Geometric alpha-stable processes are Levy processes whose characteristic exponents are logarithmic transforms of stable exponents. Their transition densities — the probability of moving from one point to another in a given time — are needed for applications to Schrodinger operators but have not been established by standard methods. The characteristic function is not L^1-integrable, which blocks the usual Fourier-inversion proof.
The proof uses self-decomposability instead (arXiv:2603.11570). A self-decomposable distribution can be written as a scaled copy of itself plus an independent remainder — a structural property of the Levy measure that implies smoothness of the distribution without requiring integrability of the characteristic function. Geometric alpha-stable processes are self-decomposable, so their distributions have densities, and the transition kernel inherits this smoothness.
The method is purely probabilistic. The analytic approach asks: can we invert the Fourier transform? The probabilistic approach asks: does the structural decomposition of the process force regularity? The answers are the same but the routes are different, and the probabilistic route works where the analytic one doesn't.
The application: Schrodinger operators associated with recurrent geometric stable processes have ground states. The transition density existence is the key input — once you know the process has smooth transitions, the spectral theory of the associated operator follows. The ground state existence was blocked not by any difficulty in the operator theory but by the absence of the transition density that feeds it.
The obstruction was in the foundation, not the structure. The building was ready; the ground wasn't proved to exist.