A probabilistic approach to a non‐local quadratic form and its connection to the Neumann boundary condition problem

Vondraček, Zoran

doi:10.1002/mana.201900061

Cited by 12 publications

(12 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Neumann problem (1.1) was first introduced in [18,20], and has been subsequently studied in several papers; see for example [1,3,14,30,39]. As explained in detail in [18], (1.1) is a natural Neumann problem for the fractional Laplacian, for several reasons:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

The Neumann problem for the fractional Laplacian: regularity up to the boundary

Audrito¹,

Felipe-Navarro²,

Ros‐Oton³

2020

Preprint

View full text Add to dashboard Cite

We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if u is a weak solution of (−∆) s u = f in Ω, Nsu = 0 in Ω c , then u is C α up tp the boundary for some α > 0. Moreover, in case s > 1 2 , we then show that u ∈ C 2s−1+α (Ω). To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections.Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result.Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…• The problem has a natural probabilistic interpretation, heuristically described in [18], and rigorously studied in [39].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Neumann problem for the fractional Laplacian: regularity up to the boundary

Audrito¹,

Felipe-Navarro²,

Ros‐Oton³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Concluding the Introduction, we like to mention various approaches in literature to define reflection from D c for jump processes and nonlocal operators, because their geometric setting, namely, the essential use of both X(t − ) and X(t), inspired our study of the shot-down processes. Here we refer the reader to [3], [18], [30] and [11]. In particular, the Introduction of [11] gives a general perspective on reflections, Neumann conditions and concatenation of Markov processes.…”

Section: Introductionmentioning

confidence: 99%

Shot-down stable processes

Bogdan¹,

Jastrzȩbski²,

Kaßmann³

et al. 2023

Preprint

View full text Add to dashboard Cite

The shot-down process is a strong Markov process which is annihilated, or shot down, when jumping over or to the complement of a given open subset of a vector space. Due to specific features of the shot-down time, such processes suggest new type of boundary conditions for nonlocal differential equations. In this work we construct the shot-down process for the fractional Laplacian in Euclidean space. For smooth bounded sets D, we study its transition density and characterize Dirichlet form. We show that the corresponding Green function is comparable to that of the fractional Laplacian with Dirichlet conditions on D. However, for nonconvex D, the transition density of the shot-down stable process is incomparable with the Dirichlet heat kernel of the fractional Laplacian for D. Furthermore, Harnack inequality in general fails for harmonic functions of the shot-down process.

show abstract

“…Here X is the Markov process associated with (E , F ). This new approach leads to a rich family of DN operators containing those for self-adjoint operators appearing in the literatures like [8,[35][36][37]. Particularly, the classical DN operator D can be recovered as follows: Consider (E , F ) = ( 1 2 D, H 1 (Ω)) on L 2 ( Ω), where Ω is the closure of Ω and…”

Section: Introductionmentioning

confidence: 99%

Generalizing Dirichlet-to-Neumann operators

Li¹

2022

Preprint

View full text Add to dashboard Cite

The aim of this paper is to study the Dirichlet-to-Neumann operators in the context of Dirichlet forms and especially to figure out their probabilistic counterparts. Regarding irreducible Dirichlet forms, we will show that the Dirichlet-to-Neumann operators for them are associated with the trace Dirichlet forms corresponding to the time changed processes on the boundary. Furthermore, the Dirichlet-to-Neumann operators for perturbations of Dirichlet forms will be also explored. It turns out that for typical cases such a Dirichlet-to-Neumann operator corresponds to a quasi-regular positivity preserving (symmetric) coercive form, so that there exists a family of Markov processes associated with it via Doob's h-transformations. CONTENTS 1. Introduction 2. DN operators for irreducible Dirichlet forms 3. DN operators for perturbations of Dirichlet forms 4. Revisiting classical DN operators 5. DN operators for Schrödinger operators 6. DN operators for perturbations supported on boundary Appendix A. Perturbations of Dirichlet forms Appendix B. Direct sum decomposition for perturbation of Dirichlet forms Appendix C. Irreducibility of lower bounded closed forms Appendix D. Quasi-regular positivity preserving (symmetric) coercive forms Appendix E. Classical Calderón's problem References 2010 Mathematics Subject Classification. Primary 31C25, 60J60.

show abstract

A probabilistic approach to a non‐local quadratic form and its connection to the Neumann boundary condition problem

Cited by 12 publications

References 18 publications

The Neumann problem for the fractional Laplacian: regularity up to the boundary

The Neumann problem for the fractional Laplacian: regularity up to the boundary

Shot-down stable processes

Generalizing Dirichlet-to-Neumann operators

Contact Info

Product

Resources

About