Min–max formulas for nonlocal elliptic operators

Abstract

In this work, we give a characterization of Lipschitz operators on spaces of C 2 (M ) functions (also C 1,1 , C 1,γ , C 1 , C γ ) that obey the global comparison property-i.e. those that preserve the global ordering of input functions at any points where their graphs may touch, often called "elliptic" operators. Here M is a complete Riemannian manifold. In particular, we show that all such operators can be written as a min-max over linear operators that are a combination of drift-diffusion and integro-differential parts. In the linear (and nonlocal) case, these operators had been characterized in the 1960's, and in the local, but nonlinear case-e.g. local Hamilton-Jacobi-Bellman operators-this characterization has also been known and used since approximately since 1960's or 1970s. Our main theorem contains both of these results as special cases. It also shows any nonlinear scalar elliptic equation can be represented as an Isaacs equation for an appropriate differential game. Our approach is to "project" the operator to one acting on functions on large finite graphs that approximate the manifold, use non-smooth analysis to derive a min-max formula on this finite dimensional level, and then pass to the limit in order to lift the formula to the original operator. This is the Director's cut, and it contains extra details for our own sanity.

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

“…Remark 1.7. We note that one subtle improvement of the current work upon our previous one in [29] is that because of a more streamlined proof for the translation invariant case, we were able to establish the non-translation invariant case, Theorem 1.9 (below), without the technical Assumption 1.3. This is purely an artifact of using an approximation scheme in [29] to treat all operators by the same method, and this turns out to have been not essential when one does not want the extra information provided by Theorems 1.11 and 1.14.…”

“…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.…”

“…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 :…”

“…Then the Lévy operators appearing in the min-max formula have continuous coefficients with common modulus of continuity of the form Cω(2(·)). Theorem 1 above is a special case of the main result in [29], and Theorems 2 and 3 are new.…”

“…First of all, the link between elliptic equations and a min-max formula for operators has a long history, and it has been exploited extensively in the case of local operators. Until [29], the connection was not known for nonlocal, nonlinear operators. Even so, the link between the two was natural enough that there are at least a few results that assumed a structure like (1.3), including [5], [35], [40], [47], [48], [51], among many others.…”