A nonlocal version of the inverse problem of Donsker and Varadhan.

Abstract:

In their seminal paper of 1976, M.D. Donsker and S.R.S. Varadhan addressed the following “inverse problem”: let consider two linear, second-order, uniformly elliptic operators L1, L2 with the form

Liφ = Div(Ai(x)Dφ) + bi(x) · Dφ, i = 1, 2.

If for every domain Ω and every smooth potential V , the operators L1 + V and L2 + V have the same principal eigenvalue in Ω, then the diffusions are equal (A1 = A2), and either L1φ = L2(uφ)/u for some L2-harmonic function u, or L1φ = L∗2(uφ)/u for some L∗2-harmonic function u.

In this talk we report a nonlocal a version of this problem, where both the diffu- sion and transport terms defining the involved operators have a fractional nature. We prove a similar conjugacy phenomena among operators having the same principal eigenvalues, by means of a minmax characterization for them, and developing new ideas to overcome the difficulties posed by the non locality.

Posted on Sep 3, 2021 in Differential Equations, Seminars