  In the paper we prove the convergence of viscosity solutions $u_{\lambda}$ as
$\lambda\rightarrow0_+$ for the parametrized degenerate viscous Hamilton-Jacobi
equation \[ H(x,d_x u, \lambda u)=\alpha(x)\Delta u,\quad \alpha(x)\geq 0,\quad
x\in \mathbb T^n \] under suitable convex and monotonic conditions on $H:
T^*M\times\mathbb R\rightarrow\mathbb R$. Such a limit can be characterized in
terms of stochastic Mather measures associated with the critical equation \[
H(x,d_x u,0)=\alpha(x)\Delta u. \]
