In this talk, we shall use the second-order differential geometry, developed originally by L. Schwartz and P.-A. Meyer, to construct second-order counterparts of classical structures like jets, symplectic and contact structures. These will allow us to establish stochastic Lagrangian and Hamiltonian mechanics and their key relations with second-order Hamilton-Jacobi-Bellman (HJB) equations, as well as a stochastic Noether's theorem. We will also apply the framework to describe relations between Schr\"odinger's variational problem and Onsager's approach to nonequilibrium statistical mechanics. Especially, we will rebuild Onsager's reciprocal relations by introducing Riemannian structures on thermodynamic spaces, and propose a new notion of entropy for nonequilibrium systems. This talk is based on joint works with Jean-Claude Zambrini.

In this talk I will discuss a family of Gibbsian measures on the set of Laguerre tessellations. These measures may be used to provide a systematic approach for constructing Gibbsian solutions to Hamilton-Jacobi PDEs by exploring the Eularian description of the shock dynamics. Such solutions depend on kernels satisfying kinetic-like equations reminiscent of the Smoluchowski model for coagulating and fragmenting particles.

Optimal control problem is typically solved by first finding the value function through Hamilton-Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control.

In this work, we propose a new formulation for the gradient of the value function (value―gradient) as a decoupled system of partial differential equations in the context of continuous―time deterministic discounted optimal control problem. We develop an efficient iterative scheme for this system of equations in parallel by utilizing the properties that they share the same characteristic curves as the HJE for the value function. Experimental results demonstrate that this new method not only significantly increases the accuracy but also improves the efficiency and robustness of the numerical estimates. This example will highlight the importance of unifying Eulerian and Lagrangian viewpoints for designing numerical schemes for high dimensional equation in computational math. This talk is mainly based on the joint work published at https://epubs.siam.org/doi/10.1137/21M1442838.

In this talk, I will briefly go over a recent line of research on spatial statistics of random field solutions to stochastic heat wave equations. This line was initiated in Lawrence by David Nualart and his former PhD student and collaborators in 2018. Since then, we have seen a growing literature (spatial ergodicity, central limit theorems, etc) for the above SPDEs driven by Gaussian noises.

After the brief overview, I will present recent works with Balan, and with Balan-Xia on (almost sure) central limit theorems for 1D stochastic wave equation driven by Lévy noise (in a finite-variance setting).