We introduce and rigorously analyze two distinct classes of Hardy-type operators on the p-adic vector space \(\mathbb {Q}_{p}^{n}\) . The first class consists of linear operators \(\{\mathcal {H}_t\}_{t>0}\) acting on \(L^{q}(\mathbb {Q}_{p}^{n})\) , \(1 \le q < \infty \) , forming a strongly continuous, translation-invariant, sub-Markovian semigroup. We establish boundedness, regularity properties, and invariance on Bruhat-Schwartz test functions. We develop a rigorous probabilistic interpretation, proving that these operators define transition kernels of a spatially homogeneous Lévy-type Markov process on \(\mathbb {Q}_p^n\) with pure jump dynamics. Numerical experiments validate exponential convergence to steady state and confirm the spectral gap. The second class comprises nonlinear operators \(\{\widetilde{\mathcal {H}}_t\}_{t>0}\) on \(C_{0}(\mathbb {Q}_p^n)\) , proven to be well-defined contractions satisfying the positive maximum principle. We establish m-dissipativity, guaranteeing the generation of a strongly continuous nonlinear semigroup. Computational examples illustrate spatial confinement and amplitude decay characteristic of dissipative ultrametric dynamics. This framework establishes rigorous connections between Hardy operator theory and stochastic processes on ultrametric spaces.
