哈工大武大随机分析与算法讨论班

In this talk, I present some recent progresses on optimal control of stochastic distributed parameter systems, including Pontryagin-type maximum principle, dynamic programming and linear quadratic optimal control problem.

McKean-Vlasov SDEs arise in many applications, including mathematical biology, social sciences, and machine learning. In this talk, wewill discuss parameter estimation for a cKean-Vlasov SDE and theassociated system of weakly interacting particles. We first analyse theasymptotic properties of the offline maximum likelihood estimator. Wethen propose a new online estimator, which evolves according to acontinuous-time stochastic gradient descent algorithm on the asymptoticog-likelihood of the interacting particle system. We obtain variousconvergence results for this estimator, under assumptions which guaranteeergodicity and uniform-in-time propagation chaos. Our theoretical resultsare supported via several numerical examples, including a toy linear meanfield model a stochastic Kuramoto model. and a stochastic opiniondynamics model.

The Dirichlet problem with oscillating coefficients requires a lot ofcalculation resources in general. We present a new algorithm invented by SArmstrong, A. Hannukainen, T. Kuusi, J-C. Mourrat to solve this problemquickly in the context of random conductance. This result makes use of thequantitative homogenization theory and the algorithm can also be generalized tothe percolation cluster setting.

In this talk, I will describe the emergence of the 4/3-exponent in two seem-ingly unrelated models: random distance of Liouville quantum gravity and correlation length for the two-dimensional random field Ising model. I will then explain that such 4/3-exponent, while being unexpected among respective communities even from a physics perspective, has in fact been hinted in Leighton-Shor(1989) and Talagrand (2014) where the 4/3-exponent emerges in a random matching problem Finally, I will present the heuristic computation which leads tothe emergence of the 4/3-exponent. Based on a joint work with Subhajit Goswami and a joint work with Mateo Wirth.

This paper concerns a long-range random walk in random environment indimension 1 + 1. where the environmental disorder is independent in space butcorrelated in time. We prove that the rescaled partition function converges weakly to the Stratonovich solution of a fractional stochastic heat equation with multi.plicative Gaussian noise which is white in space and colored in time. This is aoint work with Guanglin Rang and Meng Wang.

We view the classical Lindeberg principle in a homogeneous Markov proc-ess setting to establish a probability approximation framework by the associatedIto's formula and Markov operator. As applications, we study the error bounds ofthe following three approximations: multivariate normal approximation, approx1.mating a family of online stochastic gradient descents (SGDs) by a stochasticdifferential equation (SDE) driven by multiplicative Brownian motion. andEuler-Maruyama (EM) discretization for SDEs driven by alpha-stable process.Furthermore, we extend the above approximation framework to the inhomo-geneous case and consider the variable-step E-M approximations of regime-switching jump diffusion processes. This talk is based on the joint works with OM. Shao. Z.G.Su C. Deng and L.Xu .

In the first part of this talk, I will discuss some theoretical studies on deeplearning with a focus on approximation, generalization. In particular, I willcover error analysis with over-parameterization. In the second part, I will delveinto the concept of Gaussian stochastic interpolations and their applications,including the derivation of functional inequality with dimension-free constantsas well as sampling and generative learning.