2025/01/23 by Max Engelstein, Engelstein, Max, Ignasi Guillén-Mola +1
Economics, Econometrics and Finance · Decision Sciences · Engineering · #Stochastic processes and financial applications #Probability and Risk Models #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2501.13869
In this note we show that the support of a locally k-uniform measure in \mathbb Rn+1 satisfies a kind of unique continuation property. As a consequence, we show that locally uniformly distributed measures satisfy a weaker unique continuation property. This continues work of Kirchheim and Preiss (Math. Scand. 2002) and David, Kenig and Toro (Comm. Pure Appl. Math. 2001) and lends additional evidence to the conjecture proposed by Kowalski and Preiss (J. Reine Angew. Math. 1987) that each connected component of the support of a locally n-uniform measure in \mathbb Rn+1 is contained in the zero set of a quadratic polynomial.