Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations
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arXiv
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| Format: | Preprint |
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2025
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| author | Chaintron, Louis-Pierre Conforti, Giovanni Eichinger, Katharina |
| author_facet | Chaintron, Louis-Pierre Conforti, Giovanni Eichinger, Katharina |
| contents | A well-known consequence of the Pr{é}kopa-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 Schr{ö}dinger 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations Chaintron, Louis-Pierre Conforti, Giovanni Eichinger, Katharina Analysis of PDEs Functional Analysis Probability A well-known consequence of the Pr{é}kopa-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 Schr{ö}dinger 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. |
| title | Propagation of weak log-concavity along generalised heat flows via Hamilton-Jacobi equations |
| topic | Analysis of PDEs Functional Analysis Probability |
| url | https://arxiv.org/abs/2508.07931 |