Some remarks about the positivity of random variables on a Gaussian probability space

Let $(W,H,\mu)$ be an abstract Wiener space and $L$ be a probability density of class LlogL. Using the measure transportation of Monge-Kantorovitch, we prove that the kernel of the projection of L on the second Wiener chaos defines an (Hilbert-Schmidt) operator which is lower bounded by another Hilbert-Schmidt operator.Comment: 6 page

Flows driven by Banach space-valued rough paths

We show in this note how the machinery of C^1-approximate flows devised in the work "Flows driven by rough paths", and applied there to reprove and extend most of the results on Banach space-valued rough differential equations driven by a finite dimensional rough path, can be used to deal with rough differential equations driven by an infinite dimensional Banach space-valued weak geometric Holder p-rough paths, for any p>2, giving back Lyons' theory in its full force in a simple way.Comment: 8 page

Nonlinear Young integrals via fractional calculus

For H\"older continuous functions $W(t,x)$ and $\varphi_t$, we define nonlinear integral $\int_a^b W(dt, \varphi_t)$ via fractional calculus. This nonlinear integral arises naturally in the Feynman-Kac formula for stochastic heat equations with random coefficients. We also define iterated nonlinear integrals.Comment: arXiv admin note: substantial text overlap with arXiv:1404.758

Perimeter of sublevel sets in infinite dimensional spaces

We compare the perimeter measure with the Airault-Malliavin surface measure and we prove that all open convex subsets of abstract Wiener spaces have finite perimeter. By an explicit counter-example, we show that in general this is not true for compact convex domains

The Monge problem in Wiener Space

We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure {\gamma}

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces (X,d,m). Our main results are: - A general study of the relations between the Hopf-Lax semigroup and Hamilton-Jacobi equation in metric spaces (X,d). - The equivalence of the heat flow in L^2(X,m) generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures P(X). - The proof of density in energy of Lipschitz functions in the Sobolev space W^{1,2}(X,d,m). - A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott & Villani [30] and Sturm [39,40], and require neither the doubling property nor the validity of the local Poincar\'e inequality.Comment: Minor typos corrected and many small improvements added. Lemma 2.4, Lemma 2.10, Prop. 5.7, Rem. 5.8, Thm. 6.3 added. Rem. 4.7, Prop. 4.8, Prop. 4.15 and Thm 4.16 augmented/reenforced. Proof of Thm. 4.16 and Lemma 9.6 simplified. Thm. 8.6 corrected. A simpler axiomatization of weak gradients, still equivalent to all other ones, has been propose