Wong-Zakai approximation and support theorem for quasilinear parabolic stochastic partial differential equations

In this paper, we establish Wong-Zakai approximation and support theorem for quasilinear parabolic stochastic partial differential equations. Since this class of equations is not locally monotone, we implement a regularization strategy using a sequence of heat kernel to modify the matrix-valued diffusion term, thereby recovering the monotonicity. As a result, the problem is reduced to prove two results of further layer approximations. To this end, we further employ a kinetic formulation of the approximating solutions in the L¹ setting and fully use the doubling variables techniques.