This thesis focuses on a certain type of nonlinear parabolic partial differential equations, i.e. PME and FDE. Chapter 1 consists of a survey on results related to PME and FDE, and a short review on some works about deriving gradient estimates in probabilistic ways. In Chapter 2 we estimate gradient on space variables of solutions to the heat equation on Euclidean space. The main idea is to construct two semimartingales by letting the solution and its gradient running backward on the path space of a diffusion process. Estimates derived from decompositions of those two semimartingales are then combined to give rise to an upper bound on gradient that only involves the maximum of the initial data and time variable. In particular, it is independent of the dimension. In Chapter 3 we carry the idea in Chapter 2 onto the study of positive solutions to PME or FDE, and obtained a similar type of bound on |∇u| for local solutions to PME or FDE on Euclidean space. In existing literature there have always been constraints on m. By considering a more general form of transformation on u and introducing a family of equivalent measures on path space, we add more flexibility to our method. Thus our result is valid for a larger range of m. For global solutions, when m violates our constraint, we need two-sided bound on u to control |∇u|. In Chapter 4 we utilize maximum principle to derive Li-Yau type gradient estimate for PME on a compact Riemannian manifold with Ricci curvature bounded from below. Our result is able to yield a Harnack inequality possessing the right order in time variable when the lower bound of Ricci curvature is negative.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第19468号 / 理博第4128号 / 新制||理||1594(附属図書館) / 32504 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 矢野 孝次, 教授 上田 哲生, 教授 重川 一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Cette thèse traite de plusieurs sujets d'analyse dans les espaces métriques mesurés, en lien avec le transport optimal et des conditions de courbure-dimension. Nous considérons en particulier les équations de continuité dans ces espaces, du point de vue de fonctionnelles continues sur les espaces de Sobolev, et du point de vue de la dualité avec les courbes absolument continues dans l'espace de Wasserstein. Sous une condition de courbure-dimension, mais sans condition de doublement de mesure ou d'inégalité de Poincaré, nous montrons également l'identification des p-gradients faibles. Nous étudions ensuite les espaces de Sobolev sur le produit tordu de l'ensemble des réels et d'un espace métrique mesuré. En particulier, nous montrons la propriété Sobolev-à-Lipschitz sous une certaine condition de courbure-dimension. Enfin, sous une telle condition et dans le cadre d'une théorie non-lisse de Bakry-Emery, nous obtenons une inégalité améliorée de Bochner et proposons une définition du N-tenseur de Ricci. / This thesis concerns in some topics on calculus in metric measure spaces, in connection with optimal transport theory and curvature-dimension conditions. We study the continuity equations on metric measure spaces, in the viewpoint of continuous functionals on Sobolev spaces, and in the viewpoint of the duality with respect to absolutely continuous curves in the Wasserstein space. We study the Sobolev spaces of warped products of a real line and a metric measure space. We prove the 'Pythagoras theorem' for both cartesian products and warped products, and prove Sobolev-to-Lipschitz property for warped products under a certain curvature-dimension condition. We also prove the identification of p-weak gradients under curvature-dimension condition, without the doubling condition or local Poincaré inequality. At last, using the non-smooth Bakry-Emery theory on metric measure spaces, we obtain a Bochner inequality and propose a definition of N-Ricci tensor.