Propagation Dynamics for Multidimensional Nonlocal Diffusion Equations: A General Freidlin-Gärtner Formula

In this paper, we establish a unified geometric description for the propagation behavior of multidimensional nonlocal diffusion equations. By extending the classical Freidlin-Gärtner framework to general asymmetric, nonlocal operators, our theory naturally captures biased propagation–a regime where the intrinsic spreading set may exclude the origin. A key consequence is the representation of the spreading set as a Minkowski sum, which holds for both bounded and unbounded initial supports. Within this framework, we derive uniform spreading estimates and prove the local Hausdorff convergence of level sets in arbitrary dimensions. Our work therefore not only recovers known isotropic results but also provides a complete characterization of the biased case. Moreover, the methods developed here are readily adaptable to a broader class of diffusion problems featuring general operators and nonlinearities.