### $\left(2m\right)$-th mean behavior of solutions of stochastic differential equations under parametric perturbations.

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We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals “on-the-fly” as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We...

Let $P(M,G)$ be a principal fiber bundle and $E(M,N,G,P)$ an associated fiber bundle. Our interest is to study the harmonic sections of the projection ${\pi}_{E}$ of $E$ into $M$. Our first purpose is give a characterization of harmonic sections of $M$ into $E$ regarding its equivariant lift. The second purpose is to show a version of a Liouville theorem for harmonic sections of ${\pi}_{E}$.

We study continuous Anderson Hamiltonians with non-degenerate single site probability distribution of bounded support, without any regularity condition on the single site probability distribution. We prove the existence of a strong form of localization at the bottom of the spectrum, which includes Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) with finite multiplicity of eigenvalues, dynamical localization (no spreading of wave packets under the time evolution),...

We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the ``back to coordinates map'' may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for...