| Abstract: |
| This report examines nonlocal effects in pseudo-parabolic equations across deterministic and stochastic settings. In the deterministic case, When $\tau/D\leq1$, the pseudo-parabolic term acts as a structure-preserving regularization, and the traveling waves retain the monotonicity of classical Fisher-KPP fronts; when $\tau/D>1$, it actively induces oscillations in the wave profile near the equilibrium $u=1$. This predicts the saturation overshoot observed in porous media but absent in classical diffusion. In the stochastic counterpart, the nonlocal effect results in a bounded operator spectrum, which restricts the dissipation rate of high-frequency modes to a finite level and forces trajectory estimators to depend on long-time observations, while sustaining noise responses that make MLE asymptotic variance dimension-independent. Both manifestations reveal how nonlocal structure fundamentally reshapes wave dynamics and statistical estimability. |
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