We present a unified spectral framework for modeling discrete diffusion and Markovian dynamics on finite abelian groups via convolution semigroups, negative definite functions and pseudo-differential operators. Using discrete Fourier analysis on \(\mathbb {Z}_{N}\) , we construct families of probability measures whose evolution encodes diffusion processes governed by Feller semigroups. The associated pseudo-differential operators are shown to be m-dissipative and self-adjoint, ensuring stability and well-posedness. Explicit symbolic generators based on quadratic dispersion are introduced, revealing deep connections between harmonic structures and stochastic evolution. This approach not only advances the theory of discrete diffusion on algebraic groups but also opens avenues toward non-abelian and ultrametric generalizations, with applications in signal processing, graph-based modeling, and numerical analysis.
