Partial differential equations with oscillatory coefficients arise in many applications, such as the modeling of composite materials and flame propagations. The fine scale oscillations of the physical media are often unknown and, hence, modeled as random. The theory of stochastic homogenization amounts to exploring the self-averaging mechanism of the PDEs, and it leads to mean field approximations of large scale behaviors of the solution. The theory of random fluctuations aims at studying the difference between the solution and its homogenization limit, and it leads to higher order, and very often Gaussian, corrections to the approximation. In this talk, I will present some results on stochastic homogenization, fluctuation theory for elliptic equations with random potentials and some related numerical analysis and applications in inverse problems.

Wenjia Jing is an assistant professor at the Yau Mathematical Sciences Center of Tsinghua University. He works on partial differential equations with random coefficients, especially homogenization theory, waves in random media and their applications in inverse problems. Wenjia Jing received his B.S. degree from Peking University in 2006 and his PhD degree from Columbia University in 2011. From 2011 to 2013, he was a postdoctoral researcher at Ecole Normale Superieure at Paris. From 2013 to 2016, he was a Dickson Instructor of Mathematics at The University of Chicago.