a page on [[Non-local Lagrangian | Non-local Lagrangians]] and mention relevant examples, note Non-local Lagrangian is a term used by particle physicists and it is not necessarily the same as a [[Dirichlet form]]. |

* We need to explain further the [[Extension technique]] and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: [[Geometric Scattering Theory]], [[Ambient Metric Construction]], [[GJMS Operators]] and [[Singular Yamabe Problem]]..) | * We need to explain further the [[Extension technique]] and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: [[Geometric Scattering Theory]], [[Ambient Metric Construction]], [[GJMS Operators]] and [[Singular Yamabe Problem]]..) | ||

- | * It would be | + | * It would be good to have an article on the natural similarities between non-local operators and [[elliptic operators on graphs]]. |

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* [[nonlocal image processing]]. | * [[nonlocal image processing]]. | ||

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* [[Keller-Segel equation]]. | * [[Keller-Segel equation]]. | ||

- | * [[Active scalar equations]]. | + | * Improve the page on [[Active scalar equations]]. |

* [[Ising Models With Nonlocal Interactions]] | * [[Ising Models With Nonlocal Interactions]] | ||

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* [[Neumann Boundary Conditions]] | * [[Neumann Boundary Conditions]] | ||

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* Processes in the "limit" $\alpha=0$, [[Zero Order Operators]] | * Processes in the "limit" $\alpha=0$, [[Zero Order Operators]] | ||

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* Discussion of the [[Barlow-Bass-Chen-Kassmann Counterexample]] to Regularity of $L$-Harmonic functions. | * Discussion of the [[Barlow-Bass-Chen-Kassmann Counterexample]] to Regularity of $L$-Harmonic functions. | ||

- | * Discussion of Jean Picard's paper | + | * Discussion of Jean Picard's paper (regularity for pure jump processes with non-degenerate, translation invariant generators) |

* Add Courrège's Structure theorem for operators and other structure theorems (generators of Feller semigroups, Hunt processes... etc) | * Add Courrège's Structure theorem for operators and other structure theorems (generators of Feller semigroups, Hunt processes... etc) | ||

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+ | * A page on the nonlocal analogue of the thin film equation | ||

== (partially) Completed tasks == | == (partially) Completed tasks == | ||

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* Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]]. | * Definition of [[viscosity solutions]] for nonlocal equations. Also a discussion on existence using [[Perron's method]] and uniqueness through the [[comparison principle]]. | ||

- | * Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[ | + | * Some general regularity results like [[holder estimates]], [[Harnack inequalities]], [[Aleksandrov-Bakelman-Pucci estimates]], some reference to [[free boundary problems]]. |

* A list of [[regularity results for fully nonlinear integro-differential equations|regularity results]] for [[fully nonlinear integro-differential equations]]. | * A list of [[regularity results for fully nonlinear integro-differential equations|regularity results]] for [[fully nonlinear integro-differential equations]]. | ||

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* Having a [[list of equations]] may make it easier to navigate the wiki. | * Having a [[list of equations]] may make it easier to navigate the wiki. | ||

+ | * Jump processes via [[Subordination]] | ||

== Wiki Maintenance == | == Wiki Maintenance == |

## Latest revision as of 21:11, 25 January 2016

## Things that need to be done

The list below can be a starting point to click on links and edit each page. The following are some of the topics that should appear in this wiki.

- Some discussion on Dirichlet forms, and maybe some models from nonlocal image processing.

- Fractional curvatures in conformal geometry.

- Create a page on Non-local Lagrangians and mention relevant examples, note Non-local Lagrangian is a term used by particle physicists and it is not necessarily the same as a Dirichlet form.

- We need to explain further the Extension technique and its connection with fractional powers of the Laplacian and Conformal geometry. Required background: Geometric Scattering Theory, Ambient Metric Construction, GJMS Operators and Singular Yamabe Problem..)

- It would be good to have an article on the natural similarities between non-local operators and elliptic operators on graphs.

- Improve the page on Active scalar equations.

- Nonlocal Phase Field Equations this will include many types of evolution equations such as nonlocal reaction-diffusion, models arising from nonlocal ising models, nonlocal cahn-hilliard, nonlocal biological aggregation, sharp interface limits (giacomin-lebowitz, bates-et-al, de massi-pressutti, etc...)

- Boundary Harnack inequality it would be good to possibly include this in the description of what is different/same regarding local vs. nonlocal equations.

- Describe, with references, the connection between holder estimates of viscosity solutions of stationary and or parabolic equations and the existence and uniqueness of solutions to the relevant Martingale Problem

- Processes in the "limit" $\alpha=0$, Zero Order Operators

- Discussion of the Barlow-Bass-Chen-Kassmann Counterexample to Regularity of $L$-Harmonic functions.

- Discussion of Jean Picard's paper (regularity for pure jump processes with non-degenerate, translation invariant generators)

- Add Courrège's Structure theorem for operators and other structure theorems (generators of Feller semigroups, Hunt processes... etc)

- A page on the nonlocal analogue of the thin film equation

## (partially) Completed tasks

- A sort of Starting page that serves as a "root" for all pages we add (ideally, any page we create should be reachable from here). (UPDATE: See discussion.)

- Definition of viscosity solutions for nonlocal equations. Also a discussion on existence using Perron's method and uniqueness through the comparison principle.

- Some general regularity results like holder estimates, Harnack inequalities, Aleksandrov-Bakelman-Pucci estimates, some reference to free boundary problems.

- A list of regularity results for fully nonlinear integro-differential equations.

- Some discussion on models involving Levy processes and stochastic control.

- A page about semilinear equations.

- There is also plenty of work on Dislocation dynamics that we ought to add later on.

- Phase transitions involving non-local interactions, in particular, pages about Particle Systems, discussing the Giacomin-Lebowitz theory and the Ohta-Kawasaki functional.

- It is convenient to have a mini second order elliptic wiki inside this wiki.

- Having a list of equations may make it easier to navigate the wiki.

- Jump processes via Subordination

## Wiki Maintenance

- Get access to LocalSettings.php (ask Maorong for access maybe?) so we can change the page name of the Main Page and the Starting Page (specially since those are the names shown by a google search)

- Fill up the list of upcoming events such as conferences, workshops, summer schools.

- Nestor has started a Literature on Nonlocal Equations to dump there all papers we want to reference or are already referencing on the wiki.