Transient one-dimensional diffusions conditioned to converge to a different limit point

Let $(X_t)_{t\geq 0}$ be a regular one-dimensional diffusion that models a biological population. If one assumes that the population goes extinct in finite time it is natural to study the $Q$-process associated to $(X_t)_{t\geq 0}$. This is the process one gets by conditioning $(X_t)_{t\geq 0}$ to survive into the indefinite future. The motivation for this paper comes from looking at populations that are modeled by diffusions which do not go extinct in finite time but which go `extinct asymptotically' as $t\rightarrow \infty$. We look at transient one-dimensional diffusions $(X_t)_{t \geq 0}$ with state space $I=(\ell, \infty)$ such that $X_t\rightarrow \ell$ as $t\rightarrow \infty$, $\mathbb{P}^x$-almost surely for all $x\in I$. We `condition' $(X_t)_{t \geq 0}$ to go to $\infty$ as $t\rightarrow \infty$ and show that the resulting diffusion is the Doob $h$-transform of $(X_t)_{t\geq 0}$ with $h=s$ where $s$ is the scale function of $(X_t)_{t\geq 0}$. Finally, we explore what this conditioning does in two examples.


Introduction
Let (X t ) t≥0 be a one dimensional diffusion with state space I := (0, ∞). If one assumes that (X t ) t≥0 models a population, then the hitting time of 0, T 0 := inf{t ≥ 0 : X t = 0}, can be interpreted as the time of extinction. Throughout the paper we take (X t ) t≥0 to be the canonical process on C (I) so that our diffusion is described by a family of probability measures In mathematical biology and ecology it is of interest to know what the distribution of the population looks like before extinction. Consequently, if P x Conditions for the existence and uniqueness of the Q -process have been studied in Collet et al. (1995), Cattiaux et al. (2009) and Champagnat and Villemonais (2014) while some examples of applications to ecology and genetics appear in Lambert (2008) and Etheridge et al. (2013).
One of the simplest ways of modeling a population living in one patch is assuming it is given by a geometric Brownian motion (Y t ) t≥0 , see Evans et al. (2013) and Evans et al. (2015). Then (Y t ) t≥0 can be recovered as the solution to the following where (W t ) t≥0 is a standard Brownian motion; that is, there is exponential growth with a rate that varies stochastically in time.
, t ≥ 0. Starting from any y ∈ (0, ∞) the process (Y t ) t≥0 does not hit 0: the population will not go extinct in finite time. The long term behavior of (1.1) is determined by the stochastic growth rate µ − σ 2 2 .
• If µ − σ 2 2 > 0 then lim t→∞ Y t = ∞, P y -almost surely for all y ∈ (0, ∞). • If µ − σ 2 2 < 0 then lim t→∞ Y t = 0, P y -almost surely for all y ∈ (0, ∞). • If µ − σ 2 2 = 0 then (Y t ) t≥0 is null-recurrent. In the cases where the geometric Brownian motion (Y t ) t≥0 is transient, one type of conditioning is to consider the limit as t → ∞ of the process obtained by conditioning on the event {Y t = a} for some fixed a > 0. It is not hard to see by direct computation using transition densities that this conditional distribution does not depend on µ, that the limit exists, the limit does not depend on a, and the limit is just the (unconditional) distribution of Y with µ − σ 2 2 = 0. In general, if (X t ) t≥0 has transition densities p t (x, y) with respect to some reference measure m, then (X s ) 0≤s≤t conditional on the event {X t = a} is a time-inhomogeneous Markov process with transition densities Under appropriate conditions, the limit exists, is strictly positive, and P x [h u (X w )] = h u+w (x) for all x in the state space and all u ∈ R + ; see Chung and Walsh (2005) and Borodin and Salminen (2002). In such situations, the limit of the conditioned processes is the Doob h-transform process, a time-homogeneous Markov process with transition densities If one adds competition for resources in (1.1) then the population (Ỹ t ) t≥0 is modeled by where κ > 0 is the strength of intraspecific competition. One can show, see Evans et al. (2015), that almost surely the population (Ỹ t ) t≥0 does not go extinct in finite time and that when µ − σ 2 2 < 0 one has Pỹ-almost surely for allỹ ∈ (0, ∞). In the models defined by (1.1) and (1.2) with µ − σ 2 2 < 0 the populations stay positive for all t ≥ 0 and go extinct asymptotically as t → ∞. As a result we felt that it is interesting to study an analogue of the Q -process, where we condition that the population does not go extinct asymptotically.
Example 1.2. Suppose (X t ) t≥0 is a Brownian motion with negative drift −µ < 0. That is, where (W t ) t≥0 is a standard Brownian motion. It is well known that lim t→∞ X t = −∞ P x -almost surely for all x ∈ R. If we condition (X t ) t≥0 on the event {X T ∈ (a, ∞)} and then let T → ∞ and a → ∞ such that a T → c it is known that one gets a process (Z t ) t≥0 that is Brownian motion with drift c, In other words, if we take Brownian motion with negative drift −µ and condition it 'to go to +∞' we get a process that is Brownian motion with positive drift c > 0.
We show, in Theorem 2.2, that when we have a diffusion (X t ) t≥0 on (ℓ, ∞) such that X t → ℓ as t → ∞ almost surely for any starting point x ∈ (ℓ, ∞) and we suitably 'condition'X to go to ∞ as t → ∞ what we get is the h-transform of (X t ) t≥0 with h = s where s is the scale function of (X t ) t≥0 . The conditioning works as follows: we let ζ be an independent exponential with rate λ, condition (X t ) t≥0 on {X ζ − ∈ (a, ∞)}, kill the process at ζ , and then let a → ∞ followed by λ → ∞. Our result is similar to Proposition 3.2 from Perkowski and Ruf (2012

Limits of diffusions conditioned to go to infinity at their terminal times
We denote by C (I), C b (I) the continuous and the continuous bounded functions on I and by B(I) the Borel subsets of I.
A diffusion is a strong Markov process with continuous paths. For the sake of completeness we present some basic facts about diffusions in Appendix A.
Remark 2.1. Suppose the diffusion (X t ) t≥0 has null killing measure and scale and speed measures that are absolutely continuous with respect to Lebesgue measure If furthermore, one assumes m ′ ∈ C (I) and s ′ ∈ C 1 (I) then one can show that the infinitesimal generator G : D(G)  → C b (I) of (X t ) t≥0 is a second order differential operator The following theorem is a generalization of Example 1.2. A similar result appears in Proposition 3.2 of Perkowski and Ruf (2012).
Theorem 2.2. Let (X t ) t≥0 be a regular one-dimensional diffusion on (ℓ, ∞) with semigroup (P t ) t≥0 , scale function s and killing measure k ≡ 0. We make the following assumptions: • The boundary points {ℓ, ∞} are inaccessible.
• (X t ) t≥0 is transient and lim t→∞ X t = ℓ, P x -almost surely for all x ∈ (ℓ, ∞). • ζ is an independent exponential random variable with rate λ.
Kill (X t ) t≥0 at ζ and condition on {X ζ − ∈ (a, ∞)}. The limit as a → ∞ followed by the limit as λ ↓ 0 of the conditioned killed process is a diffusion (Z t ) t≥0 with semigroup (Q t ) t≥0 given by Furthermore if the generator of (X t ) t≥0 acts on functions with compact support in C 2 ((ℓ, ∞)) as where w ∈ I is an arbitrary reference point.

Remark 2.3.
We note that we get the same result in Theorem 2.2 if we take limits in the other order, namely if we let λ ↓ 0 followed by a → ∞.
Proof. Suppose first that (X t ) t≥0 is a one-dimensional diffusion in natural scale. By Theorem V.50.7 from Rogers and Williams (2000), the Green function (or the density of the resolvent of (X t ) t≥0 against the speed measure) is given by for a constant c λ and certain functions ψ + λ and ψ − λ . Suppose that A = (a, ∞) for some a and that ζ is an independent exponential random variable with rate λ > 0. Using the technical result, Corollary B.12 from Appendix B, we know that Assume that x < a. Then, using (2.3) and (2.2) we havē (2.4) We know that where T z is the hitting time of z. We have assumed that we are working in natural scale, but we see that this last quantity does not depend on that assumption. By assumption we have a process on the interval (ℓ, ∞) such that the endpoints are inaccessible and X t → ℓ as t → ∞. Then, by the definition of the scale function s, with similar formulas for the other hitting probabilities. So, for the original process, . (2.6) The assumption that (X t ) t≥0 wanders off to the left boundary point ℓ implies that lim u↓ℓ s(u) ̸ = −∞ and so we can assume that the scale function is chosen so that this limit is 0. Eq. (2.6) becomes . (2.7) Our limit semigroup is therefore Note that this is just the h-transform with h := s of the semigroup of (X t ) t≥0 . If (X t ) t≥0 has generator G acting on functions with compact support in C 2 ((ℓ, ∞)) as then the scale function will be while the speed measure will have density where w is an arbitrary reference point. Note that for all x ∈ (ℓ, ∞). According to Theorem B.13 from Appendix B we see that the generator associated with the semigroup (Q t ) t≥0 acts on functions with compact support in C 2 ((ℓ, ∞)) as Remark 2.4. If the diffusion (X t ) t≥0 is a strictly positive local martingale then one can see that on I = (0, ∞) the following conditions hold • The boundary points 0, ∞ are inaccessible.
• lim t→∞ X t = 0, P x -almost surely for all x ∈ I.
The last condition holds because of the following argument: A strictly positive continuous local martingale X is a supermartingale since X is bounded below by 0. Also, if By Doob's first martingale convergence theorem we get that This fact combined with (2.8) yields that Note that we can get diffusions that are strictly positive local martingales using the following proposition.
The stochastic differential equation has a solution for each y ∈ (ℓ, r) that does not explode and is unique in law. Then (Y t ) t≥0 is a regular diffusion with scale function density and speed measure density given by (2.11) By Remark 2.6 and Theorem 2.2 we have the following corollary.
Corollary 2.7. Let (X t ) t≥0 be the solution to the one dimensional stochastic differential equation

, ∞ inaccessible boundary points and such that
σ 2 (·) are locally integrable on I.
• ζ is an independent exponential with rate λ. If we condition (X t ) t≥0 on {X ζ − ∈ (a, ∞)} for a ∈ (ℓ, ∞), kill the process at ζ , and let a → ∞ followed by λ ↓ 0 we get a diffusion (Z t ) t≥0 that can be represented as the solution to the SDE

Brownian motion with negative drift
We do this type of construction with Brownian motion that has a negative drift −µ < 0 Note that in this case lim t→∞ X t = −∞ P x -almost surely for all x ∈ (−∞, ∞) and ℓ = −∞. Since b(x) = −µ and σ (x) = 1 the scale function will be given by As a result and the driftb of the conditioned limiting process

An example from population dynamics
Next, let us look what happens to the diffusion This SDE models the total population abundance of a species living in one patch. The intuitive meaning of the coefficients is the following • µ is the intrinsic growth rate of the population in the absence of stochasticity, • κ is the strength of intraspecific competition, • σ 2 is the infinitesimal variance parameter of the stochastic growth rate.
We consider the case when µ − σ 2 2 < 0. When this condition is satisfied one knows that the SDE has a strong solution which does not explode, satisfies X t > 0 for all t ≥ 0 and goes asymptotically to zero lim t→∞ X t = 0 P x -almost surely for all x ∈ R + . See for example . Using the construction above with ℓ = 0, b(z) = µz − κz 2 and σ (z) = σ z we get that the scale function is The logarithmic derivative of s(x) will be given by As a result the drift of the conditioned diffusion (Z t ) t≥0 is Let us study the asymptotics as x → ∞ ofb(x). First look at Do the change of variables As a result This implies that the limiting diffusion (Z t ) t≥0 looks, for large values of (Z t ) t≥0 , like dȲ t = (µȲ t + κȲ 2 t ) dt + σȲ t dW t .
Remark 2.8. One can show that the process (Ȳ t ) t≥0 explodes in finite time.
Let us next study the asymptotics as x → 0 ofb(x).
so that the process (Ẑ t ) t≥0 is not going to go to zero. This together with Remark 2.8 show that the process (Z t ) t≥0 explodes in finite time.

Acknowledgments
The author thanks Johannes Ruf, Steve Evans and Alison Etheridge for helpful discussions. A.H. was supported by EPSRC grant EP/K034316/1.

Appendix A
We follow Borodin and Salminen (2002) for some basic facts about diffusions. In this paper we only consider regular diffusions; that is, diffusions such that for all x, y ∈ I P x {T y < ∞} > 0 where T y := inf{t : X t = y}-any state y can be reached in finite time with positive probability from any state x.
The diffusion (X t ) t≥0 determines three basic Borel measures on the state space I: a scale measure s, a speed measure m, and a killing measure k (see Itô and McKean, 1974). From now on we will consider that there is no killing, that is k ≡ 0. It turns out to be convenient not to specify these objects absolutely but only up to a constant. If (s * , m * ) and (s * * , m * * ) are two pairs of these objects, then s * * = cs * for some strictly positive constant c, in which case m * * = c −1 m * . The scale measure s is diffuse. Both the scale measure and the speed measure have full support and assign finite mass to intervals of the form (y, z), where ℓ < y < z < r. If (P X t ) t≥0 is the transition semigroup of (X t ) t≥0 , then there exists a density p that is strictly positive, jointly continuous in all variables, and symmetric such that With a standard abuse of notation, as well as using s to denote the scale measure we write s for any scale function such that For α > 0 the Green function r α (x, y) is given by (t; x, y) dt, where p(t; x, y) is the transition density with respect to the speed measure m. Set for a function f : (a, b) → R.
The diffusion (X t ) t≥0 determines and in turn is determined by its infinitesimal generator. The infinitesimal generator is specified by the scale, speed and killing measures and by boundary conditions on functions in the domain.
Definition A.10. The (weak) infinitesimal generator of (X t ) t≥0 is the operator G defined by together with boundary conditions. See Borodin and Salminen (2002) for more details.

Appendix B
Let (X t , Ω, F , P x , θ t , (F t )) be a strong Markov process with state space a locally compact metric space E. We let Ω be the space of functions ω : R +  → E ∂ which are right continuous, admit an almost surely finite terminal time ζ < ∞, and have left limits. We can define (X t ) t≥0 on this probability space by X t (ω) = ω(t), ω ∈ Ω, t ≥ 0. The probability space Ω comes equipped with the family of shift operators (θ t ) t≥0 , where for any fixed t ≥ 0, θ t : Ω → Ω and (θ t ω)(s) = ω(s + t) for all ω ∈ Ω and s ≥ 0.
We let F 0 t be the natural filtration on Ω: t and for an initial law µ let F µ denote the completion of F 0 relative to P µ and let N µ denote the P µ -null sets in F µ .
Define then The process X will be described by the probability family (P x ) x∈E for which We assumed X has an almost surely finite terminal time ζ < ∞. This means that ζ has the property ζ = s + ζ • θ s , on the event {ζ > s}. In other words, if the process has not died by time s, then the decision about when to die comes from looking at the future piece of path as though we are starting at time zero.
We call a function f : E → R + ∪ {∞} excessive if the following two conditions are satisfied for all x ∈ E. Theorem B.11. Letf : E → R + be excessive. The operators (P t ) t≥0 defined as define a submarkovian semigroup for a family of probability measures (Q x ) x∈I on Ω. The process X is strong (sub)Markov under (Q x ) x∈I .

Proof.
Sincef is excessive we can use Theorem 11.9 from page 325 in Chung and Walsh (2005) to say that under (Q x ) x∈I the process X is a right-continuous killed strong Markov process which has left limits except possibly at its death time.
The next result shows that the process X conditioned to be in a set A ⊂ E right before the terminal time ζ is strong Markov.
Corollary B.12. Assume that ζ is an independent exponential with rate λ > 0 and that A ⊂ E is such that P x {X ζ − ∈ A} > 0 for all x ∈ E. Then under the probability family defined by X is a strong Markov process with transition semigroup (B.13) Proof. Let f (x) := P x {X ζ − ∈ A} and note that where U λ is the λ-resolvent of X . Since 1 A is a bounded positive function we can apply Proposition 2 from page 46 of Chung and Walsh (2005) to conclude thatf is λ-excessive for X . As a resultf is excessive for X killed at the terminal time ζ . The result now follows by applying Theorem B.11.
The following result tells us how the characteristics of a diffusion change under an h-transform.
Theorem B.13. Let (X t ) t≥0 be a regular, transient diffusion living on I with null killing measure, speed measure m and scale function s. Suppose that h is a strictly positive excessive function such that h(x 0 ) = 1 for some x 0 in the state space and that the boundary points of I are inaccessible. The Doob h-transform of (X t ) t≥0 is a regular diffusion (X h t ) t≥0 with the following characteristics: • Scale measure s h (dy) = h −2 (y) s(dy). (B.14) • Speed measure m h (dy) = h 2 (y) m(dy).

(B.15)
• If m ′ ∈ C (I) and h, s ′ ∈ C 1 (I) the generator of (X h t ) t≥0 acts on functions with compact support in C 2 ((ℓ, ∞)) as Proof. For a proof of this fact see Evans and Hening (in preparation). This result also seems to be known in the folk-lore but we were not able to find a proof for the general result.