Nonconservative Diffusions on [0, 1] with Killing and Branching: Applications to Wright-Fisher Models with or without Selection
Abstract
We consider nonconservative diffusion processes xt on the unit interval, so with absorbing barriers. Using Doob-transformation techniques involving superharmonic functions, we modify the original process to form a new diffusion process presenting an additional killing rate part d > 0. We limit ourselves to situations for which is itself nonconservative with upper bounded killing rate. For this transformed process, we study various conditionings on events pertaining to both the killing and the absorption times. We introduce the idea of a reciprocal Doob transform: we start from the process , apply the reciprocal Doob transform ending up in a new process which is xt but now with an additional branching rate b > 0, which is also upper bounded. For this supercritical binary branching diffusion, there is a tradeoff between branching events giving birth to new particles and absorption at the boundaries, killing the particles. Under our assumptions, the branching diffusion process gets eventually globally extinct in finite time. We apply these ideas to diffusion processes arising in population genetics. In this setup, the process xt is a Wright-Fisher diffusion with selection. Using an exponential Doob transform, we end up with a killed neutral Wright-Fisher diffusion . We give a detailed study of the binary branching diffusion process obtained by using the corresponding reciprocal Doob transform.
1. Introduction
We consider diffusion processes on the unit interval with a series of elementary stochastic models arising chiefly in population dynamics in mind. These connections found their way over the last sixty years, chiefly in mathematical population genetics. In this context, we refer to [1] and to its extensive and nonexhaustive list of references for historical issues in the development of modern mathematical population genetics (after Wright, Fisher, Crow, Kimura, Nagylaki, Maruyama, Ohta, Watterson, Ewens, Kingman, Griffiths, and Tavaré, to cite only a few). See also the general monographs [2–6].
Special emphasis is put on Doob-transformation techniques of the diffusion processes under concern. Most of the paper′s content focuses on the specific Wright-Fisher (WF) diffusion model and some of its variations, describing the evolution of one two-locus colony undergoing random mating, possibly under the additional actions of mutation, selection, and so on. We now describe the content of this work in more detail.
Section 2 is devoted to generalities on one-dimensional diffusions on the unit interval [0, 1]. It is designed to fix the background and notations. Special emphasis is put on the Kolmogorov backward and forward equations, while stressing the crucial role played by the boundaries in such one-dimensional diffusion problems. Some questions such as the meaning of speed and scale functions, existence of an invariant measure, and validity of detailed balance are addressed in the light of the Feller classification of boundaries. When the boundaries are absorbing, the important problem of evaluating additive functionals along sample paths is then briefly discussed, emphasizing the prominent role played by the Green function of the model; several simple illustrative examples are supplied. So far, we have dealt with a given process, say xt, and recalled the various ingredients for computing the expectations of various quantities of interest, summing up over the history of paths. In this setup, there is no distinction among paths with different destinations, nor did we allow for annihilation or creation of paths inside the domain before the process reached one of the boundaries. The Doob transform of paths allows to do so. We, therefore, describe the transformation of sample paths techniques deriving from superharmonic additive functionals. Some Doob transformations of interest are then investigated, together with the problem of evaluating additive functionals of the transformed diffusion process itself. Roughly speaking, the transformation of paths procedure allows to select sample paths of the original process with, say, a fixed destination and/or, more generally, to kill certain sample paths that do not fit the integral criterion encoded by the additive functional. As a result, this selection of paths procedure leads to a new process described by an appropriate modification of the infinitesimal generator of the original process including a multiplicative killing part rate of the sample paths inside the interval. It turns out, therefore, that the same diffusion methods used in the previous discussions apply to the transformed processes, obtained after a change of measure.
Let us be more specific. In this work, we limit ourselves to nonconservative diffusion processes xt on the unit interval and so with absorbing barriers. Using Doob-transformation techniques involving superharmonic functions α, we modify the original process to form a new diffusion process presenting an additional killing rate part d > 0. We further limit ourselves to situations for which is itself nonconservative with bounded above killing rate. For a large class of diffusion processes, the exponential function or some linear combinations of exponential functions are admissible superharmonic functions α, leading to the required property on d. The full transformed process has two stopping times: the time to absorption to the boundaries and the killing time inside the domain. We study various conditionings of the transformed process: conditioning on events leading to both random stopping times occurring after the current time or only in the remote future and conditioning on events leading to either killing or absorption time occurring first. We give the relevant quasistationary limit laws, in the spirit of Yaglom [7]. This is made possible thanks to the existence of an harmonic function for the full infinitesimal generator of the transformed process.
We next introduce the idea of a reciprocal Doob transform: we start now from the process , apply the reciprocal Doob transform ending up in a new process which is xt but now with an additional branching rate b > 0, which is bounded. Under this reciprocal technique, the particles are not killed, rather they are allowed either to survive or split. The transformed process is a binary branching diffusion. For this supercritical binary branching diffusion process, there is a tradeoff between branching events giving birth to new particles and absorption at the boundaries, killing the particles. Under our assumptions, the branching diffusion process gets eventually globally extinct in finite time.
We next apply these general ideas to diffusion processes arising in population genetics.
In Section 3 we start recalling that Wright-Fisher diffusion models with various drifts are continuous space-time models which can be obtained as scaling limits of a biased discrete Galton-Watson model with a conservative number of offspring over the generations. Sections 4 and 5 are devoted to a detailed study of both the neutral WF diffusion process and the WF diffusion with selection, respectively.
In Section 6, we apply the Doob-transformation techniques to these processes: The starting point process xt is a Wright-Fisher diffusion with selection differential σ > 0. We use the exponential Doob kernel α = e−σx. The transformed process accounts for a neutral Wright-Fisher evolution for the allele 1 frequency, subject to the additional possibility of the extinction of the population itself due to killing at rate d proportional to its heterozygosity. This model is of importance in population genetics as it first appeared in [8, Page 272] as a scaling limit of a discrete population genetics model of recombination. We particularize the relevant Yaglom limit laws obtained after conditionings on events pertaining to both the killing or the absorption times occurring first. The computations of the quasistationary distributions are explicit here. Our approach relies on the spectral expansion of the transition probability kernels of both xt and which are known (from the works of Kimura) to involve oblate spheroidal wave functions and Gegenbauer polynomials, respectively.
In Section 7, we follow the general reciprocal path indicated in Section 2 and apply it to the particular models under concern, thereby illustrating and developing the idea of a reciprocal Doob transform. We give a detailed study of the binary branching diffusion process obtained by using the corresponding reciprocal Doob transform eσx when the starting point process is now a neutral Wright-Fisher diffusion process. We end up in a globally subcritical branching particle system, each diffusing according to the WF model with selection. This problem is amenable to the results obtained in [9, 10].
2. Diffusion Processes on The Unit Interval: A Reminder
We start with generalities on one-dimensional diffusions exemplifying our study to the Wright-Fisher model and its relatives. For more technical details, we refer to [8, 11–13].
2.1. Generalities on One-Dimensional Diffusions on the Interval [0,1]
2.2. Natural Coordinate, Scale, and Speed Measure
Examples 2.2 (from population genetics). (i) Assume that f(x) = 0 and g2(x) = x(1 − x). This is the neutral Wright-Fisher (WF) model discussed at length later. This diffusion is already in natural scale and φ(x) = x, m(x) = [x(1−x)]−1. The speed measure is not integrable.
(ii) With u1, u2 > 0, assume f(x) = u1 − (u1 + u2)x and g2(x) = x(1 − x). This is the Wright-Fisher model with mutation. The parameters u1, u2 can be interpreted as mutation rates. The drift vanishes when x = u1/(u1 + u2) which is an attracting point for the dynamics. Here,,, with φ(0) = −∞ and φ(1) = +∞ if u1, u2 > 1/2. The speed measure density is and so is always integrable.
(iii) With σ ∈ R, assume a model with quadratic logistic drift f(x) = σx(1 − x) and local variance g2(x) = x(1 − x). This is the WF model with selection. For this diffusion (see [15]), φ(x) = ((1 − e−2σx)/(1 − e−2σ)) and m(x) ∝ [x(1−x)]−1e2σx are not integrable. Here, σ is a selection or fitness parameter. We shall return at length to this model and its neutral version later.
2.3. The Transition Probability Density
In this case, a sample path of (xt; t ≥ 0) can reach ∘ from the inside of I in finite time but cannot reenter. The sample paths are absorbed at ∘. There is an absorption at ∘ at time τx,∘ = inf (t > 0 : xt = ∘|x0 = x) and P(τx,∘ < ∞) = 1. Whenever both boundaries {0,1} are absorbing, the diffusion xt should be stopped at τx : = τx,0∧τx,1. Would none of the boundaries {0,1} be absorbing, then τx = +∞, which we rule out.
Examples of diffusion with exit boundaries are WF model and WF model with selection. In the WF model including mutations, the boundaries are entrance boundaries and so are not absorbing.
When the boundaries are absorbing, p(x; t, y) is a subprobability. Letting , we clearly have ρt(x) = P(τx > t). Such models are nonconservative.
2.4. Additive Functionals Along Sample Paths
Some Examples. (1) Assume that c = 1 and d = 0: here, α = E(τx) is the mean time of absorption (average time spent in (0,1) before absorption), solution to
(2) Whenever both {0,1} are exit boundaries, it is of interest to evaluate the probability that xt first hits [0,1] (say) at 1, given x0 = x. This can be obtained by choosing c = 0 and d(∘) = 1(∘ = 1).
Let then α = :α1(x) = P(xt first hits [0,1] at 1∣x0 = x). α1(x) is a G-harmonic function solution to G(α1) = 0, with boundary conditions α1(0) = 0 and α1(1) = 1. Solving this problem, we get
(3) Let y ∈ I and put c = (1/2ε) 1 (x ∈ (y − ε, y + ε)) and d = 0. As ε → 0, c converges weakly to δy(x) and, is the Green function, solution to
(4) Also of interest are the additive functionals of the type
Whenever c(x) = δy(x), d = 0, then
2.5. Transformation of Sample Paths (Doob-Transform) and Killing
In the preceding subsections, we have dealt with a given process and recalled the various ingredients for the expectations of various quantities of interest, summing over the history of paths. In this setup, there is no distinction among paths with different destinations nor did we allow for annihilation or creation of paths inside the domain before the process reached one of the boundaries. The Doob transform of paths allows to do so.
Consider a one-dimensional diffusion (xt; t ≥ 0) as in (2.1) with absorbing barriers. Let p(x; t, y) be its transition probability, and let τx be its absorption time at the boundaries.
In the sequel, we shall limit ourselves to the cases for which the following additional conditions hold on the transformed process.
(i) Nonconservativeness of <!--${ifMathjaxEnabled: 10.1155%2F2011%2F605068}-->x̃t<!--${/ifMathjaxEnabled:}--><!--${ifMathjaxDisabled: 10.1155%2F2011%2F605068}--><!--${/ifMathjaxDisabled:}-->. We will next suppose that the boundaries ∘: = 0 or 1 are both exit (or absorbing) boundaries for the new process in (2.38). From the Feller criterion for exit boundaries, this will be the case if for all y0 ∈ (0,1)
(ii) Boundedness of the Killing Rate d. In some examples, the killing rate d = −G(α)/α is bounded above. For example, suppose that the drift of the diffusion process (xt; t ≥ 0) is bounded above by f* = max x(f(x)) > 0. (If the drift of (xt; t ≥ 0) is bounded below by f* < 0, we are led to the same conclusions while considering the process 1 − xt instead of xt.) Then, choosing α(x) = e−ax, a > 0, − G(α) = (af − (a2/2)g2)α < af*α. Thus, d = −G(α)/α is bounded above by af*. Because −G(α) = c ≥ 0, all this makes sense if, for all x, af(x) − (a2/2)g2(x) ≥ 0 or −∂U : = 2f/g2 ≥ a (the opposite of the gradient of the potential function U in (2.12) is bounded below).
Let (ak; k ≥ 1) be a nonincreasing sequence of [0,1]-valued real numbers. Let (αk; k ≥ 1) be a sequence of nonnegative real numbers such that for all x ∈ (0,1)
Therefore, for a large class of diffusion processes, the exponential function or some linear combinations of exponential functions are superharmonic functions α, leading to a bounded above killing rate d = −G(α)/α.
2.6. Normalizing and Conditioning
Because the transformed process is nonconservative, it is of interest to inspect various conditionings in the sense of Yaglom, [7].
(ii) Under our assumptions, in the transformation of paths process, the transformed process can both be absorbed at the boundaries and be killed. So, both and are finite with positive probability. We wish to understand the processes conditioned on the events or , (see [16]).
Note also that ] are the transition probability densities of conditioned on the event [of conditioned on the event ]. They are the Yaglom limits of both conditioned processes.
Additive Functionals of the Transformed Process. for the new process , it is also of interest to evaluate additive functionals along their own sample paths. Let then be such an additive functional where the functions and are themselves both nonnegative. It solves
Specific transformations of interest. (i) The case c = 0 deserves a special treatment. Indeed, in this case, and so , the absorption time for the process governed by the new SDE (2.38). Here, . Assuming α solves −G(α) = 0 if x ∈ I with boundary conditions α(0) = 0 and α(1) = 1 (α(0) = 1 and α(1) = 0, resp.), the new process is just (xt; t ≥ 0) conditioned on exiting at x = 1 (at x = 0, resp.). In the first case, the boundary 1 is exit, whereas 0 is entrance; α reads
Example 2.1. Consider the WF model on [0,1] with selection for which, with σ ∈ R, f(x) = σx(1 − x) and g2(x) = x(1 − x). Assume that α solves −G(α) = 0 if x ∈ (0,1) with α(0) = 0 and α(1) = 1; one gets, α(x) = (1 − e−2σx)/(1 − e−2σ). The diffusion corresponding to (2.38) has the new drift: , independently of the sign of σ. It models the WF diffusion with selection conditioned on exit at ∘ = 1.
(ii) Assume that α now solves −G(α) = 1 if x ∈ I with boundary conditions α(0) = α(1) = 0. In this case study, one selects sample paths of (xt; t ≥ 0) with a large mean absorption time α(x) = E(τx). Sample paths with large sojourn time in I are favored. We have
(iii) Assume that α now solves −G(α) = δy(x) if x ∈ I with boundary conditions α(0) = α(1) = 0. In this case study, one selects sample paths of (xt; t ≥ 0) with a large sojourn time density at y recalling . The stopping time of occurs at rate δy(x)/𝔤(x, y). It is a killing time when the process is at y for the last time after a geometrically distributed number of passages there with rate 1/𝔤(x, y) (or with success probability 1/(1 + 𝔤(x, y))). Let . Then, solves , with explicit solution
The Green function at y0 ∈ (0,1) of the transformed process is solution to . It takes the simple form
(iv) Let λ1 be the smallest non-null eigenvalue of the infinitesimal generator G. Let α = u1 be the corresponding eigenvector, that is, satisfying, −Gu1 = λ1u1 with boundary conditions u1(0) = u1(1) = 0. Then, c = λu1. The new KB operator associated to the transformed process is
When dealing for example with the neutral Wright-Fisher diffusion, it is known that λ1 = 1 with u1 = x(1 − x) and v1 ≡ 1 (see Section 4.3, example (ii)). The Q-process in this case obeys
2.7. Branching and the Reciprocal Doob Transform
The process with infinitesimal generator is now a pure binary branching diffusion process. For this class of models, an initial particle started at x obeys a diffusion process with infinitesimal generator G, absorbed when it hits the boundaries. At some random (mean b*) exponential time, this particle dies, giving birth in the process to a random number M(x) (either 1 or 2) of daughter particles started where the mother particle died and diffusing independently as their mother did and so forth for the subsequent generation particles. We have EM(x) = μ(x).
The process with infinitesimal generator is, thus, a branching diffusion with supercritical binary splitting mechanism (μ(x) > 1). There is, therefore, a competition between the branching phenomenon that leads to an exponential increase of the number of particles in the system and the absorption at the boundaries of the living particles.
3. The Wright-Fisher Example
In this section, we briefly and informally recall that the celebrated WF diffusion process with or without a drift may be viewed as a scaling limit of a simple two alleles discrete space-time branching process preserving the total number N of individuals in the subsequent generations (see [8, 12, 21] for example).
3.1. The Neutral Wright-Fisher Model
Equation (3.3) is a one-dimensional diffusion as in (2.1) on [0,1], with zero drift f(x) = 0 and volatility . This diffusion is already in natural coordinate, and so φ(x) = x. The scale function is x and the speed measure [x(1−x)]−1dx. One can check that both boundaries are exit in this case: the stopping time is τx = τx,0∧τx,1 where τx,0 is the extinction time and τx,1 the fixation time. The corresponding infinitesimal generators are and .
3.2. Nonneutral Cases
For instance, taking pN(x) = (1 − π2,N)x + π1,N(1 − x), where (π1,N, π2,N) are small (N-dependent) mutation probabilities from A1 to A2 (A2 to A1, resp.). Assuming that , leads after scaling to the drift of WF model with positive mutations rates (u1, u2).
4. The Neutral WF Model
In this section, we particularize the general ideas developed in the introductory Section 2 to the neutral WF diffusion (3.3) and draw some straightforward conclusions most of which are known which illustrate the use of Doob transforms.
4.1. Explicit Solutions of the Neutral KBE and KFE
As shown by Kimura in [22], it turns out that both Kolmogorov equations are exactly solvable, in this case, using spectral theory. Indeed, the solutions involve a series expansion in terms of eigenfunctions of the KB and KF infinitesimal generators with discrete eigenvalues spectrum. We now consider the specific neutral WF model.
With z ∈ (−1,1), let (Pk(z); k ≥ 0) be the degree-(k + 1) Gegenbauer polynomials solving with ; we let P0(z) : = (1 − z)/2. When k ≥ 1, we have Pk(±1) = 0 and so Pk(z) = (1 − z2)Qk(z), where Qk(z) is a polynomial with degree k − 1 satisfying Qk(−1) = (−1)k−1 and Qk(1) = 1. With x ∈ (0,1), let (uk(x); k ≥ 0) be defined by: uk(x) = Pk(1 − 2x). These polynomials clearly constitute a system of eigenfunctions for the KB operator with eigenvalues λk = (k(k + 1))/2, k ≥ 0, thus with −G(uk(x)) = λkuk(x). In particular, u0(x) = x, u1(x) = x − x2, u2(x) = x − 3x2 + 2x3, u3(x) = x − 6x2 + 10x3 − 5x4, u4(x) = x − 10x2 + 30x3 − 35x4 + 14x5, …. With k ≥ 1, we have uk(0) = uk(1) = 0 and and .
The eigenfunctions of the KF operator are given by vk(y) = m(y) · uk(y), k ≥ 0, where the Radon measure of weights m(y)dy is the speed measure: m(y)dy = dy/(y(1 − y)), for the same eigenvalues. For instance, v0(y) = 1/(1 − y), v1(y) = 1, v2(y) = 1 − 2y, v3(y) = 1 − 5y + 5y2, v4(y) = 1 − 9y + 21y2 − 14y3, ….
Although λ0 = 0 really constitutes an eigenvalue, only v0(y) is not a polynomial. When k ≥ 1, from their definition, the uk(x) polynomials satisfy uk(0) = uk(1) = 0 in such a way that vk(y) = m(y) · uk(y), k ≥ 1 is a polynomial with degree k − 1.
4.2. Additive Functionals for the Neutral WF
As a Few Examples (1) Let c = 1 and d = 0: here, α(x) = E(τx) is the mean time of absorption (average time spent in I before absorption). The solution is (the Crow and Kimura formula, see [2])
(2) Let c = 0 and d(∘) = 1(∘ = 1). Let α(x) = P(xt first hits [0,1] at 1 | x0 = x). Then, α(x) is a G-harmonic function solution to G(α) = 0, with boundary conditions α(0) = 0 and α(1) = 1. The solution for WF model is: α(x) = x. Stated differently, x = P(τx,1 < τx,0) is the probability that the exit time at ∘ = 1 is less than the one at ∘ = 0, starting from x.
On the contrary, choosing α(x) to be a G-harmonic function with boundary conditions α(0) = 1 and α(1) = 0, α(x) = P(xt first hits [0,1] at 0 | x0 = x) = 1 − x. Thus, 1 − x = P(τx,0 < τx,1).
(3) Let c(xs) = 2xs(1 − xs) measure the heterozygosity of the WF process at time s and assume d(0) = d(1) = 1. A remarkable thing is that the average heterozygosity over the sample paths is
4.3. Transformation of WF Sample Paths, [3]
The stopping time of is just its killing time when the process is at y for the last time with a geometrically number of passages at y with rate 1 (or success probability 1/2).
5. The WF Model with Selection
Define and where m(x) = e2σx/(x(1 − x)) is the speed measure density of the WF model with selection (3.8).
6. From the WF Model with Selection to the Neutral WF Model: Doob Transform and Killing
We shall consider the following transformation of paths for the WF model with selection. Consider the Wright-Fisher diffusion with selection (3.8): , x0 = x ∈ (0,1). For this model, and both boundaries are exit.
Assume that σ > 0 so that the drift term is bounded above by f* = σ/4, together with 2f/g2 being bounded below (as a constant function here equal to 2σ). We are then in the general framework of the problems under study in this paper. This suggests that for some admissible choice of a superharmonic exponential function α = e−ax, the α-Doob transform of xt could lead to a transformed process with bounded killing rate d = −G(α)/α. We shall choose a = σ for its interesting features.
The transformed process accounts for a neutral evolution of the allele A1 frequency subject to the additional extinction opportunity of the population itself due to killing at rate proportional to its heterozygosity. Leaving aside the fact that it can be obtained after a suitable Doob transformation, this model is of importance in population genetics: it first appeared in ([8, Page 272]) as a scaling limit of a population genetics model of recombination.
From the general study of Section 2, we obtain the following.
7. From the Neutral WF Model to the WF Model with Selection: Reciprocal Doob Transform and Branching
We now follow the general path indicated in Section 2.7 and apply it to the particular models under concern. We, therefore, illustrate and develop the idea of a reciprocal Doob transform on the specific example of interest.
The density of the transformed process is . It is exactly known because p is known from (7.1).
The transformed process (with infinitesimal backward generator ) accounts for a branching diffusion (BD), where a diffusing mother particle (with generator and started at x) lives a random exponential time with constant rate b*. When the mother particle dies, it gives birth to a spatially dependent random number M(x) of particles (with mean μ(x)). If M(x) ≠ 0, M(x) independent daughter particles are started where their mother particle died; they move along a WF diffusion with selection and reproduce, independently and so on.
(i), uniformly in x,
(ii) there exists a constant γ > 0:, uniformly in x,
In the statement (ii), the quantity 1 − P(Nt(x) = 0) = P(Nt(x) > 0) is also P(T(x) > t) where T(x) is the global extinction time of the particle system descending from an Eve particle started at x. The number −λ1 is the usual Malthus decay rate parameter. From has a natural interpretation in terms of the propensity of the particle system to survive to its extinction fate: the so-called reproductive value in demography.
(iii) with ψ = 1 reads giving an interpretation of the constant γ (which may be hard to evaluate in practise).
Remark 7.1. At time t, let denote the positions of the BD particle system. Let stand for the functional generating function (|z| ≤ 1) of the measure-valued branching particle system. u(x, t; z) obeys the nonlinear (quadratic) Kolmogorov-Petrovsky-Piscounoff PDE, [27]
In particular, if , u(x, t) obeys the linear backward PDE