Relative Mather discrepancy on arc spaces

Given any generically étale morphism of varieties $f\text{\hspace{0.17em}⁣}:X\to Y$, we define the relative Mather discrepancy function on the arc space $X_{\mathrm{\infty }}$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_{\mathrm{\infty }}\text{\hspace{0.17em}⁣}:X_{\mathrm{\infty }}\to Y_{\mathrm{\infty }}$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat{K}$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat{K}$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.