Forward-backward SDEs with distributional coefficients

Forward-backward stochastic differential equations (FBSDEs) have attracted significant attention since they were introduced almost 30 years ago, due to their wide range of applications, from solving non-linear PDEs to pricing American-type options. Here, we consider two new classes of multidimensional FBSDEs with distributional coefficients (elements of a Sobolev space with negative order). We introduce a suitable notion of a solution, show existence and uniqueness of a strong solution of the first FBSDE, and weak existence for the second. We establish a link with PDE theory via a nonlinear Feynman-Kac representation formula. The associated semi-linear second order parabolic PDE is the same for both FBSDEs, also involves distributional coefficients and has not previously been investigated; our analysis uses mild solutions, Sobolev spaces and semigroup theory.Comment: 40 pages, no figures - new improved version with shorter proof of Thm 18, extended results in Thm 25 and Thm 27. Other minor clarifications adde

Fast synchrotron X-ray tomographic quantification of dendrite evolution during the solidification of Mg-Sn alloys

The evolution of dendritic microstructures during the solidification of a Mg-15 wt%Sn alloy was investigated in situ via fast synchrotron X-ray microtomography. To enable these in situ observations a novel encapsulation method was developed and integrated into a fast, pink beam, imaging beamline at Diamond Light Source. The dendritic growth was quantified with time using: solid volume fraction, tip velocity, interface specific surface area, and surface curvature. The influence of cooling rate upon these quantities and primary phase nucleation was investigated. The primary dendrites grew with an 18-branch, 6-fold symmetry structure, accompanied by coarsening. The coarsening process was assessed by the specific surface area and was compared with the existing models. These results provide the first quantification of dendritic growth during the solidification of Mg alloys, confirming existing analytic models and providing experimental data to inform and validate more complex numeric models