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

Abstract

In this paper, we develop a unified geometric framework for the propagation dynamics of multidimensional nonlocal Fisher--KPP equations with asymmetric dispersal. In dimensions $N⩾2$, asymmetry leads to a multidimensional propagation geometry determined by the full family of minimal wave speeds, and the intrinsic spreading set may even fail to contain the origin. We show that the mean dispersal drift $\mathbf{p}={\int }_{{\mathbb{R}}^N}x\mathcal{J}(x)dx$ provides a natural center for the spreading geometry, yielding a new drift-centered radial description. For unbounded initial supports, we prove that the spreading set $\mathcal{W}$ is given by the Minkowski-sum formula. Although $\mathcal{W}$ need not be convex, it remains star-shaped with respect to $\mathbf{p}$ and admits a Freidlin--Gärtner-type variational characterization. We further establish locally truncated Hausdorff convergence under a hypothesis on directions, and a global $o(t)$-Hausdorff approximation under a finite-thickness assumption on the initial support. Thus, our results extend the classical Freidlin--Gärtner picture to genuinely biased multidimensional nonlocal propagation.