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Theses

Regularization phenomena for stochastic (partial) differential equations via Itô- and pathwise stochastic calculi

Abstract : In this thesis, we study three instances of regularization phenomena for stochastic (partial) differential equations (SPDEs). We first study semilinear SPDEs with unbounded diffusion terms: By deriving a generalization to the maximal inequality for stochastic convolutions we are able to establish existence of strong solutions in the subcritical regime. We moreover use the associated sequence of subcritical solutions to establish existence of a martingale solution in the critical case via the Flandoli-Gatarek compactness method. Secondly, we establish a law of large numbers for interacting particle systems without imposing independence or finite moment assumptions on the initial conditions: Towards this end, we establish a non-closed equation satisfied by the associated empirical measure in a mild sense that differs from the expect limiting McKean-Vlasov PDE only by a certain noise term. In treating said noise term, we employ pathwise rough path bounds and arguments based on Itô-calculus in a complementary fashion that allow to establish the desired law of large numbers.Finally we investigate regularization phenomena through averaging along curves. Based on recent space-time regularity estimates for local times of fractional Brownian motion in one dimension, we study averaged transport equations in passing by their associated regularized characteristics. By employing a fixed point argument on the level of transport equations, we are able to subsequently pass to a Burgers' type equation averaged along paths of fractional Brownian motion. The arguments at each step are conditional on the Hurst parameter satisfying explicitly established conditions.
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Submitted on : Wednesday, December 15, 2021 - 4:01:33 PM
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Florian Bechtold. Regularization phenomena for stochastic (partial) differential equations via Itô- and pathwise stochastic calculi. Probability [math.PR]. Sorbonne Université, 2021. English. ⟨NNT : 2021SORUS160⟩. ⟨tel-03481974⟩

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