We define the operator \(D^+_VD^-_W:=\Delta _{W,V}\) 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 \(\Delta _{W,V}\) . 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 càdàg 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.
