Hopf's lemma for viscosity solutions to a class of non-local equations with applications

We consider a large family of non-local equations featuring Markov generators of L\'evy processes, and establish a non-local Hopf's lemma and a variety of maximum principles for viscosity solutions. We then apply these results to study the principal eigenvalue problems, radial symmetry of the positive solutions, and the overdetermined non-local torsion equation.Comment: 16 page

Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of L\'{e}vy processes

We introduce a class of L\'{e}vy processes subject to specific regularity conditions, and consider their Feynman-Kac semigroups given under a Kato-class potential. Using new techniques, first we analyze the rate of decay of eigenfunctions at infinity. We prove bounds on $\lambda$-subaveraging functions, from which we derive two-sided sharp pointwise estimates on the ground state, and obtain upper bounds on all other eigenfunctions. Next, by using these results, we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions. We establish the relationships of these properties, derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the L\'{e}vy density and the potential at infinity, define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations. These results are amply illustrated by key examples.Comment: Published at http://dx.doi.org/10.1214/13-AOP897 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Universal Constraints on the Location of Extrema of Eigenfunctions of Non-Local Schr\"odinger Operators

We derive a lower bound on the location of global extrema of eigenfunctions for a large class of non-local Schr\"odinger operators in convex domains under Dirichlet exterior conditions, featuring the symbol of the kinetic term, the strength of the potential, and the corresponding eigenvalue, and involving a new universal constant. We show a number of probabilistic and spectral geometric implications, and derive a Faber-Krahn type inequality for non-local operators. Our study also extends to potentials with compact support, and we establish bounds on the location of extrema relative to the boundary edge of the support or level sets around minima of the potential.Comment: 30 pages, To appear in Jour. Diff. Equation, 201

Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schr\"odinger equations with exterior conditions

We consider Dirichlet exterior value problems related to a class of non-local Schr\"odinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove elliptic and parabolic Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain existence and uniqueness of weak solutions. Next we prove a refined maximum principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also, we prove a weak anti-maximum principle in the sense of Cl\'ement-Peletier, valid on compact subsets of the domain, and a full anti-maximum principle by restricting to fractional Schr\"odinger operators. Furthermore, we show a maximum principle for narrow domains, and a refined elliptic ABP-type estimate. Finally, we obtain Liouville-type theorems for harmonic solutions and for a class of semi-linear equations. Our approach is probabilistic, making use of the properties of subordinate Brownian motion.Comment: 35 pages, Liouville-type theorems for semi-linear equations adde

Fall-off of eigenfunctions for non-local Schr\"odinger operators with decaying potentials

We study the spatial decay of eigenfunctions of non-local Schr\"odinger operators whose kinetic terms are generators of symmetric jump-paring L\'evy processes with Kato-class potentials decaying at infinity. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps. We find that the decay rates of eigenfunctions depend on the process via specific preference rates in particular jump scenarios, and depend on the potential through the distance of the corresponding eigenvalue from the edge of the continuous spectrum. We prove that the conditions of the jump-paring class imply that for all eigenvalues the corresponding positive eigenfunctions decay at most as rapidly as the L\'evy intensity. This condition is sharp in the sense that if the jump-paring property fails to hold, then eigenfunction decay becomes slower than the decay of the L\'evy intensity. We furthermore prove that under reasonable conditions the L\'evy intensity also governs the upper bounds of eigenfunctions, and ground states are comparable with it, i.e., two-sided bounds hold. As an interesting consequence, we identify a sharp regime change in the decay of eigenfunctions as the L\'evy intensity is varied from sub-exponential to exponential order, and dependent on the location of the eigenvalue, in the sense that through the transition L\'evy intensity-driven decay becomes slower than the rate of decay of L\'evy intensity. Our approach is based on path integration and probabilistic potential theory techniques, and all results are also illustrated by specific examples.Comment: to appear in Potential Analysis, 40 page

Transition in the decay rates of stationary distributions of L\'evy motion in an energy landscape

The time evolution of random variables with L\'evy statistics has the ability to develop jumps, displaying very different behaviors from continuously fluctuating cases. Such patterns appear in an ever broadening range of examples including random lasers, non-Gaussian kinetics or foraging strategies. The penalizing or reinforcing effect of the environment, however, has been little explored so far. We report a new phenomenon which manifests as a qualitative transition in the spatial decay behavior of the stationary measure of a jump process under an external potential, occurring on a combined change in the characteristics of the process and the lowest eigenvalue resulting from the effect of the potential. This also provides insight into the fundamental question of what is the mechanism of the spatial decay of a ground state

Contractivity and ground state domination properties for non-local Schr\"odinger operators

We study supercontractivity and hypercontractivity of Markov semigroups obtained via ground state transformation of non-local Schr\"odinger operators based on generators of symmetric jump-paring L\'evy processes with Kato-class confining potentials. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps, and the related operators include pseudo-differential operators of interest in mathematical physics. We refine these contractivity properties by the concept of $L^p$-ground state domination and its asymptotic version, and derive sharp necessary and sufficient conditions for their validity in terms of the behaviour of the L\'evy density and the potential at infinity. As a consequence, we obtain for a large subclass of confining potentials that, on the one hand, supercontractivity and ultracontractivity, on the other hand, hypercontractivity and asymptotic ultracontractivity of the transformed semigroup are equivalent properties. This is in stark contrast to classical Schr\"odinger operators, for which all these properties are known to be different.Comment: 15 page

Zero-energy bound state decay for non-local Schr\"odinger operators

We consider solutions of the eigenvalue equation at zero energy for a class of non-local Schr\"odinger operators with potentials decreasing to zero at infinity. Using a path integral approach, we obtain detailed results on the spatial decay of both $L^2$ and resonance solutions at infinity. We highlight the interplay of the kinetic term and the potential in these decay behaviours, and identify the decay mechanisms resulting from specific balances of global lifetimes with or without the potential.Comment: 35 page