We build two models of population genetics with continuous geography. Rather than defining the models directly in terms of the dynamics of populations undergoing random change, we instead define associated coalescing processes that model lineages traced backwards in time. We then construct forwards-in-time models that have our coalescing processes as their duals. At each location in the "geography space" E (the circle of unit circumference), there is a population whose individuals have types from a "type space" K. Each population undergoes changes due to random mating, reproduction, and migration to other locations.;One model has as its dual a system of coalescing particles on E built from sticky Levy flows; the other has as its dual a system of particles on E performing independent copies of some Levy process and coalescing according to their local times together. Both processes (call them X and Xˆ respectively) live on a state space xi consisting of probability measure-valued functions; for each time t and for each point e ∈ E, Xt( e) and Xˆt(e) are measures on K describing the composition of the population at the point e at time t. For both models, we are able to deduce continuity in the time variable t of sample paths with respect to an appropriate topology on xi.;We then examine the case where the underlying Levy process of the sticky Levy flow and of the independent coalescing particles is a standard Brownian motion. We are able to analyze how the generator---in the sense of a martingale problem---of Xˆ behaves when applied to the algebra of functions generated by linear functionals xi → R of the form nm deye ne dkck , where psi is a C2 function on E, chi is a bounded function on K, and m is Lebesgue measure on E. We use this to show that Xˆ has a representative that is jointly continuous in both the time variable t and the geography variable e. Finally, we discover that the domain of the generator of X includes the above linear functionals, but does not include all of the algebra they generate.
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机译：我们用连续的地理学建立了两种人口遗传模型。我们不是直接根据经历随机变化的种群动态来定义模型，而是定义相关联的合并过程，该过程对沿时间追溯的谱系进行建模。然后，我们构建将合并过程作为对偶的及时转发模型。在“地理空间” E（单位圆周的圆）中的每个位置，都有一个种群，其个体具有来自“类型空间” K的类型。每个种群都因随机交配，繁殖和迁移到其他位置而发生变化。 。;一个模型具有双重系统，该系统将由粘性Levy流构建的E上的粒子聚结；另一个作为对偶系统，在E上有一个粒子系统，该系统执行Levy过程的独立副本，并根据它们的本地时间合并在一起。这两个过程（分别称为X和Xˆ）都生活在由概率度量值函数组成的状态空间xi上。对于每个时间t和每个点e∈E，Xt（e）和Xˆt（e）是对K的度量，描述了在时间t点e处的总体组成。对于这两个模型，我们都能够针对xi上的适当拓扑推断出样本路径的时间变量t的连续性;然后我们研究了粘性Levy流和独立凝聚粒子的潜在Levy过程是标准的布朗运动。我们可以分析Xˆ的生成器（就a问题而言）在将其应用于由nm deye ne dkck形式的线性函数xi→R生成的函数的代数时的行为，其中psi是a E上的C2函数，K上的chi是有界函数，m是E上的Lebesgue测度。我们用它来证明Xˆ具有在时间变量t和地理变量e上共同连续的代表。最后，我们发现X的生成器的域包括上述线性泛函，但不包括它们生成的所有代数。
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