Lévy-Type Dirichlet Problems on the Half-Line: Probabilistic Mild Solutions and Weighted Energy Estimates

This paper studies Dirichlet problems for one-dimensional Lévy-type nonlocal elliptic equations on the half-line. The equation Lμν(x)=f(x),x>0,ν(x)=0,x≤0 is transformed into a weighted nonlocal equation associated with a multiplicative jump process. Under basic structural assumptions on the Lévy measure, the transformed generator is realized through a martingale problem, and the associated exponential killing representation gives a probabilistic mild solution with an immediate L∞-estimate. For the one-dimensional fractional Laplacian, the transformed process is exactly multiplicative. This yields a new approach in which solution estimates are derived from the stochastic equation of the transformed process; smooth-data resolvent solutions are estimated in weighted Lp-spaces and extended to general data by approximation. For more general Lévy measures, a smooth weighted energy estimate is proved. The key analytic input is a weighted adjoint integral inequality for the transformed generator, verified for subordinate Brownian motions associated with Bernstein functions and for non-unimodal logarithmically perturbed stable-type operators.