2025/08/11 by Louis-Pierre Chaintron, Chaintron, Louis-Pierre, Giovanni Conforti +3 · 5 citations
Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Biology Tumor Growth #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2508.07931
openalex publication_date 2025/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A well-known consequence of the Prékopa-Leindler inequality is the preservation of logconcavity by the heat semigroup. Unfortunately, this property does not hold for more general semigroups. In this paper, we exhibit a slightly weaker notion of log-concavity that can be propagated along generalised heat semigroups. As a consequence, we obtain logsemiconcavity properties for the ground state of Schrödinger operators for non-convex potentials, as well as propagation of functional inequalities along generalised heat flows. We then investigate the preservation of weak log-concavity by conditioning and marginalisation, following the seminal works of Brascamp and Lieb. To our knowledge, our results are the first of this type in non log-concave settings. We eventually study generation of log-concavity by parabolic regularisation and prove novel two-sided log-Hessian estimates for the fundamental solution of parabolic equations with unbounded coefficients, which can be made uniform in time. These properties are obtained as a consequence of new propagation of weak convexity results for quadratic Hamilton-Jacobi-Bellman (HJB) equations. The proofs rely on a stochastic control interpretation combined with a second order analysis of reflection coupling along HJB characteristics.