Min-max formulas for nonlocal elliptic operators

Guillen, Nestor; Schwab, Russell W.

doi:10.48550/arxiv.1606.08417

Cited by 7 publications

(28 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will review the proof of this result in the context of Euclidean space, where many of the arguments simplify greatly. Moreover, we prove two refinements of the main result from [29] relevant to the Euclidean case, one involving translation invariant operators and one for operators that behave continuously with respect to translation operators. Stated informally, our results are the following: Theorem 1.…”

Section: Introductionmentioning

confidence: 82%

See 1 more Smart Citation

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

Guillen

Schwab

2020

Nonlinear Analysis

Self Cite

View full text Add to dashboard Cite

An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal and nonlinear ones. In previous work, the authors considered such operators in Riemannian manifolds and proved they can be represented by a min-max formula in terms of Lévy operators. In this note we revisit this theory in the context of Euclidean space. With the intricacies of the general Riemannian setting gone, the ideas behind the original proof of the min-max representation become clearer. Moreover, we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity.Date: Monday 18th February, 2019 (This is the revised version per referee suggestions.) 2010 Mathematics Subject Classification. 35J99, 35R09, 45K05, 46T99, 47G20, 49L25, 49N70, 60J75, 93E20 .

show abstract

Section: Introductionmentioning

confidence: 82%

“…In our previous work, [29], we showed such a min-max representation in (1.3). The result in [29] in fact dealt with a more general situation where I :…”

Section: Introductionmentioning

confidence: 82%

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

Guillen

Schwab

2020

Nonlinear Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…Equations of the form (1.1) arise in stochastic optimal control and stochastic differential games where the operators are the generators of pure jump processes. In a work by one of the authors and Schwab [14] it is proved that (roughly speaking) that the class of operators given by a min-max as in (1.1) is the same as the class of operators satisfying the global comparison property.…”

Section: Introductionmentioning

confidence: 99%

Coupling Lévy measures and comparison principles for viscosity solutions

Guillen

Mou

Święch

2019

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The other motivation of studying regularity for the integro-differential operator I in (1.1) comes from the generality of the operator. Indeed it has been proved that if I maps C 2 functions to C 0 functions and moreover satisfies the degenerate ellipticity assumption then I should have the form in (1.1), see [9,19].…”

Section: Introductionmentioning

confidence: 99%