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Orientadores: Paulo Regis Caron Ruffino, Xue-Mei Hairer Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica Resumo: Seja $M$ uma variedade compacta munida de um par de folheações complementares (vertical e horizontal). O objetivo desta tese é estudar decomposições de fluxos de difeomorfismos em um contexto de baixa regularidade. Provamos que dado um semimartingale $Z_t$ (o qual pode ter infinitos saltos em intervalos compactos), então, até um tempo de parada $\tau$, um fluxo de difeomorfismo em $M$ dirigido por $Z_t$ pode ser decomposto em um processo no grupo de Lie de difeomorfismos cujas trajetórias caminham ao longo das folhas horizontais composto com um processo no grupo de difeomorfismos cujas trajetórias caminham ao longo das folhas verticais. Equações para estes processos são determinadas. Os processos estocásticos com componentes de saltos são gerados por equações de Marcus (como em Kurtz, Pardoux and Protter, Annal. I.H.P., section B, 31 (1995)). Generalizamos ainda mais este contexto geométrico para quaisquer tipo de semimartingales. Mostramos também que esta decomposição também funciona para soluções de equações diferenciais de Young e exploramos alguns aspectos geométricos da integral de Young. No contexto de saltos, nossa técnica é baseada em uma extensão da fórmula de Itô-Ventzel-Kunita para processos com saltos. No contexto de integrais de Young, fazemos uma aplicação de uma fórmula de Itô-Ventzel-Kunita para caminhos $\alpha$-H{\"o}lder Contínuos proposta por Castrequini e Catuogno (Chaos Solitons Fractals, 2022). Algumas obstruções geométricas e topológicas para decomposições também são consideradas Abstract: Let $M$ be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical. This thesis aims to study a decomposition of flows of diffeomorphisms in the low regularity context. Namely, we prove that given a general semimartingale $Z_t$ (which can have an infinity number of jumps in compact intervals) up to a stopping time $\tau$, a stochastic flow of local diffeomorphisms in $M$ driven by $Z_t$ can be decomposed into a process in the Lie group of diffeomorphisms which trajectories remain along the horizontal leaves composed with a process in the Lie group of diffeomorphisms which trajectories remain along the vertical leaves. SDEs of these processes are shown. The stochastic flows with jumps are generated by the classical Marcus equation (as in Kurtz, Pardoux and Protter, Annal. I.H.P., section B, 31 (1995)). We enlarge the scope of this geometric decomposition and consider flows driven by arbitrary semimartingales with jumps. We show that this decomposition also holds for solutions of Young differential equations exploring the geometry of Young integrals. In the jump context, our technique is based on our extension of the Itô-Ventzel-Kunita formula for stochastic flows, which may jump infinitely many times. In the Young integral context, we apply a Young Itô-Kunita formula for $\alpha$-H{\"o}lder paths proved by Castrequini and Catuogno (Chaos Solitons Fractals, 2022). Geometrical and other topological obstructions for the decomposition are also considered, e.g., sufficient conditions for the existence of global decomposition for all $t\geq 0$ Doutorado Matemática Doutor em Matemática CAPES 001

In this thesis we propose to give some applications and generalizations of the theory of controlled rough paths, which can be summarized under three headings :- Getting a "good" notions of rough sheet which allows the construction of two dimensional integral driven by an irregular noise andobtain a change of variable formula for the fractional Brownian sheet or more generally a Gaussian sheet.- Construction of local and global solution for a large class of dispersive equations with an irregular modulation in the dispersion term using the nonlinear Young integral and the notion of controlled path.- Rigourous Interpretation of the stochastic quantization equation in the 3 dimensional torus and the construction of a local solution for this equation using the notion of controlled distribution.; Dans cette thèse nous nous proposons de donner certaines applications et généralisations de la théorie des chemins rugueux contrôlé, qu’on peut résumer en trois thèmes : - Obtention d’une « bonne » notions de draps rugueux ce qui permet la construction d’une intégrale plane diriger par des bruit très irrégulier et obtenir une formule de changement de variable pour un drap Brownien fractionnaire ou plus généralement un drap Gaussien.- Construction de solution local et global pour une large classe d’équations aux dérivée partielle dispersive présentant des modulation irrégulière en utilisant l’intégrale de Young non linéaire et la notion de chemin contrôlé. - Interprétation rigoureuse de l’équation de quantisation stochastique en dimension 3 et la construction d’une solution local pour cette dernière en utilisant le notion de distribution contrôlé.

In this thesis we propose to give some applications and generalizations of the theory of controlled rough paths, which can be summarized under three headings :- Getting a "good" notions of rough sheet which allows the construction of two dimensional integral driven by an irregular noise andobtain a change of variable formula for the fractional Brownian sheet or more generally a Gaussian sheet.- Construction of local and global solution for a large class of dispersive equations with an irregular modulation in the dispersion term using the nonlinear Young integral and the notion of controlled path.- Rigourous Interpretation of the stochastic quantization equation in the 3 dimensional torus and the construction of a local solution for this equation using the notion of controlled distribution.; Dans cette thèse nous nous proposons de donner certaines applications et généralisations de la théorie des chemins rugueux contrôlé, qu’on peut résumer en trois thèmes : - Obtention d’une « bonne » notions de draps rugueux ce qui permet la construction d’une intégrale plane diriger par des bruit très irrégulier et obtenir une formule de changement de variable pour un drap Brownien fractionnaire ou plus généralement un drap Gaussien.- Construction de solution local et global pour une large classe d’équations aux dérivée partielle dispersive présentant des modulation irrégulière en utilisant l’intégrale de Young non linéaire et la notion de chemin contrôlé. - Interprétation rigoureuse de l’équation de quantisation stochastique en dimension 3 et la construction d’une solution local pour cette dernière en utilisant le notion de distribution contrôlé.

In this thesis we propose to give some applications and generalizations of the theory of controlled rough paths, which can be summarized under three headings :- Getting a "good" notions of rough sheet which allows the construction of two dimensional integral driven by an irregular noise andobtain a change of variable formula for the fractional Brownian sheet or more generally a Gaussian sheet.- Construction of local and global solution for a large class of dispersive equations with an irregular modulation in the dispersion term using the nonlinear Young integral and the notion of controlled path.- Rigourous Interpretation of the stochastic quantization equation in the 3 dimensional torus and the construction of a local solution for this equation using the notion of controlled distribution.