We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space $\mathcal{V}$ (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming--Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over $\mathcal{V}$.
Distribution dependence generally prevents McKean-Vlasov state solution maps from satisfying a cocycle identity, while the relevant Sobolev embedding is noncompact on unbounded domains. For a class of stochastic reaction-diffusion equations with a dissipative polynomial reaction, a one-sided monotone state-law coupling...
We study the long-term distributional dynamics and small-noise asymptotics of two-dimensional nonautonomous stochastic primitive equations driven by Gaussian and multiplicative L\'evy noise, together with moderate deviations for the purely jump model. For sufficiently small noise, uniform moment bounds and an exponenti...
We study limit solutions of the one-phase free boundary problem on noncollapsed Ricci limit spaces. We introduce a quantitative stratification $\mathcal{S}^k_{\varepsilon,r}$ of the singular set of the free boundary that simultaneously captures the approximate splitting of the ambient space and the approximate homogene...
We establish general conditions for the weak convergence of regular one-dimensional diffusion processes when their scale functions, speed measures, state spaces and boundary conditions may substantially vary. Our approach is based on the classical It\^o--McKean representation of regular diffusions via time changes and...
We prove weak existence and uniqueness in law of $[0,1]$-valued solutions for a class of scalar stochastic heat equations with measurable drifts $b : [0,1] \to \mathbb{R}$, driven by Wright--Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities $-Cu\leq b(u)\le...
We investigate the pathwise asymptotic dynamics of a nonlocal lattice equation with fixed delay on $\ell^2(\mathbb Z)$ driven by an infinite-dimensional stationary weakly geometric rough path. Using a rough-flow Doss-Sussmann transformation and the method of steps, we establish global well-posedness in the controlled-p...
