Consider a sequence of Markov processes $X^1, X^2,...$ with state space $E$, where $X^N$ has a strong drift to $D \subseteq E$, such that $Φ(X^N)$ is slow for some appropriate $Φ: E\to D$. Using the method of martingale problems, we give a limit result, such that $Φ(X^N) \xRightarrow{N\to\infty} Z$ in the space of càdlàg paths, and $X^N \xRightarrow{N\to\infty} X$ in measure. \\ We apply the general limit result to models for copy number variation of genetic elements in a diploid Moran model of size $N$. The population by time $t$ is described by $X^N \in \mathcal P(\mathbb N_0)$, where $X^N_k$ is the frequency of individuals with copy number $k$, and $Φ: \mathcal P(\mathbb
Slow-fast systems arise frequently in probabilistic models (see e.g. Ball et al., 2006;Berglund and Gentz, 2006;Li and Sieber, 2022;Kifer, 2024;Champagnat and Hass, 2025). We study the situation of a fast evolving sequence of Markov processes X N , such that (i) Z N := Φ(X N ) evolves slowly and (ii) X N is pushed fast towards a slow subset of the state space. A similar situation was studied by Katzenberger (1991) using semi-martingale techniques. However, we do not show convergence of X N in path space using Lyapunov functions, but rather use tightness and martingale techniques in order to show convergence of X N in measure, and of Φ(X N ) in path space. Actually, this approach has appeared in a special situation in Pfaffelhuber and Wakolbinger (2023), but is here carried out in full generality.
As our main application, we use a population genetic model, where reproduction involves two parents and each individual has a type in N 0 , counting the number of genetic elements it carries. The model is fully specified once we specify the distribution 1 University of Freiburg, Germany. E-mail: samuel.adeosun@stochastik.uni-freiburg.de 2 University of Freiburg, Germany. E-mail: p.p@stochastik.uni-freiburg.de of genetic elements a parent gives to its offspring. Such models build on the two-parental Moran model and have e.g. been studied in Coron and Le Jan (2022); Otto et al. (2022); Otto and Wiehe (2023); Pfaffelhuber and Wakolbinger (2023); Omole and Czuppon (2025).
Recall that a process X = (X t ) t≥0 with complete and separable metric state space (E, r) solves the (G, D)-martingale problem for some linear G :
is a martingale for all f ∈ D. A Markov process is the unique solution to its martingale problem (when taking D large enough), and if there is a unique such solution, it is a strong Markov process (see e.g. Theorems 4.3.1 and 4.3.2 in Ethier and Kurtz (1986)).
Usually, such a process has càdlàg paths [0, ∞) → E and we denote the set of such paths by D(E); see Theorem 4.3.6 in Ethier and Kurtz (1986).
Assume we have a sequence of Markov processes X 1 , X 2 , … with state space (E, r E ) such that the generator G N of X N has domain D E ⊆ C b (E) and is of the form
(2.1)
We are interested in the weak limit of X N as N → ∞, in the special situation that D is another Polish space and Φ : E → D is such that, for some
G 1 (g • Φ) = 0.
(2.2) (In other words, the dynamics given by G N 1 change X N fast, but does not change Φ(X N ).) Recall that there are two kinds of convergence on D(E). First, the usual Skorohod convergence; see e.g. Chapter 3 in Ethier and Kurtz (1986). Second, there is convergence in measure: Define the weighted occupation measure of ξ ∈ D(E), as the probability measure
where t ≥ 0 and A is a measurable subset of E. Following Kurtz (1991) we say that a sequence (ξ N ) in D(E) converges in measure to ξ ∈ D(E) if the sequence of probability measures Γ ξ N converges weakly to Γ ξ . We assume that (Z, X), where convergence to Z is with respect to the Skorohod topology in D(E), and to X in measure.
For A1 to hold along a subsequence, it suffices to assume that Φ(X N ) N =1,2,… is tight (in the space of càdlàg paths on D), and (X N ) N =1,2,… are tight in measure, i.e. the sequence of occupation measures is tight.
A3 There is Ξ :
x ∈ E, then x = Ξ(Φ(x)). (In other words, we can recover x if we are given Φ(x) and G 1 f (x) = 0.) Then, we have the following Theorem 2.1. Let (E, r E ) and (D, r D ) be complete and separable spaces,
weakly in the space of càdlàg paths, and X N N →∞ = === ⇒ X weakly in measure. Then, for f ∈ D ′ E , using (2.1),
is a martingale, so the right hand side is a martingale with bounded variation, hence vanishes, i.e. G 1 f (X s ) = 0 for Lebesgue almost all t. From A3, this implies that X t = Ξ(Φ(X t )) for Lebesgue almost all t ≥ 0.
In order to put everything together, consider g ∈ D D and note that g
is a martingale. In particular, Φ(X) solves the martingale problem for g → (G 0 (g
Let us consider the following simple example:
So X N is some Brownian motion in R 2 with a strong force to the diagonal. Taking Φ(x, y) = 1 2 (x + y) and
x is on the diagonal, i.e. x = Ξ(Φ(x)) with Ξ(z) := (z, z). So, we have that the limit Φ(X), has generator
So, as anticipated, the limit of
Here is an extension of the population model for diploid organisms from Pfaffelhuber and Wakolbinger (2023), which we will study in detail in the remainder of the paper:
• A diploid population of constant size N consists of individuals, each carrying a certain number of genetic elements.
• Reproduction events occur at rate N 2 . Upon a reproduction event, choose individuals a, b, c. Individual c dies, and is replaced by offspring from a and b.
• For probability distributions p N k , k = 0, 1, 2, …, if individual a has k genetic elements, it transfers a random number of genetic elements to its offspring, distributed like p N k . Individual b inherits an independent number of gen
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