We define the operator $D^+_VD^-_W:=Δ_{W,V}$ on the one-dimensional torus $\mathbb{T}$. Here, $W$ and $V$ are functions inducing (possibly atomic) positive Borel measures on $\mathbb{T}$, and the derivatives are generalized lateral derivatives. For the first time in this work, the space of test functions $C^{\infty}_{W,V}(\mathbb{T})$ emerges as the natural regularity space for solutions of the eigenproblem associated with $Δ_{W,V}$. Moreover, these spaces are essential for characterizing the energetic space $H_{W,V}(\mathbb{T})$ as a Sobolev-type space. By observing that the Sobolev-type spaces $H_{W,V}(\mathbb{T})$ with additional Dirichlet conditions are reproducing kernel Hilbert spaces, we introduce the so-called $W$-Brownian bridges as mean-zero Gaussian processes with associated Cameron-Martin spaces derived from these spaces. This framework allows us to introduce $W$-Brownian motion as a Feller process with a two-parameter semigroup and càdlàg sample paths, whose jumps are subordinated to the jumps of $W$. We establish a deep connection between $W$-Brownian motion and these Sobolev-type spaces through their associated Cameron-Martin spaces. Finally, as applications of the developed theory, we demonstrate the existence and uniqueness of related deterministic and stochastic differential equations.