Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures. (Featured Article in SIAM Journal on Mathematical Analysis)

Abstract: We will discuss how certain Hamilton-Jacobi-Bellman (HJB) equations in spaces of probability measures can be approximated by finite dimensional equations. The most interesting cases are convergence of viscosity solutions of HJB equations corresponding either to deterministic optimal control problems for systems of $n$ particles or to stochastic optimal control problems for systems of $n$ particles with a common noise, to the viscosity solution of a limiting HJB equation in the space of probability measures. The limiting HJB equation is interpreted in its ``lifted" form in a Hilbert space, which has a unique viscosity solution. When the Hamiltonian is convex in the gradient variable and equations are of first order, it can be proved that the viscosity solutions of the finite dimensional problems converge to the value function of a variational problem in $\mathcal{P}_2(\R^d)$ thus providing a representation formula for the solution of the limiting first order HJB equation. The talk will also contain an overview of existing works and various approaches to partial differential equations in abstract spaces, including spaces of probability measures and Hilbert spaces. The talk is based on a joint work with W. Gangbo and S. Mayorga.

Featured Article Authors: W. Gangbo, S. Mayorga, A. Swiech

analysis of PDEs

Audience: researchers in the topic

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Seminar In the Analysis and Methods of PDE (SIAM PDE)

Series comments: The Editorial Board of SIMA and the officers of the SIAM Activity Group on Analysis of Partial Differential Equations (SIAG/APDE) are launching the Seminar In the Analysis and Methods of PDE (SIAM PDE) to be held on the first Thursday of the month at 11:30am ET, except in August and January. The COVID-19 pandemic and consequent social distancing call for online venues of research dissemination. This webinar will serve as a way to recognize achievements in our area, and it will be expected to help promoting the standing of both SIMA and APDE. Further, these webinars should be beneficial to mathematicians who are in relatively isolated places.

This webinar is intended to continue well beyond this social distancing period. There are two type of SIAM PDE webinars: (i) Colloquium-type webinars by outstanding speakers presenting a survey of results on a topic where critical progress was recently achieved (and not necessarily published in SIMA); (ii) Webinars by outstanding authors based on some of the best articles accepted by SIMA. To launch the webinar series the initial six seminars will be of the Colloquium-type.

The talks are open to the public and, due to security reasons, all attendees have to register by following the link siam.zoom.us/webinar/register/WN_wShvDipBQw6WafNN6QO5wg

The link will contain detailed information about the talks. The registration is quick (asks only for your name and email) and, once registered, you will receive an email with the link to the webinars, which is unique to you, so please do not share that email. The registration is valid for all this webinar series, and you will receive a reminder about future talks.

Further, you will receive a follow-up email up to 7 days after each webinar, with a link to the on-demand SIAM PDE YouTube playlist recording.

Updated information and schedule will be available at sinews.siam.org/Details-Page/seminar-in-the-analysis-and-methods-of-pde

The SIAM PDE Webinar was initiated on June 2020:

By the SIMA Editors:

Robert Lipton, SIMA Editor-in-Chief (Louisiana State University, USA)

Athanasios Tzavaras (KAUST, Saudi Arabia)

By the SIAG/APDE Officers:

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