标题： 非交换变量中的粘性解 显示英文标题

标题： Viscosity Solutions in Non-commutative Variables

摘要： 受平均场博弈与随机矩阵理论之间类比的启发，我们发展了在非交换变量设定下的随机最优控制问题和哈密顿-雅可比方程的粘性解。 而不是实向量，方程的输入是来自迹 von Neumann 代数的自伴算子元组。 平均场博弈中的个体噪声被自由半圆形布朗运动所取代，该过程描述了自伴矩阵空间上布朗运动的大型$n$极限。 我们同样将平均场博弈中的经典共同噪声引入非交换设定中，使这些问题能够结合经典和非交换的随机性。 显示英文摘要

摘要： Motivated by parallels between mean field games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equations in the setting of non-commutative variables. Rather than real vectors, the inputs to the equation are tuples of self-adjoint operators from a tracial von Neumann algebra. The individual noise from mean field games is replaced by a free semi-circular Brownian motion, which describes the large-$n$ limit of Brownian motion on the space of self-adjoint matrices. We introduce a classical common noise from mean field games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness.