Backward stochastic differential equations associated with the vorticity equations

In this paper we derive a non-linear version of the Feynman–Kac formula for the solutions of the vorticity equation in dimension 2 with space periodic boundary conditions. We prove the existence (global in time) and uniqueness for a stochastic terminal value problem associated with the vorticity equation in dimension 2. A particular class of terminal values provide, via these probabilistic methods, solutions for the vorticity equation.     

Generalized stochastic flows and applications to incompressible viscous fluids

We introduce a notion of generalized stochastic flows on manifolds, that extends to the viscous case the one defined by Brenier for perfect fluids. Their kinetic energy extends the classical kinetic energy to Brownian flows, defined as the L²$L^2$ norm of their drift. We prove that there exists a generalized flow which realizes the infimum of the kinetic energy among all generalized flows with prescribed initial and final configuration. We also construct generalized flows with prescribed drift and kinetic energy smaller than the L²$L^2$ norm of the drift.     

The temperature dependent amide I band of crystalline acetanilide

The temperature dependent anomalous peak in the amide I band of crystalline acetanilide is thought to be due to self-trapped states. On the contrary, according to the present model, the anomalous peak comes from the fraction of ACN molecules strongly hydrogen-bonded to a neighboring ACN molecule, and its intensity decreases because, on average, this fraction decreases as temperature increases. This model provides, for the first time, an integrated and theoretically consistent view of the temperature dependence of the full amide I band and a qualitative explanation of some of the features of nonlinear pump–probe experiments.     

On a Non-Periodic Modified Euler Equation: Well-Posedness and Quasi-Invariant Measures

Potential Analysis - We consider a modified Euler equation on $\mathbb {R}^{2}$ . We prove existence of weak global solutions for bounded (and fast decreasing at infinity) initial conditions and...     

Influence of the nonlinearity and dipole strength on the amide I band of protein alpha-helices

Vibrational energy storage and propagation are simulated in a fully atomic model of an alpha-helix by combining the AMBER force field for proteins with an extended version of the Davydov/Scott model for amide I vibrational transfer [A. Scott, Phys. Rep. 217, 1 (1992)]. Dipole-dipole interactions between transition dipole moments of amide I and its on-site energies are calculated from the corresponding three-dimensional atomic positions. The comparison of the theoretically calculated absorption line shapes with the experimentally measured ones leads to a putative value of the nonlinearity parameter of -30 pN.     

Navier–Stokes equations and forward–backward SDEs on the group of diffeomorphisms of a torus

We establish a connection between the strong solution to the spatially periodic Navier–Stokes equations and a solution to a system of forward–backward stochastic differential equations (FBSDEs) on the group of volume-preserving diffeomorphisms of a flat torus. We construct representations of the strong solution to the Navier–Stokes equations in terms of diffusion processes.     

Unicité de solutions d'équations différentielles sur l'espace de Wiener

Unicity of the flow associated to a good vector field on the Wiener space.     

Non-existence of infinitesimally invariant measures on loop groups

Let G be a compact Lie group, L(G) the associated loop group, ω the canonical symplectic form on L(G). Set H the Hamiltonian function for which the associated ω-Hamiltonian vector field is the infinitesimal rotation. Then H generates a canonical semi-definite Riemannian structure on L(G), which induces a Riemannian structure on the free loop groupL(G)/G=L₀(G). This metric corresponds to the Sobolev norm H¹. Using orthonormal frame methodology the positivity and finiteness of the Ricci curvature of L₀(G) is proved. By studying the dissipation towards high modes of a unitary group valued SDE it is proved that the loop group does not have any infinitesimally invariant measure.