Nonconservative Diffusions on [0,1] with Killing and Branching: Applications to Wright-Fisher Models with or without Selection

Abstract

We consider nonconservative diffusion processes 𝑥_𝑡 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 𝑥_𝑡 presenting an additional killing rate part 𝑑>0. We limit ourselves to situations for which 𝑥_𝑡 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 𝑥_𝑡, apply the reciprocal Doob transform ending up in a new process which is 𝑥_𝑡 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 𝑥_𝑡 is a Wright-Fisher diffusion with selection. Using an exponential Doob transform, we end up with a killed neutral Wright-Fisher diffusion 𝑥_𝑡. 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 𝑥_𝑡, 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 𝑥_𝑡 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 𝑥_𝑡 presenting an additional killing rate part 𝑑>0. We further limit ourselves to situations for which 𝑥_𝑡 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 𝑥_𝑡, apply the reciprocal Doob transform ending up in a new process which is 𝑥_𝑡 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 𝑥_𝑡 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 𝑥_𝑡 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 𝑥_𝑡 and 𝑥_𝑡 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 𝑒^𝜎𝑥 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]

Let (𝑤_𝑡;𝑡≥0) be a standard one-dimensional Brownian (Wiener) motion. Consider a one-dimensional Itô diffusion driven by (𝑤_𝑡;𝑡≥0) on the interval say [0,1]; see [14]. We will let 𝐼=(0,1). Assume that it has locally Lipschitz continuous drift (𝑥) and local standard deviation (volatility) 𝑔(𝑥), namely, consider the stochastic differential equation (SDE)𝑑𝑥_𝑡=𝑓𝑥_𝑡𝑑𝑡+𝑔𝑥_𝑡𝑑𝑤_𝑡,𝑥₀=𝑥∈(0,1). (2.1) The condition on 𝑓(𝑥) and 𝑔(𝑥) guarantees in particular that there is no point 𝑥_∗ in 𝐼 for which |𝑓(𝑥)| or |𝑔(𝑥)| would blow up and diverge as |𝑥−𝑥_∗|→0.

The Kolmogorov backward infinitesimal generator of (2.1) is 𝐺=𝑓(𝑥)𝜕_𝑥+(1/2)𝑔²(𝑥)𝜕²_𝑥. As a result, for all suitable 𝜓 in the domain of the operator 𝑆_𝑡∶=𝑒^𝑡𝐺, 𝑢∶=𝑢(𝑥,𝑡)=𝐄𝜓(𝑥_{𝑡∧𝜏𝑥}) satisfies the Kolmogorov backward equation (KBE) 𝜕_𝑡𝑢=𝐺(𝑢);𝑢(𝑥,0)=𝜓(𝑥). (2.2) In the definition of the mathematical expectation 𝑢, we have 𝑡∧𝜏_𝑥∶=inf(𝑡,𝜏_𝑥), where 𝜏_𝑥 indicates a random time at which the process should possibly be stopped (absorbed), given the process was started in 𝑥. The description of this (adapted) absorption time is governed by the type of boundaries which {0,1} are to (𝑥_𝑡;𝑡≥0).

2.2. Natural Coordinate, Scale, and Speed Measure

For such Markovian diffusions, it is interesting to consider the 𝐺-harmonic coordinate 𝜑∈𝐶² belonging to the kernel of 𝐺 that is, satisfying 𝐺(𝜑)=0. For 𝜑 and its derivative 𝜑^′∶=𝑑𝜑/𝑑𝑦, with (𝑥₀,𝑦₀)∈(0,1), one finds 𝜑^′𝑦=𝜑^′𝑦₀𝑒^{−2∫𝑦𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧},𝜑(𝑥)=𝜑𝑥₀+𝜑^′𝑦₀^𝑥_𝑥0𝑒^{−2∫𝑦𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧}𝑑𝑦. (2.3) One should choose a version of 𝜑 satisfying 𝜑^′(𝑦)>0, 𝑦∈𝐼. The function 𝜑 kills the drift 𝑓 of (𝑥_𝑡;𝑡≥0) in the sense that considering the change of variable 𝑦_𝑡=𝜑(𝑥_𝑡),𝑑𝑦_𝑡=𝜑^′𝑔𝜑⁻¹𝑦_𝑡𝑑𝑤_𝑡,𝑦₀=𝜑(𝑥). (2.4) The driftless diffusion (𝑦_𝑡;𝑡≥0) is often termed the diffusion in natural coordinates with state-space [𝜑(0),𝜑(1)]. Its volatility is 𝑔(𝑦)∶=(𝜑^′𝑔)(𝜑⁻¹(𝑦)). The function 𝜑 is often called the scale function.

Whenever 𝜑(0)>−∞ and 𝜑(1)<+∞, one can choose the integration constants defining 𝜑(𝑥) so that 𝜑(𝑥)=∫^𝑥₀𝑒^{−2∫𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧}𝑑𝑦∫¹₀𝑒^{−2∫𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧},𝑑𝑦 (2.5) with 𝜑(0)=0 and 𝜑(1)=1. In this case, the state-space of (𝑦_𝑡;𝑡≥0) is again [0,1], the same as for (𝑥_𝑡;𝑡≥0).

Finally, considering the random time change 𝑡→𝜃_𝑡 with inverse: 𝜃→𝑡_𝜃 defined by 𝜃_𝑡𝜃=𝜃 and 𝜃=^𝑡𝜃₀𝑔²𝑦_𝑠𝑑𝑠, (2.6) the novel diffusion (𝑤_𝜃∶=𝑦_𝑡𝜃;𝜃≥0) is easily checked to be identical in law to a standard Brownian motion. Let now 𝛿_𝑦(⋅)= weak-lim_𝜀↓0(1/2𝜀)𝟏(⋅∈(𝑦−𝜀,𝑦+𝜀)) stand for the Dirac delta mass at 𝑦. The random time 𝜃_𝑡 can be expressed as 𝜃_𝑡=¹₀𝑑𝑥⋅𝑚(𝑥)^𝑡₀𝛿_𝜑(𝑥)𝑤_𝑠𝑑𝑠=^𝑡₀𝑚𝜑⁻¹𝑤_𝑠𝑑𝑠, (2.7) where 𝑚(𝑥)∶=1/(𝑔²𝜑^′)(𝑥) is the (positive) speed density at 𝑥=𝜑⁻¹(𝑦) and 𝐿_𝑡(𝑦)∶=lim_𝜀↓0(1/2𝜀)∫^𝑡₀𝟏(𝑤_𝑠∈(𝑦−𝜀,𝑦+𝜀))𝑑𝑠 the local time at 𝑦 of the Brownian motion before time 𝑡. Both the scale function 𝜑 and the speed measure 𝑑𝜇=𝑚(𝑥)⋅𝑑𝑥 are, therefore, essential ingredients to reduce the original stochastic process (𝑥_𝑡;𝑡≥0) to the standard Brownian motion (𝑤_𝑡;𝑡≥0). Indeed, it follows from the above arguments that if 𝜃_𝑡=∫^𝑡₀𝑚(𝑥_𝑠)𝑑𝑠, then (𝜑(𝑥_𝜃𝑡);𝑡≥0) is a Brownian motion. The Kolmogorov backward infinitesimal generator 𝐺 may be written in Feller form 𝐺(⋅)=12𝑑𝑑𝑑𝜇⋅.𝑑𝜑 (2.8)

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,𝜑^′(𝑦)=𝜑^′(𝑦₀)𝑦^−2𝑢1(1−𝑦)^−2𝑢2,𝜑(𝑥)=𝜑(𝑥₀)+𝜑^′(𝑦₀)∫^𝑥_𝑥0𝑦^−2𝑢1(1−𝑦)^−2𝑢2𝑑𝑦, with 𝜑(0)=−∞ and 𝜑(1)=+∞ if 𝑢₁,𝑢₂>1/2. The speed measure density is 𝑚(𝑥)∝𝑥^2𝑢1−1(1−𝑥)^2𝑢2−1 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)]⁻¹𝑒^2𝜎𝑥 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

Assume that 𝑓(𝑥) and 𝑔(𝑥) are now differentiable in 𝐼. Let then 𝑝(𝑥;𝑡,𝑦) stand for the transition probability density function of 𝑥_𝑡 at 𝑦 given 𝑥₀=𝑥. Then, 𝑝∶=𝑝(𝑥;𝑡,𝑦) is the smallest solution to the Kolmogorov forward (Fokker-Planck) equation (KFE) 𝜕_𝑡𝑝=𝐺^∗𝑝,𝑝𝑥;0,𝑦=𝛿_𝑦(𝑥), (2.9) where 𝐺^∗(⋅)=−𝜕_𝑦(𝑓(𝑦)⋅)+(1/2)𝜕²_𝑦(𝑔²(𝑦)⋅) is the adjoint of 𝐺 (𝐺^∗ acts on the terminal value 𝑦, whereas 𝐺 acts on the initial value 𝑥). The way one can view this PDE depends on the type of boundaries that {0,1} are.

We will next suppose that the boundaries ∘∶=0 or 1 are both exit (or absorbing) boundaries. From the Feller classification of boundaries, this will be the case if forall𝑦₀∈(0,1)(i)𝑚𝑦∉𝐿₁𝑦₀,∘,(ii)𝜑^′𝑦^𝑦_𝑦0𝑚(𝑧)𝑑𝑧∈𝐿₁𝑦₀,∘, (2.10) where a function 𝑓(𝑦)∈𝐿₁(𝑦₀,∘) if ∫^∘_𝑦0|𝑓(𝑦)|𝑑𝑦<+∞.

In this case, a sample path of (𝑥_𝑡;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 𝜏_𝑥,∘=inf(𝑡>0∶𝑥_𝑡=∘|𝑥₀=𝑥) and 𝐏(𝜏_𝑥,∘<∞)=1. Whenever both boundaries {0,1} are absorbing, the diffusion 𝑥_𝑡 should be stopped at 𝜏_𝑥∶=𝜏_𝑥,0∧𝜏_𝑥,1. Would none of the boundaries {0,1} be absorbing, then 𝜏_𝑥=+∞, 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 𝜌_𝑡(𝑥)∶=∫¹₀𝑝(𝑥;𝑡,𝑦)𝑑𝑦, we clearly have 𝜌_𝑡(𝑥)=𝐏(𝜏_𝑥>𝑡). Such models are nonconservative.

For one-dimensional diffusions, the transition density 𝑝(𝑥;𝑡,𝑦) is reversible with respect to the speed density ([8, Chapter 15, Section  13]) and so detailed balance holds 𝑚(𝑥)𝑝𝑥;𝑡,𝑦=𝑚𝑦𝑝𝑦;𝑡,𝑥,0<𝑥,𝑦<1. (2.11) The speed density 𝑚(𝑦) satisfies 𝐺^∗(𝑚)=0. It may be written as a Gibbs measure with density: 𝑚(𝑦)∝(1/𝑔²(𝑦))𝑒^{−𝑈(𝑦)}, where the potential function 𝑈(𝑦) reads 𝑈𝑦∶=−2^𝑦₀𝑓(𝑧)𝑔²(𝑧)𝑑𝑧,0<𝑦<1, (2.12) and with the measure 𝑑𝑦/𝑔²(𝑦) standing for the reference measure.

Further, if 𝑝(𝑠,𝑥;𝑡,𝑦) is the transition probability density from (𝑠,𝑥) to (𝑡,𝑦), 𝑠<𝑡, then −𝜕_𝑠𝑝=𝐺(𝑝), with terminal condition 𝑝(𝑡,𝑥;𝑡,𝑦)=𝛿_𝑦(𝑥) and so 𝑝(𝑠,𝑥;𝑡,𝑦) also satisfies the KBE when looking at it backward in time. The Feller evolution semigroup being time homogeneous, one may as well observe that with 𝑝∶=𝑝(𝑥;𝑡,𝑦), operating the time substitution 𝑡−𝑠→𝑡, 𝑝 itself solves the KBE 𝜕_𝑡𝑝=𝐺𝑝,𝑝𝑥;0,y=𝛿_𝑦(𝑥). (2.13) In particular, integrating over 𝑦, 𝜕_𝑡𝜌_𝑡(𝑥)=𝐺(𝜌_𝑡(𝑥)), with 𝜌₀(𝑥)=𝟏(𝑥∈(0,1)).

𝑝(𝑥;𝑡,𝑦) being a sub-probability, we may define the normalized conditional probability density 𝑞(𝑥;𝑡,𝑦)∶=𝑝(𝑥;𝑡,𝑦)/𝜌_𝑡(𝑥), now with total mass 1. We get 𝜕_𝑡𝜕𝑞=−^𝑡𝜌_𝑡(𝑥)𝜌_𝑡(𝑥)⋅𝑞+𝐺^∗𝑞,𝑞𝑥;0,𝑦=𝛿_𝑦(𝑥). (2.14) The term 𝑏_𝑡(𝑥)∶=−𝜕_𝑡𝜌_𝑡(𝑥)/𝜌_𝑡(𝑥)>0 is the time-dependent birth rate at which mass should be created to compensate the loss of mass of the original process due to absorption of (𝑥_𝑡;𝑡≥0) at the boundaries. In this creation of mass process, a diffusing particle started in 𝑥 dies at rate 𝑏_𝑡(𝑥) at point (𝑡,𝑦), where it is duplicated in two new independent particles both started at 𝑦 (resulting in a global birth) evolving in the same diffusive way (consider a diffusion process with forward infinitesimal generator 𝐺^∗ governing the evolution of 𝑝(𝑥;𝑡,𝑦). Suppose that a sample path of this process has some probability that it will be killed or create a new copy of itself and that the killing and birth rates 𝑑 and 𝑏 depend on the current location 𝑦 of the path. Then, the process with the birth and death opportunities of a path has the infinitesimal generator 𝜆(𝑦)⋅+𝐺^∗(⋅), where 𝜆(𝑦)=𝑏(𝑦)−𝑑(𝑦). The rate can also depend on 𝑡 and 𝑥). The birth rate function 𝑏_𝑡(𝑥) depends here on 𝑥 and 𝑡, not on 𝑦.

When the boundaries of 𝑥_𝑡 are absorbing, the spectra of both −𝐺 and −𝐺^∗ are discrete (see [8, Page 330]): There exist positive eigenvalues (𝜆_𝑘)_𝑘≥1 ordered in ascending sizes and eigenvectors (𝑣_𝑘,𝑢_𝑘)_𝑘≥1 of both −𝐺^∗ and −𝐺 satisfying −𝐺^∗(𝑣_𝑘)=𝜆_𝑘𝑣_𝑘 and −𝐺(𝑦_𝑘)=𝜆_𝑘𝑢_𝑘 such that with ⟨𝑢_𝑘,𝑣_𝑘⟩∶=∫¹₀𝑢_𝑘(𝑥)𝑣_𝑘(𝑥)𝑑𝑥 and 𝑏_𝑘∶=⟨𝑢_𝑘,𝑣_𝑘⟩⁻¹, the spectral representation 𝑝𝑥;𝑡,𝑦=_𝑘≥1𝑏_𝑘𝑒^{−𝜆𝑘𝑡}𝑢_𝑘(𝑥)𝑣_𝑘𝑦 (2.15) holds.

Let 𝜆₁>𝜆₀=0 be the smallest nonnull eigenvalue of the infinitesimal generator −𝐺^∗ (and of −𝐺). Clearly, −(1/𝑡)log𝜌_𝑡(𝑥)→_𝑡→∞𝜆₁ and by L' Hospital rule, therefore, 𝑏_𝑡(𝑥)→_𝑡→∞𝜆₁. Putting 𝜕_𝑡𝑞=0 in the latter evolution equation, independently of the initial condition 𝑥𝑞𝑥;𝑡,𝑦⟶_𝑡→∞𝑞_∞𝑦=𝑣₁𝑦, (2.16) where 𝑣₁ is the eigenvector of −𝐺^∗ associated to 𝜆₁, satisfying −𝐺^∗𝑣₁=𝜆₁𝑣₁. The limiting probability 𝑣₁/norm (after a proper normalization) is called the Yaglom limit law of (𝑥_𝑡;𝑡≥0) conditioned on being currently alive at all time 𝑡 (see [7]).

2.4. Additive Functionals Along Sample Paths

Let (𝑥_𝑡;𝑡≥0) be the diffusion model defined by (2.1) on the interval 𝐼, where both endpoints are assumed absorbing (exit). This process is, thus, transient and nonconservative. We wish to evaluate the nonnegative additive quantities 𝛼(𝑥)=𝐄^𝜏𝑥₀𝑐𝑥_𝑠𝑑𝑠+𝑑𝑥_𝜏𝑥, (2.17) where the functions 𝑐 and 𝑑 are both assumed nonnegative on 𝐼 and 𝜕𝐼={0,1}. The functional 𝛼(𝑥)≥0 solves the Dirichlet problem −𝐺(𝛼)=𝑐if𝑥∈𝐼,𝛼=𝑑if𝑥∈𝜕𝐼, (2.18) and 𝛼 is a superharmonic function for 𝐺, satisfying −𝐺(𝛼)≥0.

Some Examples.
(1) Assume that 𝑐=1 and 𝑑=0∶ here, 𝛼=𝐄(𝜏_𝑥) is the mean time of absorption (average time spent in (0,1) before absorption), solution to −𝐺(𝛼)=1if𝑥∈𝐼,𝛼=0if𝑥∈𝜕𝐼. (2.19)
(2) Whenever both {0,1} are exit boundaries, it is of interest to evaluate the probability that 𝑥_𝑡 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(𝑥_𝑡firsthits[0,1]at1∣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 get𝛼₁(𝑥)=𝜑(𝑥)−𝜑(0)𝜑(1)(0)=∫−𝜑^𝑥0𝑑𝑦𝑒^{−2∫𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧}∫¹₀𝑑𝑦𝑒^{−2∫𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧}. (2.20) On the contrary, choosing 𝛼₀(𝑥) to be a 𝐺-harmonic function with boundary conditions 𝛼₀(0)=1 and 𝛼₀(1)=0, 𝛼₀(𝑥)=𝐏(𝑥_𝑡firsthits[0,1]at0∣𝑥₀=𝑥)=1−𝛼₁(𝑥).
(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 𝛿_𝑦(𝑥) and, 𝛼=∶𝔤(𝑥,𝑦)=𝐄(lim(1/2𝜀)∫^𝜏𝑥₀𝟏(𝑥_𝑠∈(𝑦−𝜀,𝑦+𝜀))𝑑𝑠)=∫^∞₀𝑝(𝑥;𝑠,𝑦)𝑑𝑠 is the Green function, solution to−𝐺(𝔤)=𝛿_𝑦(𝑥)if𝑥∈𝐼,𝔤=0if𝑥∈𝜕𝐼. (2.21)𝔤 is, therefore, the mathematical expectation of the local time at 𝑦, starting from 𝑥 (the sojourn time density at 𝑦). The solution is known to be (see [8, page 198] or [5, page 280])𝔤𝜑(𝑥)(0)𝜑(1)𝑦𝑥,𝑦=2−𝜑−𝜑𝑔²𝜑^′𝑦𝜑(1)−𝜑(0)if𝔤𝜑(1)(𝑥)𝜑𝑦(0)𝑥≤𝑦,𝑥,𝑦=2−𝜑−𝜑𝑔²𝜑^′𝑦𝜑(1)−𝜑(0)if𝑥≥𝑦. (2.22) The Green function is of particular interest to solve the general problem of evaluating additive functionals 𝛼(𝑥). Indeed, as is well known, see [8], for example, the integral operator with respect to the Green kernel inverts the second-order operator −𝐺 leading to𝛼(𝑥)=_𝐼𝔤𝑥,𝑦𝑐𝑦𝑑𝑦if𝑥∈𝐼,𝛼=𝑑if𝑥∈𝜕𝐼. (2.23) Under this form, 𝛼(𝑥) appears as a potential function and all potential function is superharmonic. Note that for all harmonic function ℎ≥0 satisfying −𝐺(ℎ)=0, 𝛼_ℎ(𝑥)∶=_𝐼𝔤𝑥,𝑦𝑐𝑦𝑑𝑦+ℎ(𝑥) (2.24) is again superharmonic because −𝐺(𝛼_ℎ)=𝑐≥0.
(4) Also of interest are the additive functionals of the type 𝛼_𝜆(𝑥)=𝐄^𝜏𝑥₀𝑒^−𝜆𝑠𝑐𝑥_𝑠𝑑𝑠+𝑑𝑥_𝜏𝑥, (2.25) where the functions 𝑐 and 𝑑 are again both assumed to be nonnegative. The functional 𝛼_𝜆(𝑥)≥0 solves the Dynkin problem, [8] (𝜆𝐼−𝐺)𝛼_𝜆=𝑐if𝛼𝑥∈𝐼,_𝜆=𝑑if𝑥∈𝜕𝐼 (2.26) involving the action of the resolvent operator (𝜆𝐼−𝐺)⁻¹ on 𝑐.
Whenever 𝑐(𝑥)=𝛿_𝑦(𝑥),𝑑=0, then𝛼_𝜆=∶𝔤_𝜆𝑥,𝑦=𝐄^𝜏𝑥₀𝑒^−𝜆𝑠𝛿_𝑦𝑥_𝑠𝑑𝑠=^∞₀𝑒^−𝜆𝑠𝑝𝑥;𝑠,𝑦𝑑𝑠 (2.27) is the 𝜆-potential function, solution to (𝜆𝐼−𝐺)𝔤_𝜆=𝛿_𝑦(𝑥)if𝔤𝑥∈𝐼,_𝜆=0if𝑥∈𝜕𝐼. (2.28)𝔤_𝜆 is, therefore, the mathematical expectation of the exponentially damped local time at 𝑦, starting from 𝑥 (the temporal Laplace transform of the transition probability density from 𝑥 to 𝑦 at 𝑡), with 𝑔₀=𝑔. Then, it holds that 𝛼_𝜆(𝑥)=_𝐼𝔤_𝜆𝑥,𝑦𝑐𝑦𝑑𝑦if𝛼𝑥∈𝐼,_𝜆=𝑑if𝑥∈𝜕𝐼. (2.29) The 𝜆-potential function is also useful in the computation of the distribution of the first-passage time 𝜏_𝑥,𝑦 to 𝑦 starting from 𝑥. From the convolution formula, 𝑝𝑥;𝑡,𝑦=^𝑡₀𝐏𝜏_𝑥,𝑦∈𝑑𝑠𝑝𝑦;𝑡−𝑠,𝑦, (2.30) and taking the Laplace transform of both sides with respect to time, we obtain the Laplace-Stieltjes transform (LST) of the law of 𝜏_𝑥,𝑦 as 𝐄𝑒^{−𝜆𝜏𝑥,𝑦}=𝔤_𝜆𝑥,𝑦𝔤_𝜆.𝑦,𝑦 (2.31) We have 𝐏(𝜏_𝑥,𝑦<∞)=𝔤(𝑥,𝑦)/(𝔤(𝑦,𝑦))∈(0,1) as a result of both terms in the ratio being finite and 𝑥,𝑦 belonging to the same transience class of the process (under our assumptions that the boundaries are absorbing). Note that from the reversibility property 𝑚(𝑥)𝔤𝑥,𝑦=𝑚𝑦𝔤𝑦,𝑥. (2.32)

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 (𝑥_𝑡;𝑡≥0) as in (2.1) with absorbing barriers. Let 𝑝(𝑥;𝑡,𝑦) be its transition probability, and let 𝜏_𝑥 be its absorption time at the boundaries.

Let 𝛼(𝑥)∶=𝐄(∫^𝜏𝑥₀𝑐(𝑥_𝑠)𝑑𝑠+𝑑(𝑥_𝜏𝑥)) be a nonnegative additive functional solving −𝐺(𝛼)=𝑐if𝑥∈𝐼,𝛼=𝑑if𝑥∈𝜕𝐼. (2.33) Recall the functions 𝑐 and 𝑑 are both chosen nonnegative so that so is 𝛼.

Define a new transformed stochastic process (𝑥_𝑡;𝑡≥0) by its transition probability 𝑝=𝛼𝑦𝑥;𝑡,𝑦𝛼(𝑥)𝑝𝑥;𝑡,𝑦. (2.34) In this construction of (𝑥_𝑡;𝑡≥0) through a change of measure, sample paths of (𝑥_𝑡;𝑡≥0) for which 𝛼(𝑦) is large are favored. This is a selection of paths procedure due to Doob (see [11]).

Now, the KFE for 𝑝 clearly is 𝜕_𝑡𝑝=𝐺^∗(𝑝), with 𝑝(𝑥;0,𝑦)=𝛿_𝑦(𝑥) and 𝐺^∗(𝑝)=𝛼(𝑦)𝐺^∗(𝑝/𝛼(𝑦)). The Kolmogorov backward operator of the transformed process is, therefore, by duality 𝐺(⋅)=1𝛼(𝑥)𝐺(𝛼(𝑥)⋅). (2.35) Developing, with 𝛼^′(𝑥)∶=𝑑𝛼(𝑥)/𝑑𝑥 and 𝐺(⋅)∶=(𝛼^′/𝛼)𝑔²𝜕_𝑥(⋅)+𝐺(⋅), we get 𝐺(⋅)=1𝛼𝐺(𝛼)𝐺(⋅)𝑐⋅+=−𝛼⋅+𝐺(⋅), (2.36) and the new KB operator can be obtained from the latter by adding a drift term (𝛼^′/𝛼)𝑔²𝜕_𝑥 to the one in 𝐺 of the original process to form a new process (𝑥_𝑡;𝑡≥0) with the KB operator 𝐺 and by killing its sample paths at death rate 𝑑(𝑥)∶=(𝑐/𝛼)(𝑥) (provided 𝑐≠0). Note that 𝑑(𝑥)=1(1/(𝑐(𝑥)))∫¹₀𝔤𝑐𝑦.𝑥,𝑦𝑑𝑦 (2.37) In others words, with 𝑓(𝑥)∶=𝑓(𝑥)+(𝛼^′/𝛼)𝑔²(𝑥), the novel time-homogeneous SDE to consider is 𝑑𝑥_𝑡=𝑓𝑥_𝑡𝑑𝑡+𝑔𝑥_𝑡𝑑𝑤_𝑡,𝑥₀=𝑥∈(0,1), (2.38) possibly killed at rate 𝑑=𝑐/𝛼 as soon as 𝑐≠0. Whenever (𝑥_𝑡;𝑡≥0) is killed, it enters conventionally into some coffin state {𝜕} added to the state-space. Let 𝜏_𝑥 be the new absorption time at the boundaries of (𝑥_𝑡;𝑡≥0) started at 𝑥 (with 𝜏_𝑥=∞ would the boundaries be inaccessible to the new process 𝑥_𝑡 which we ruled out). Let 𝜏_𝑥,𝜕 be the killing time of (𝑥_𝑡;𝑡≥0) started at 𝑥 (the hitting time of 𝜕), with 𝜏_𝑥,𝜕=∞ if 𝑐=0. Then, 𝜏_𝑥∶=𝜏_𝑥∧𝜏_𝑥,𝜕 is the novel stopping time of (𝑥_𝑡;𝑡≥0). The SDE for (𝑥_𝑡;𝑡≥0), together with its global stopping time 𝜏_𝑥 characterize the new process (𝑥_𝑡;𝑡≥0) with generator 𝐺 to consider.

In the sequel, we shall limit ourselves to the cases for which the following additional conditions hold on the transformed process.

(i) Nonconservativeness of 𝑥_𝑡.
We will next suppose that the boundaries ∘∶=0 or 1 are both exit (or absorbing) boundaries for the new process 𝑥_𝑡 in (2.38). From the Feller criterion for exit boundaries, this will be the case if forall𝑦₀∈(0,1)𝑚𝑦∉𝐿₁𝑦₀,∘,𝜑^′𝑦^𝑦_𝑦0𝑚(𝑧)𝑑𝑧∈𝐿₁𝑦₀,∘, (2.39) where 𝑚(𝑦)=1/((𝑔²𝜑^′)(𝑦)) is the new speed measure density for 𝑥_𝑡 and 𝜑 its scale function. Recalling 𝑓=𝑓+(𝛼^′/𝛼)𝑔² and 𝑔²=𝑔², we have 𝜑^′𝑦=𝜑^′𝑦₀𝑒^{−2∫𝑦𝑦0(}𝑓_(𝑧)/(𝑔²_{(𝑧)))𝑑𝑧}=𝜑^′𝑦₀𝛼²𝑦𝜑^′𝑦,𝜑(𝑥)=𝜑𝑥₀+𝜑^′𝑦₀^𝑥_𝑥0𝛼⁻²𝑦𝑒^{−2∫𝑦𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧}𝑑𝑦. (2.40) So, we assume here that 𝑥_𝑡 obeys itself a nonconservative diffusion.

(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 (𝑥_𝑡;𝑡≥0) is bounded above by 𝑓_∗=max_𝑥(𝑓(𝑥))>0. (If the drift of (𝑥_𝑡;𝑡≥0) is bounded below by 𝑓_∗<0, we are led to the same conclusions while considering the process 1−𝑥_𝑡 instead of 𝑥_𝑡.) 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, forall𝑥, 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 (𝑎_𝑘;𝑘≥1) be a nonincreasing sequence of [0,1]-valued real numbers. Let (𝛼_𝑘;𝑘≥1) be a sequence of nonnegative real numbers such that for all 𝑥∈(0,1)𝛼(𝑥)=_𝑘≥1𝛼_𝑘𝑒^{−𝑎𝑘𝑥}<∞. (2.41) Whenever 𝑓 is bounded above and, forall𝑥, 2𝑓/𝑔²≥𝑎₁, we have −𝐺(𝛼)=_𝑘≥1𝛼_𝑘𝑎_𝑘𝑎𝑓−^2𝑘2𝑔²𝑒^{−𝑎𝑘𝑥}≤𝑓_∗𝑘≥1𝛼_𝑘𝑎_𝑘𝑒^{−𝑎𝑘𝑥}<𝑎₁𝑓_∗𝑘≥1𝛼_𝑘𝑒^{−𝑎𝑘𝑥}=𝑎₁𝑓_∗𝛼. (2.42) Thus, 𝑑=−𝐺(𝛼)/𝛼 is bounded above by 𝑎₁𝑓_∗.
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 𝑥_𝑡 is nonconservative, it is of interest to inspect various conditionings in the sense of Yaglom, [7].

(i) Consider again the process with infinitesimal generator 𝐺 losing mass due to killing and/or absorption at the boundaries. Integrating over 𝑦, with 𝜌_𝑡(𝑥)∶=∫_𝐼𝑝(𝑥;𝑡,𝑦)𝑑𝑦=𝐏(𝜏_𝑥>𝑡), we have 𝜕_𝑡𝜌_𝑡(𝑥)=𝐺𝜌_𝑡(𝑥)=−𝑑(𝑥)𝜌_𝑡(𝑥)+𝐺𝜌_𝑡(𝑥), (2.43) with 𝜌₀(𝑥)=𝟏(𝑥∈(0,1)). This gives the tail distribution of the full stopping time 𝜏_𝑥.

Defining the conditional probability density 𝑞(𝑥;𝑡,𝑦)∶=𝑝(𝑥;𝑡,𝑦)/𝜌_𝑡(𝑥), now with total mass 1, with 𝑞(𝑥;0,𝑦)=𝛿_𝑦(𝑥), we get 𝜕_𝑡𝜕𝑞=−^𝑡𝜌_𝑡(𝑥)𝜌_𝑡(𝑥)⋅𝑞+𝐺^∗𝑞=𝑏_𝑡(𝑥)−𝑑𝑦⋅𝑞+𝐺^∗𝑞. (2.44) The term 𝑏_𝑡(𝑥)=−𝜕_𝑡𝜌_𝑡(𝑥)/𝜌_𝑡(𝑥)>0 is the rate at which mass should be created to compensate the loss of mass of the process (𝑥_𝑡;𝑡≥0) due to its possible absorption at the boundaries and/or killing. Again, we have 𝑏_𝑡(𝑥)→𝜆₁, where 𝜆₁ is the smallest positive eigenvalue of −𝐺, and therefore, putting 𝜕_𝑡𝑞=0 in the latter evolution equation, we get that independently of the initial condition 𝑥𝑞𝑥;𝑡,𝑦⟶_𝑡→∞𝑞_∞𝑦, (2.45) where 𝑞_∞(𝑦) is the solution to −𝐺^∗𝑞_∞=𝜆₁−𝑑𝑦⋅𝑞_∞,or−𝐺^∗𝑞_∞=𝜆₁⋅𝑞_∞. (2.46) With 𝑣₁ the eigenvector of −𝐺^∗ associated to 𝜆₁, 𝑞_∞(𝑦) is of the product form 𝑞_∞𝑦=𝛼𝑦𝑣₁𝑦𝛼,𝑣₁, (2.47) where ⟨𝛼,𝑣₁⟩=∫¹₀𝛼(𝑦)𝑣₁(𝑦)𝑑𝑦. This results directly from the fact that 𝐺^∗(⋅)=𝛼(𝑦)𝐺^∗(⋅/𝛼(𝑦)) and that 𝑣₁ is the stated eigenvector of −𝐺^∗. A different way to see this is as follows. We have 𝜌_𝑡(𝑥)=1𝛼(𝑥)¹₀𝛼𝑦𝑝𝑥;𝑡,𝑦𝑑𝑦, (2.48) and the conditional density of 𝑥_𝑡 given 𝜏_𝑥>𝑡 is, therefore,𝑞=𝛼𝑦𝑝𝑥;𝑡,𝑦𝑥;𝑡,𝑦∫¹₀𝛼𝑦𝑝.𝑥;𝑡,𝑦𝑑𝑦 (2.49) The rest follows from observing that, to the leading order in 𝑡 in (2.15), for large time 𝑝𝑥;𝑡,𝑦∼𝑏₁𝑒^−𝜆1𝑡⋅𝑢₁(𝑥)𝑣₁𝑦, (2.50) where 𝑢₁ (𝑣₁,resp.) is the eigenvector of −𝐺 (−𝐺^∗,resp.) associated to 𝜆₁ and 𝑏₁=⟨𝑢₁,𝑣₁⟩⁻¹. From this, it is clear that −(1/𝑡)log𝜌_𝑡(𝑥)→_𝑡→∞𝜆₁ and 𝑞∼𝑒𝑥;𝑡,𝑦^−𝜆1𝑡⋅𝛼𝑦𝑣₁𝑦𝑒^−𝜆1𝑡⋅𝛼,𝑣₁=𝑞_∞𝑦. (2.51) The limiting probability 𝑞_∞=𝛼𝑣₁/norm can, therefore, be interpreted as the Yaglom limit law of (𝑥_𝑡;𝑡≥0) conditioned on the event 𝜏_𝑥>𝑡.

(ii) Under our assumptions, in the transformation of paths process, the transformed process (𝑥_𝑡;𝑡≥0) can both be absorbed at the boundaries and be killed. So, both 𝜏_𝑥 and 𝜏_𝑥,𝜕 are finite with positive probability. We wish to understand the processes (𝑥_𝑡;𝑡≥0) conditioned on the events {𝜏_𝑥<𝜏_𝑥,𝜕} or {𝜏_𝑥,𝜕<𝜏_𝑥}, (see [16]).

The probability mass cumulated at the boundaries {0,1} by time 𝑡 clearly is [17] 𝐏𝜏_𝑥≤𝑡=𝐏𝜏_𝑥,𝜕=1>𝑡2^𝑡₀𝑝(𝑥;𝑠,0)+𝑝(𝑥;𝑠,1)𝑑𝑠. (2.52)

As 𝑡→∞, this probability tends to 𝐏(𝜏_𝑥,𝜕=∞)=𝐏(𝜏_𝑥<𝜏_𝑥,𝜕)=∶𝛽(𝑥). Note that 𝛽(𝑥)=𝐏𝑥_𝜏𝑥∈{0,1}=𝐄𝟏𝑥_𝜏𝑥∈{0,1}. (2.53)

Now (assuming 𝑥≠{0,1}), 𝐺𝐏𝜏_𝑥,𝜕=1>𝑡2^𝑡₀𝐺𝑝(𝑥;𝑠,0)+𝐺𝑝()=1𝑥;𝑠,1𝑑𝑠2^𝑡₀𝜕_𝑠𝑝(𝑥;𝑠,0)+𝜕_𝑠𝑝(=1𝑥;𝑠,1)𝑑𝑠2𝑝(𝑥;𝑡,0)+𝑝(𝑥;𝑡,1)⟶_𝑡→∞0. (2.54)

Thus, 𝛽 is defined by 𝐺𝛽(𝑥)=0, (2.55) [or 𝐺(𝛽(𝑥))=𝑑(𝑥)𝛽(𝑥)], with boundary conditions 𝛽(0)=𝛽(1)=1. It serves as a positive harmonic function for 𝐺. This is a Sturm-Liouville problem to be solved for each case study.

The density of the process (𝑥_𝑡;𝑡≥0) conditioned on the event {𝜏_𝑥<𝜏_𝑥,𝜕} is 𝑝_𝑎=𝛽𝑦𝑥;𝑡,𝑦𝛽(𝑥)𝑝=𝛽𝑦𝑥;𝑡,𝑦𝛽(𝑥)𝛼𝑦𝛼(𝑥)𝑝𝑥;𝑡,𝑦. (2.56) The density of the process (𝑥_𝑡;𝑡≥0) conditioned on the event {𝜏_𝑥,𝜕<𝜏_𝑥} is 𝑝_𝜕=𝑦𝑥;𝑡,𝑦1−𝛽1−𝛽(𝑥)𝑝=𝑦𝑥;𝑡,𝑦1−𝛽(𝑥)𝛼𝑦1−𝛽𝛼(𝑥)𝑝𝑥;𝑡,𝑦. (2.57) Note that ¹₀𝑝_𝑎𝑥;𝑡,𝑦𝑑𝑦=𝐏𝑡_𝑥̃>𝑡∣𝜏_𝑥<𝜏_𝑥,𝜕,₁₀𝑝_𝜕𝑥;𝑡,𝑦𝑑𝑦=𝐏𝜏_𝑥,𝜕>𝑡∣𝜏_𝑥,𝜕<𝜏_𝑥, (2.58) and, with 𝜏_𝑥=𝜏_𝑥∧𝜏_𝑥,𝜕,𝐏𝜏_𝑥>𝑡=𝛽(𝑥)𝐏𝜏_𝑥>𝑡∣𝜏_𝑥<𝜏_𝑥,𝜕+1−𝛽(𝑥)𝐏𝜏_𝑥,𝜕>𝑡∣𝜏_𝑥,𝜕<𝜏_𝑥=𝛽(𝑥)¹₀𝛽𝑦𝛽(𝑥)𝑝𝑥;𝑡,𝑦𝑑𝑦+1−𝛽(𝑥)¹₀1−𝛽𝑦1−𝛽(𝑥)𝑝=𝑥;𝑡,𝑦𝑑𝑦₁₀𝑝𝑥;𝑡,𝑦𝑑𝑦, (2.59) as required, because this is the probability that 𝑥_⋅ is neither in {0,1} nor in state 𝜕 at time 𝑡.

Note also that 𝑞_𝑎(𝑥;𝑡,𝑦)=𝑝_𝑎(𝑥;𝑡,𝑦)/∫¹₀𝑝_𝑎(𝑥;𝑡,𝑦)𝑑𝑦[resp.,𝑞_𝜕(𝑥;𝑡,𝑦)=𝑝_𝜕(𝑥;𝑡,𝑦)/∫¹₀𝑝_𝜕(𝑥;𝑡,𝑦)𝑑𝑦] are the transition probability densities of (𝑥_𝑡;𝑡≥0) conditioned on the event {𝜏_𝑥<𝜏_𝑥,𝜕and𝜏_𝑥>𝑡} [of (𝑥_𝑡;𝑡≥0) conditioned on the event {𝜏_𝑥,𝜕<𝜏_𝑥and𝜏_𝑥,𝜕>𝑡,resp.}]. They are the Yaglom limits of both conditioned processes.

The backward infinitesimal generators of both processes with transition probability densities 𝑝_𝑎 and 𝑝_𝜕 are, respectively, given by 𝐺_𝑎(⋅)=1𝛽(𝑥)𝐺𝛽(𝑥)⋅,𝐺_𝜕(⋅)=11−𝛽(𝑥)𝐺1−𝛽(𝑥)⋅. (2.60) We get, respectively, 𝐺_𝑎(⋅)=𝐺(⋅)+𝛽^′𝑔²𝛽(𝑥)𝜕_𝑥(⋅),𝐺_𝜕(⋅)𝑑=−𝐺(⋅)−𝛽1−𝛽⋅+^′𝑔²(𝑥)𝜕1−𝛽^𝑥(⋅). (2.61) Thus, in 𝐺_𝑎(⋅), there is no multiplicative part (no killing) and a shift in the drift, showing that the associated conditioned process (𝑥_𝑎,𝑡;𝑡≥0) obeys the SDE 𝑑𝑥_𝑎=𝑓_𝑎𝑥_𝑎𝑑𝑡+𝑔𝑥_𝑎𝑑𝑤_𝑡, (2.62) with drift 𝑓_𝑎(𝑥)=𝑓(𝑥)+𝛽^′𝑔²𝛽(𝑥)(𝑥)+𝛼=𝑓^′𝑔²𝛼(𝑥)+𝛽^′𝑔²𝛽(𝑥)=𝑓(𝑥)+𝑔²(𝑥𝛼)^′𝛼(𝑥)+𝛽^′𝛽(𝑥). (2.63) This process is ultimately absorbed at {0,1}.

In 𝐺_𝜕(⋅), there is a killing multiplicative part which is enhanced 𝑑/(1−𝛽)>𝑑 and a shift in the drift, showing that the associated conditioned process (𝑥_𝜕,𝑡;𝑡≥0) exhibits a faster killing rate, but the drift shift guarantees that (𝑥_𝜕,𝑡;𝑡≥0) is not absorbed at the boundaries. We have 𝑓_𝜕(𝑥)=𝑓(𝑥)+𝑔²(𝑥𝛼)^′𝛼(𝑥)−𝛽^′(𝑥.1−𝛽) (2.64)

Additive Functionals of the Transformed Process.
for the new process (𝑥_𝑡;𝑡≥0), it is also of interest to evaluate additive functionals along their own sample paths. Let then 𝛼(𝑥)∶=E(∫^𝜏𝑥₀𝑐(̃𝑥_𝑠)𝑑𝑠+𝑑(𝑥_𝜏𝑥)) be such an additive functional where the functions 𝑐̃ and 𝑑 are themselves both nonnegative. It solves −𝐺(𝛼)=𝑐̃if𝑑𝑥∈𝐼,𝛼=if𝑥∈𝜕𝐼. (2.65) Then, recalling the expression of the Green function 𝔤(𝑥,𝑦) of (𝑥_𝑡;𝑡≥0) in (2.22), we find explicitly 𝛼(𝑥)=1𝛼(𝑥)_𝐼𝔤𝑥,𝑦𝛼𝑦𝑐̃𝑦𝑑𝑦. (2.66)

Specific transformations of interest.
(i) The case 𝑐=0 deserves a special treatment. Indeed, in this case, 𝜏_𝑥,𝜕=∞ and so 𝜏_𝑥∶=𝜏_𝑥, the absorption time for the process (𝑥_𝑡;𝑡≥0) governed by the new SDE (2.38). Here, 𝐺=𝐺. 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 (𝑥_𝑡;𝑡≥0) is just (𝑥_𝑡;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 𝛼(𝑥)=∫^𝑥₀𝑒^{−2∫𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧}𝑑𝑦∫¹₀𝑒^{−2∫𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧},𝑑𝑦 (2.67) with 𝑓(𝑥)(𝑥)+𝑔=𝑓²(𝑥)𝑒^{−2∫𝑥0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧}∫^𝑥₀𝑒^{−2∫𝑦0(𝑓(𝑧)/(𝑔2(𝑧)))𝑑𝑧},𝑑𝑦 (2.68) giving the new drift. In the second case, 𝛼(𝑥)=(∫¹_𝑥𝑒^{−2∫𝑦0(𝑓(𝑧)/𝑔2(𝑧))𝑑𝑧}𝑑𝑦/(∫¹₀𝑒^{−2∫𝑦0(𝑓(𝑧)/𝑔2(𝑧))𝑑𝑧}𝑑𝑦)), and the boundary 0 is exit, whereas 1 is entrance. Thus, 𝜏_𝑥 is just the exit time at 𝑥=1 (at 𝑥=0,resp.). Let 𝛼(𝑥)∶=𝐄(𝜏_𝑥). Then, 𝛼(𝑥) solves −𝐺(𝛼)=1, whose explicit solution is 𝛼(𝑥)=1𝛼(𝑥)_𝐼𝔤𝑥,𝑦𝛼𝑦𝑑𝑦, (2.69) in terms of 𝔤(𝑥,𝑦), the Green function of (𝑥_𝑡;𝑡≥0).
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: 𝑓(𝑥)=𝜎𝑥(1−𝑥)coth(𝜎𝑥), 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 (𝑥_𝑡;𝑡≥0) with a large mean absorption time 𝛼(𝑥)=𝐄(𝜏_𝑥). Sample paths with large sojourn time in 𝐼 are favored. We have 𝛼(𝑥)=_𝐼𝔤𝑥,𝑦𝑑𝑦, (2.70) where 𝔤(𝑥,𝑦) is the Green function (2.22). The boundaries of (𝑥_𝑡;𝑡≥0) are now both entrance boundaries and so 𝜏_𝑥=∞. (𝑥_𝑡;𝑡≥0) is not absorbed at the boundaries. The stopping time 𝜏_𝑥 of (𝑥_𝑡;𝑡≥0) is just its killing time 𝜏_𝑥,𝜕. Let 𝛼(𝑥)∶=𝐄(𝜏_𝑥,𝜕). Then, 𝛼(𝑥) solves −𝐺(𝛼)=1, 𝛼(0)=𝛼(1)=0, with explicit solution 𝛼(𝑥)=1𝛼(𝑥)_𝐼𝔤𝑥,𝑦𝛼𝑦𝑑𝑦. (2.71)
(iii) Assume that 𝛼 now solves −𝐺(𝛼)=𝛿_𝑦(𝑥) if 𝑥∈𝐼 with boundary conditions 𝛼(0)=𝛼(1)=0. In this case study, one selects sample paths of (𝑥_𝑡;𝑡≥0) with a large sojourn time density at 𝑦 recalling 𝛼(𝑥)=∶𝔤(𝑥,𝑦)=𝐄(∫^𝜏𝑥₀𝛿_𝑦(𝑥_𝑠)𝑑𝑠). The stopping time 𝜏_𝑦(𝑥) of (𝑥_𝑡;𝑡≥0) occurs at rate 𝛿_𝑦(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 𝛼_𝑦(𝑥)∶=𝐄(𝜏_𝑦(𝑥)). Then, 𝛼_𝑦(𝑥) solves −𝐺(𝛼)=1, with explicit solution 𝛼_𝑦(𝑥)=1𝔤𝑥,𝑦_𝐼𝔤(𝑥,𝑧)𝔤𝑧,𝑦𝑑𝑧. (2.72) when 𝑥=1/𝑁,𝛼_𝑦(1/𝑁) may be viewed as the age of a mutant currently observed to present frequency 𝑦, see [18].
The Green function at 𝑦₀∈(0,1) of the transformed process (𝑥_𝑡;𝑡≥0) is 𝔤_𝑦(𝑥,𝑦₀) solution to −𝐺(𝔤_𝑦)=𝛿_𝑦0(𝑥). It takes the simple form 𝔤_𝑦𝑥,𝑦₀=1𝔤𝑥,𝑦_𝐼𝔤(𝑥,𝑧)𝔤𝑧,𝑦𝛿_𝑦0(𝑧)𝔤𝑦𝑑𝑧=⁰,𝑦𝔤𝔤𝑥,𝑦𝑥,𝑦₀. (2.73)
(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 (𝑥_𝑡;𝑡≥0) is 𝐺(⋅)=1𝛼𝐺(𝛼)⋅+𝐺(⋅)=−𝜆₁⋅+𝐺(⋅), (2.74) obtained while killing the sample paths of the process (𝑥_𝑡;𝑡≥0) governed by 𝐺 at constant death rate 𝑑=𝜆₁. The transition probability of the transformed stochastic process (𝑥_𝑡;𝑡≥0) is 𝑝=𝑢𝑥;𝑡,𝑦¹𝑦𝑢₁(𝑥)𝑝𝑥;𝑡,𝑦. (2.75) Define 𝑝(𝑥;𝑡,𝑦)=𝑒^𝜆1𝑡𝑝(𝑥;𝑡,𝑦). It is the transition probability of the process (𝑥_𝑡;𝑡≥0) governed by 𝐺; it corresponds to the original process (𝑥_𝑡;𝑡≥0) conditioned on never hitting the boundaries {0,1} (the so-called 𝑄-process of (𝑥_𝑡;𝑡≥0), see [19]). It is simply obtained from (𝑥_𝑡;𝑡≥0) by adding the additional drift term (𝑢^′₁/𝑢₁)𝑔² to 𝑓, where 𝑢₁ is the eigenvector of 𝐺 associated to its smallest non-null eigenvalue. The determination of 𝛼=𝑢₁ is a Sturm-Liouville problem. When 𝑡 is large, to the dominant order 𝑝𝑥;𝑡,𝑦∼𝑒^−𝜆1𝑡𝑢₁(𝑥)𝑣₁𝑦𝑢₁,𝑣₁, (2