Limit of solutions for semilinear Hamilton-Jacobi equations with degenerate viscosity

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 $$H(x,d_x u,0)=\alpha(x)\Delta u.$$