Convexity properties of discrete Schrödinger evolutions

In this paper, we give log-convexity properties for solutions to discrete Schrödinger equations with different discrete versions of Gaussian decay at two different times. For free evolutions, we use complex analysis arguments to derive these properties, while in a perturbative setting we use a preliminar log-convexity statement in order to conclude our result.The author was supported by the projects PGC2018-094528-B-I00 funded by (AEI/FEDER, UE) and acronym “IHAIP,” ERCEA Advanced Grant 669689-HADE, and the Basque Government IT641-1

Discrete Carleman estimates and three balls inequalities

We prove logarithmic convexity estimates and three balls inequalities for discrete magnetic Schrödinger operators. These quantitatively connect the discrete setting in which the unique continuation property fails and the continuum setting in which the unique continuation property is known to hold under suitable regularity assumptions. As a key auxiliary result which might be of independent interest we present a Carleman estimate for these discrete operators.Acknowledgements: The first author is supported by ERCEA Advanced Grant 2014 669689—HADE, by the project PGC2018-094528-B-I00 (AEI/FEDER, UE) and acronym “IHAIP”, and by the Basque Government through the project IT1247-19. The second author is supported by the Basque Government through the BERC 2018-2021 program, by the Spanish Ministry of Science, Innovation, and Universities MICINNU: BCAM Severo Ochoa excellence accreditation SEV-2017-2018 and through project PID2020-113156GB-I00. She also acknowledges the RyC project RYC2018-025477-I and IKERBASQUE. The fourth author is supported by the Spanish research project PGC2018-094522-B-I00 from the MICINNU and by the project VP42 “Ecuaciones de evolución no lineales y no locales y aplicaciones” from the Cons. de Univ., Igualdad, Cultura y Deporte, Cantabria, Spain. The authors would like to thank Sylvain Ervedoza for pointing out the optimal scaling in τh ≤ δ0 in the Carleman inequality

Hardy’s uncertainty principle and unique continuation property for stochastic heat equations

The goal of this paper is to prove a qualitative unique continuation property at two points in time for a stochastic heat equation with a randomly perturbed potential, which can be considered as a variant of Hardy’s uncertainty principle for stochastic heat evolutions