Recent Results on a Generalization of the Laplacian

In this paper we discuss recent results regarding a generalization of the Laplacian. To be more precise, fix a function$W(x_1,\ldots,x_d) = \sum_{k=1}^d W_k(x_k)$, where each $W_k: \bb R \to \bb R$ is a right continuous with left limits and strictly increasing function.Using $W$, we construct the generalized laplacian $\mc L_W = \sum_{i=1}^d \partial_{x_i}\partial_{W_i}$, where $\partial_{W_i}$ is a generalized differentialoperator induced by the function $W_i$.We present results on spectral properties of $\mc L_W$, Sobolev spaces induced by $\mc L_W$ ($W$-Sobolev spaces), generalized partial differential equations, generalized stochastic differential equations andstochastic homogenization.