Conditional propagation of chaos and a class of quasilinear PDE's (Q1902957)

The author considers the following quasilinear equation in \(\mathbb{R}^d\), \(d \leq 3\), of parabolic type: \[ \begin{cases} b \bigl( x,u (x,t) \bigr) {\partial \over \partial t} u(x,t) & = \sum^d_{i,j = 1} {\partial \over \partial x_i} \left[ a^{ij} \bigl( x,u (x,t) \bigr) {\partial \over \partial x_j} u(x,t) \right], \\ u(x,0) & = u_0(x). \end{cases} \tag{1} \] After some transformations leading to associated equations, the conditional propagation of chaos is used to solve the above problem. The author introduces a class of diffusion processes and proves the tightness of their law; then, he identifies the limit. One of the interests of the method is that it only needs weak conditions on \(b\). Stefan's problem can be reduced to a special case of (1).