The authors study the convergence for the viscosity solution \(u^{\varepsilon}:[0,T]\times \mathbb R^{k}\to \mathbb R^{k}\) of the variational inequality \[ -\partial_{t}u^{\varepsilon}( t,x) -L_{\varepsilon}u^{\varepsilon}( t,x) -f(x,u^{\varepsilon}(t,x))\in(\partial\varphi) (u^{\varepsilon}(t,x)), \] \(u^{\varepsilon}( t,x) \in\overline{ \text{Dom} (\varphi)}\), \(t\in [0,T]\), \(~u^{\varepsilon}(0,x) =g(x)\), \(x\in \mathbb R\), when \(\varepsilon\downarrow0\). Here \(\partial\varphi\) denotes the subdifferential of the lower semi-continuous, proper convex function \(\varphi:\mathbb R^{k} \to(-\infty,+\infty]\) and \(L_{\varepsilon}\) is the generator of a \(d\)-dimensional diffusion process \(X^{\varepsilon}\) which converges in law to some diffusion process \(X\) with generator \(L\). They prove that, under suitable assumptions on the coefficients, there is a limit function \(u\) that satisfies the same variational inequality as \(u^{\varepsilon}\), only with \(L_{\varepsilon}\) replaced by \(L\). For this they use the generalized Feynman-Kac formula \(u^{\varepsilon}(t,x)=Y_{t}^{\varepsilon}(t,x)\) associating \(u^{\varepsilon}\) to the solution \((X^{\varepsilon},Y^{\varepsilon },K^{\varepsilon})\) of a reflected backward stochastic differential equation [see E. Pardoux and A. Răşcanu, Stochastic Processes Appl. 76, 191–215 (1998; Zbl 0932.60070)] and show the convergence in law of \((X^{\varepsilon},Y^{\varepsilon},K^{\varepsilon})\) to the solution of a reflected backward equation driven by \(X\).
The proof of this convergence is based on a double approximation scheme: the Yosida approximation for the reflection term and the usual homogenization approximation, see E. Pardoux and Yu. A. Veretennikov [Stochastics Stochastics Rep. 60, 255–270 (1997; Zbl 0891.60053)]. Finally, under the assumption of smoothness on the coefficients of the \(X^{\varepsilon}\)’s and for \(k=1\), the convergence result is also obtained in the case of semi-linear PDEs with solution in Sobolev sense.