Research Projects

The primary objective of the proposal is to address some fundamental question on Mean field Game (MFG) with nonlocal diffusion. MFG is a coupled system of nonlinear equation which describes a class of differential game problems with very large players, having wide range of applications in various fields including crowd dynamics, biology, economics, finance and social sciences. Despite of having diverse applications, the mathematical study of MFG problems are relatively new, only after the model equation introduced by J. M. Lasry and P. L. Lions in 2006. Since then, the topic received a major attention and the research progressed rapidly, but most of the results are only with local diffusions. Nonlocal operators arise naturally in applications to model long range interaction phenomenon. We aim to provide a comprehensive study on MFG problem involving nonlocal/fractional order diffusion generating from pure jump L\'evy processes. Nonlocal MFG model with congestion and models on bounded domain is couple of such important direction with wide practical relevance, but analytical and numerical study for such cases are missing. Building on the research experience of the PI, the plan is to address 4 to 5 primary objectives - wellposedness of nonlocal MFG with congestion, wellposedness of nonlocal MFG in bounded domain with different boundary condition, numerical convergence analysis of such problems. Another important part of the project is to study the numerical approximation of MFG with fully nonlinear local and nonlocal diffusion, generated by controlled L\'evy diffusion. In two recent articles, PI jointly with E. Jakobsen and M. Krupski established the wellposedness theory for such problem. The finite difference approximation and convergence analysis will be carried out by considering difference-quadrature schemes. We further implement the scheme for fractional Laplace operators with polynomial Hamiltonian growth, where powers of discrete Laplacian will be considered as the monotone discretization. In summary, although the objectives are related, addressing each objective require different techniques - nonlocal generalization of some existing techniques and new ideas specific to counter long jump interaction along with small jump diffusion as seen in PI's earlier research works. The expertise in research with nonlocal operators for various problems, along with recent experience in some class of nonlocal MFG (refer to the technical document section 5), PI is confident to complete these aforementioned directions successfully. The results will be of interest to the leading researchers around the globe, both academically and in terms of application. The filed being new and emerging, it will also be beneficial for PhD students related to this project to progress further in their respective career.