2.1. Hypothesis

We will consider two different sets of assumptions, including or rejecting the possibility for deleterious mutations to invade the population.

First, the assumptions [H] below are generally in force throughout the paper, although sometimes some of them may not be involved (we will mention when this is the case):

1. [H1] The function $f\in \mathcal{C}^0\big( \mathbb{R}_+^*, \,\mathbb{R}_+\big)$ is positive.

2. [H2] The function $g\in \mathcal{C}^0\big( \mathbb{R}^d \times \mathbb{R}^d, \,\mathbb{R}_+\big)$ is bounded on any $K\times \mathbb{R}^d$ , where K is a compact set of $\mathbb{R}^d$ .

3. [H3] The function r is locally Lipschitz continuous on $\mathbb{R}^d$ , and r(x) tends to $ -\infty$ as $\|x\|$ tends to $\infty$ .

4. [H4] We have $\nu\big(\mathbb{R}^d\big) < \infty$ . Moreover, there exist $\theta, \nu_\wedge > 0$ and $\eta \in (0, \theta)$ such that

   $$\nu(dw) \ge \nu_\wedge \; \mathbf{1}_{B(\theta + \eta)\setminus B(\theta - \eta)}\; dw,$$
   where B(R), for $R>0$ , denotes the open ball of radius R centered at the origin.

5. [H5] Provided $d\ge 2$ , $\nu(dw)<< dw$ , and the density $g(x, w)\, \nu(w)$ (for a jump from x to $x+w$ ) of the jump size law with respect to the Lebesgue measure satisfies

   $$ \begin{equation*}\forall \,x_\vee>0,\quad \sup \left \lbrace \dfrac{g(x, w)\, \nu(w)}{\int_{\mathbb{R}^d} g(x, w^{\prime})\, \nu(w^{\prime})\, dw^{\prime}};\,\|x\| \le x_\vee,\, w\in \mathbb{R}^d \right \rbrace < \infty.\end{equation*}$$

When we allow deleterious mutations to invade the population, we actually mean that the rate of invasion is always positive, leading to the following assumption:

1. [D] The function g is positive.

Otherwise, we consider the case where deleterious mutations are forbidden, in the sense that the rate is zero for mutations that would induce an increase in $\|X\|$ . The invasion rate of advantageous mutations, however, is still assumed to be positive. This is stated in Assumption [A], below, as the alternative to Assumption [D]:

1. [A] For any $x, w\in \mathbb{R}^d$ , $\|x+w\| < \|x\| $ implies $g(x, w) > 0$ , while $\|x+w\| \ge \|x\|$ implies $g(x, w) = 0.$

2.2. Statement of the main theorems

First we need to ensure that the model specified by Equation (S) properly defines a unique solution. This is stated in the next proposition.

Proposition 2.1. Suppose that the assumptions [H] hold. Then, for any initial condition $(x, y) \in \mathbb{R}^d\times \mathbb{R}_+^*$ , there is a unique strong solution $\big(X_t, Y_t\big)_{t\ge 0}$ in the Skorokhod space satisfying (S) for any $t < \tau_\partial$ , and $X_t = Y_t = 0$ for $t\ge \tau_\partial$ , where the extinction time is expressed as $\tau_\partial\,:\!=\, \textstyle{\sup_{\{n\ge 1\} } }\, T_Y^n\wedge \textstyle{\sup_{\{n\ge 1\} } }\, T_X^n,$ where

$$T_Y^n\,:\!=\, \inf\{t \ge 0, \; Y_t \le 1/n \} \quad \text{ and } \quad T_X^n\,:\!=\, \inf\{t \ge 0, \; \|X_t\| \ge n\}.$$

We exploit the notion of uniform exponential quasi-stationary convergence as previously introduced in [Reference Velleret32, Section 2.3].

Definition 1. For any linear, positive, and bounded semigroup $(P_t)_{t\ge 0}$ acting on a Polish state space $\mathcal{Z}$ , we say that P displays a uniform exponential quasi-stationary convergence with characteristics $(\alpha, h, \lambda) \in \mathcal{M}_1(\mathcal{Z})\mathbin{\! \times \!} B(\mathcal{Z})\mathbin{\! \times \!} \mathbb{R}$ if $\langle \alpha \, \big| \, h\rangle = 1$ and there exist $C, \gamma>0$ such that for any $t>0$ and for any measure $\mu\in \mathcal{M}(\mathcal{Z})$ with $\left\| \mu\right\|_{TV}\le 1$ ,

(2.1) $$ \begin{equation} \left\| e^{\lambda t} \mu P_t(ds) - \langle \mu\, \big| \, h\rangle \alpha(ds)\right\|_{TV} \le C e^{-\gamma t}. \end{equation}$$

Our main theorem is stated as follows, with $\mathcal{Z}\,:\!=\, \mathbb{R}^d\mathbin{\! \times \!} \mathbb{R}_+^*$ .

In [Reference Velleret3a, Section 2.3.2] there is also an analysis of the so-called Q-process, whose properties are as follows.

Remarks.

• ⋆ For the total variation norm, it is equivalent to consider either (X, Y) or (X, N).

• ⋆ The constant $\langle \mu\, \big| \, 1/h\rangle $ in (2.3) is optimal up to a factor of 2, in the sense that for any $u>0$ , we have $\|\mu - u\,\alpha\|_{1/h} \ge \|\mu - \langle\mu\, \big| \, 1/h \rangle \beta\|_{1/h} / 2$ (cf. [Reference Velleret32, Fact 2.3.8]).

• ⋆ Since r tends to $-\infty$ as $\|x\|$ tends to $\infty$ , it is natural to assume that mutations leading X to be large have a very small probability of fixation. Notably, it means that we strongly expect the upper bound of g in [H2], uniform over w.

• ⋆ Under Assumption [A], one may expect the real probability of fixation g(x, w) to be at most of order $O(\|w\|)$ for small values of w (and locally in x). In such a case, we can allow $\nu$ to satisfy a smaller integrability condition than [H4] while forbidding observable accumulation of mutations.

Proof of Corollary 2.1. (X, Y) is a solution of (S) if and only if it is a solution of

where $\widetilde M$ is a Poisson random measure of intensity $ds \; \widetilde \nu(dw) \; du_f\; d\widetilde{u_g}$ , with

$$ \begin{equation*}\widetilde \nu(dw)\,:\!=\, \nu(dw)/(\|w\|\wedge 1),\quad \widetilde \varphi( x,\, y,\, w,\, u_f,\, \widetilde{u_g})= \varphi ( x,\, y,\, w,\, u_f,\, \widetilde{u_g}\cdot(\|w\|\wedge 1)),\end{equation*}$$

and with $\widetilde \varphi$ defined as $\varphi$ with g replaced by $\widetilde g$ .

Thanks to the condition on $\nu$ , [H4] holds with $\widetilde \nu$ instead of $\nu$ . Thanks to the condition on g, [H2] still holds with $\widetilde g$ instead of g. Assumptions [A] and [H5] are equivalent for the systems $(g,\nu)$ and $(\widetilde g, \widetilde \nu)$ . Consequently, if we prove Theorem 2.1 and Theorem 2.2 with [H2] and [H4], the results follow under the assumptions of Corollary 2.1.

2.3. Eco-evolutionary implications of these results

One of the major motivations for the present analysis is to make a distinction, as rigorously as possible, between an environmental change to which the population can spontaneously adapt and a change that imposes too much pressure. We recall that in [Reference Nassar and Pardoux24], the authors obtain a clear and explicit threshold for the speed of this environmental change. Namely, above this threshold, the Markov process that they consider is transient, whereas below the threshold it is recurrent. Thus, it might seem a bit frustrating that such a distinction (depending on the speed value v) cannot be observed in the previous theorems. At least, these results prove that the distinction is not based on the existence or the uniqueness of the QSD, nor even on the exponential convergence per se.

In fact, the reason why this threshold is so distinct in [Reference Nassar and Pardoux24] is that the model of [Reference Nassar and Pardoux24] is based on the following underlying assumption: the poorer the current adaptation is, the more efficiently mutations are able to fix, provided that they are then beneficial. In our case, a population that is too poorly adapted is almost certainly doomed to rapid extinction, because the population size cannot be maintained at large values. Instead, long-term survival is triggered by dynamics that maintain the population as adapted. Looking back at the history of surviving populations, it is likely that the process was mostly kept confined outside of deadly areas.

In order to establish this distinction between environmental changes that are sustainable and those that endanger the population, we need a criterion that quantifies the stability of such core regions. Our results provide two exponential rates whose comparison is enlightening: if the extinction rate is of the same order as the convergence rate, or larger, this means that the dynamics is strongly dependent upon the initial condition. If the convergence is much faster, the dynamics will rapidly become similar, regardless of the initial condition. At least, this is the case for initial conditions that are not too risky (i.e. where h is not too small). This criterion takes into account the intrinsic sustainability of the mechanisms involved in the adaptation to the current environmental change, but does not involve the specific initial state of adaptation.

Looking at the simulation results, the convergence in total variation indeed appears to happen at some exponential rate, provided that extinction does not abruptly wipe out a large part of the distribution at a given time. However, it appears computationally expensive and not very meaningful to use the decay in total variation to obtain a generic estimate of the exponential rate at which the effect of the initial condition is lost. Although they are not as clearly justified, it seems more practical to exploit the decay in time of the correlations of X and/or N starting from the QSD profile. On the other hand, it does not seem very difficult to compare the extinction rate from this estimate. This is especially true in the case where $\mathcal{X}$ is of dimension one, as one can directly estimate the dynamics of the density and thus the extinction rate. Furthermore, it is quite reassuring to see that the choice between including and forbidding deleterious mutations (for which the invasion probability is expected be positive but very small) is not crucial in the present proof. We do not see much difference when looking at the simulations.

Much more can be said if we look at the simulation estimates of the QSD, the quasi-ergodic distribution (QED), and the survival capacity. We plan to detail these simulation results in a later article, but let us already give some insights into the comparison between the QSDs and the QEDs provided in Appendix B.

We see that although the QSDs look very different at the three different values of mutation rates, the QEDs are in fact very similar. When extinction plays a notable role, a tail appears on the QSD from the area of concentration of the QED to an area where the population size is close to zero. From the shape of the tail and the fact that it does not appear on the QED or for larger mutation rate, we infer that it corresponds in some sense to a path towards rapid extinction.

These regions are clearly more unstable than the core areas where the Q-process is kept confined. This is probably due to this decline in population size when the level of maladaptation becomes more pronounced. This confinement caused by conditioning upon survival only weakens in the recent past. It is noticeable that the QSD may give mass to conditions (x, n) most likely leading to extinction, provided the delay is sufficiently large before extinction actually occurs.

2.4. Quasi-ergodicity of related models

The current paper completes the illustrations given in Subsection 4.2 of [Reference Velleret31] and Sections 4–5 of [Reference Velleret32]. Supposing the model of the current paper was in fact the original motivation for the techniques presented in those two papers, we can focus more closely on each of the difficulties identified thanks to these illustrations. In each of them, the adaptation of the population to its environment is described by some process X which is a solution of some stochastic differential equation of the form

$$ \begin{equation*}X_t = x - \int_0^t V_s\, ds + \int_0^t \Sigma_s \cdot dB_s+ \int_{[0,t] \times \mathbb{R}^d \times \mathbb{R}_+ }w \; \mathbf{1}_{\left \lbrace u \le U_s(w)\right \rbrace}\, M(ds, dw, du),\end{equation*}$$

where B is an $\mathcal{F}_t$ -adapted Brownian motion and M an $\mathcal{F}_t$ -adapted Poisson random measure. A priori, $V_s$ and $\Sigma_s$ depend on $X_s$ , while $U_s$ depends on $X_{s-}$ and possibly on a coupled process $N_t$ describing the population size. Like the product $f(Y_s)\, g(X_{s-}, w)$ in Equation (S), one specifies in $U_s(w)$ the rate at which a mutation of effect w invades the population at time s. The quantity $V_s$ relates both to the speed of the environmental change and to the mean effects of the mutations invading the population at time s in a limit of very frequent mutations of very small effects. The quantity $\Sigma_s$ then relates to the undirected fluctuations both in the environment and in the effects of this large number of small fixing mutations.

We can relate the current coupling of X and N to an approximation given by the autonomous dynamics of a process Y similar to X. For the approximation to be as valid as possible, the law of Y should be biased by some extinction rate (depending at time t on the value $Y_t$ ), and its jump rate should be adjusted. By these means, we implicitly account for what would be the fluctuations of N if X were around the value of $Y_t$ . This approximation is particularly reasonable when the characteristic fluctuations of N around its quasi-equilibrium are much quicker than the effect of the growth rate changing over time with the adaptation. Its validity is less clear when the extinction has a strong effect on the establishment of the quasi-equilibrium.

The exponential quasi-stationary convergence is treated in Subsection 4.2 of [Reference Velleret31] for a coupling (X, N) that behaves as an elliptic diffusion, while Sections 4–5 of [Reference Velleret32] deal with some cases of a biased autonomous process Y that behaves as a piecewise deterministic process. For such a process with jumps, it is manageable yet technical to deal with restrictions on the allowed directions or sizes of jumps, while requiring $V_t$ to stay at zero actually makes the proof harder than choosing $V_t \,:\!=\, v\cdot t$ .

In the current article, we treat the following two technical difficulties. Firstly, we handle the combination of techniques specific to diffusion with those specific to piecewise deterministic processes. Secondly, we treat more general restrictions on the jump effects, possibly multidimensional, even in a case where a region of the state space is transitory.

While the proofs of (A1) and (A3) strongly depend on such local properties of the dynamics, those of (A2) for these semigroups rely on a common intuition. Although we allow X to live in an unbounded domain, the maladaptation of the process when it is far from the optimal position constrains X to be kept confined conditionally upon survival. This effect of the maladaptation has been modeled either directly on the growth rate of the coupled process $N_t$ or using some averaged description in terms of extinction rate. Such a confinement property for the coupled process is in fact the main novelty of [Reference Velleret31] and is notably illustrated in [Reference Velleret31, Subsection 4.2.4].

For simplicity, we dealt there with a locally elliptic process, for which the Harnack inequality is known to greatly simplify the proof, as observed previously for instance in [Reference Champagnat and Villemonais13]. The proof of this confinement is actually simpler with Y behaving as an autonomous process under the pressure of a death rate, provided this rate goes to infinity outside of compact sets (by adapting the proof of (A2) from [Reference Velleret31, Subsection 4.1.2]).

Assume for now that the fluctuations of N are much quicker than the change of the growth rate in the domain where the population is well adapted. Then we conjecture that considering the autonomous process Y (including the bias by the extinction rate) instead of the coupled process (X, N) would produce very similar results: the extinction rates and the rates of stabilization to equilibrium in the two models should be close, while the QSD profile of X should be similar to that of Y.

The drop in the quality of the approximation when extinction has a crucial contribution can have only a limited effect for our purpose, which is to compare the extinction rate to the rate of stabilization to equilibrium; see Subsection 2.3. Indeed, as long as the extinction rate is not considerably larger than the rate of stabilization to equilibrium, such domains of maladaptation are strongly avoided when looking at the past of surviving populations. On the other hand, the population is almost certainly doomed when it enters these domains, so that we should be able to neglect the contribution to the extinction rate of the dynamics of the process there.

2.5. The mathematical perspective on quasi-stationarity

The subject of quasi-stationarity is now quite vast, and a considerable literature is dedicated to it, as suggested by the bibliography collected by Pollett [Reference Pollett28]. Some insights into the subject can be found in general surveys like [Reference Collet, Martnez and San Martn9] or [Reference Van Doorn and Pollett29], or more specifically for population dynamics in [Reference Méléard and Villemonais23]. However, it appears that much remains to be done for the study of strong Markov processes both on a continuous space and in continuous time, without any property of reversibility. For general recent results, apart from [Reference Velleret31] and [Reference Velleret32], which we exploit, we refer the reader to [Reference Champagnat and Villemonais14], [Reference Bansaye, Cloez, Gabriel and Marguet2], [Reference Cloez and Gabriel8], [Reference Ferré, Rousset and Stoltz18], or [Reference Guillin, Nectoux and Wu19]. The difficulty is increased when the process is discontinuous (because of the jumps in X) and multidimensional, since the property of reversibility becomes all the more stringent and new difficulties arise (cf. e.g. [Reference Chazottes, Collet and Méléard6, Appendix A]).

Thus, ensuring the existence and uniqueness of the QSD is already a breakthrough, and we are even able to ensure an exponential rate of convergence in total variation to the QSD, as well as similar results on the Q-process. This model is in fact a very interesting illustration of the new technique which we exploit. Notably, we see how conveniently our conditions are suited for exploiting the Girsanov transform as a way to disentangle couplings (here between X and N, which are respectively the evolutionary component and the demographic one).

Our approach relies on the general result presented in [Reference Velleret32], which, as a continuation of [Reference Velleret31], was originally motivated by this problem. In [Reference Velleret31], the generalization of the Harris recurrence property at the core of the results of [Reference Champagnat and Villemonais10] is extended to deal with exponential convergence which is not uniform with respect to the initial condition. The fine control thus obtained over the MCNE opens the way for the approach developed in [Reference Velleret32] to deal with continuous-time and continuous-space strong Markov processes with discontinuous trajectories.

After their seminal article [Reference Champagnat and Villemonais10], Champagnat and Villemonais obtained quite a number of extensions, for instance with multidimensional diffusions [Reference Champagnat, Coulibaly-Pasquier, Villemonais, Donati-Martin, Lejay, Rouault and Springer7], processes that are inhomogeneous in time [Reference Champagnat and Villemonais11], and various examples of processes in a countable space, notably with the use of Lyapunov functions; cf. [Reference Champagnat and Villemonais13] or [Reference Champagnat and Villemonais14]. Exploiting the result of [Reference Champagnat and Villemonais14], it may be possible to ensure the properties of exponential quasi-ergodicity for a discontinuous process such as that of this article, keeping a certain dependence on the initial condition. At least, the conditions they provide as well as the ones from [Reference Bansaye, Cloez, Gabriel and Marguet2] are necessarily implied by our convergence result (cf. [Reference Champagnat and Villemonais12, Theorem 2.3] or [Reference Bansaye, Cloez, Gabriel and Marguet2, Theorem 1.1]). Yet, in the approach of [Reference Champagnat and Villemonais14] for continuous-time and continuous-space Markov process, the rather abstract assumption (F3) appears tightly bound to the Harnack inequality. The similar Assumption (A4) in [Reference Bansaye, Cloez, Gabriel and Marguet2] is also left without further guidance, while the assumption of a strong Feller property in [Reference Ferré, Rousset and Stoltz18] and [Reference Guillin, Nectoux and Wu19] appears too restrictive. For discontinuous processes, these two properties generally do not hold true, which is what motivated us to look for an alternative statement in [Reference Velleret32]. This technique is very efficient here.

This dependence on the initial condition is biologically expected, although its crucial importance becomes apparent when the population is already highly susceptible to extinction. For a broader comparison of this approach with the general literature, we refer to the introduction of [Reference Champagnat and Villemonais14] and the comparison with the literature provided in [Reference Velleret31] and [Reference Velleret32].

4.1. General criteria for the proof of exponential quasi-stationary convergence

The proof of Theorem 2.1 relies on the set of assumptions $\mathbf{(AF)}$ presented in [Reference Velleret32], which we recall next. The assumptions $\mathbf{(AF)}$ are stated in the general context of a process Z that is right-continuous with left limits (càdlàg) on a Polish state space $\mathcal{Z}$ , with extinction at a time still denoted by $\tau_\partial$ . The notation is changed from that of [Reference Velleret32] to prevent confusion with the current notation, Z corresponding now to the couple (X, Y).

We introduce the following notation for the exit and first entry times for any set $\mathcal{D}$ :

(4.1) $$ \begin{equation} T_{\mathcal{D}}\,:\!=\, \inf\left \lbrace t \ge 0;\, Z_t \notin \mathcal{D} \right \rbrace,\quad \tau_\mathcal{D}\,:\!=\, \inf\left \lbrace t \ge 0;\, Z_t \in \mathcal{D} \right \rbrace.\end{equation}$$

The assumptions involved in $\mathbf{(AF)}$ are the following:

1. (A0) There exists a sequence $(\mathcal{D}_\ell)_{\ell\ge 1}$ of closed subsets of $\mathcal{Z}$ such that for any $\ell\ge 1$ ,

   $ \mathcal{D}_\ell \subset int(\mathcal{D}_{\ell+1})$ (with $int(\mathcal{D})$ denoting the interior of $\mathcal{D}$ ).

2. (A1) There exists a probability measure $\zeta \in \mathcal{M}_1(\mathcal{Z})$ such that, for any $\ell\ge 1$ , there exist $L>\ell$ and $c, t>0$ such that

   $$ \begin{equation*} \forall \,z \in \mathcal{D}_{\ell},\; \hspace{.5cm} \textrm{P}_z \left[ {Z}_{t}\in dx;\, t < \tau_\partial \wedge T_{\mathcal{D}_L} \right] \ge c\; \zeta(dz). \end{equation*}$$

3. (A2) We have $\textstyle{\sup_{\{z\in \mathcal{Z}\} } }\, \textrm{E}_{z} \left( \exp\left[\rho\, (\tau_\partial\wedge \tau_E) \right] \right) < \infty.$

4. (A3 _F ) For any $\epsilon\in (0,\, 1)$ , there exist $t_{\barwedge}, c >0$ such that for any $z \in E$ there exist two stopping times $U_H$ and V with the property

   (4.2) $$ \begin{equation} \textrm{P}_{z} \big(Z(U_H) \in dz^{\prime};\, U_H < \tau_\partial \big) \le c \,\textrm{P}_{\zeta} \big(Z(V) \in dz^{\prime} ;\, V < \tau_\partial\big), \end{equation}$$
   as well as the following conditions on $U_H$ : $\left \lbrace\tau_\partial \wedge t_{\barwedge} \le U_H \right \rbrace = \left \lbrace U_H = \infty\right \rbrace$ , and
   (4.3) $$ \begin{equation} \textrm{P}_{z} \big(U_H = \infty, \, t_{\barwedge}< \tau_\partial\big) \le \epsilon\, \exp\big({-}\rho\, t_{\barwedge}\big). \end{equation}$$
   We further require that there exist a stopping time $U_H^\infty$ extending $U_H$ in the following sense:

• ⋆ We have $U_H^\infty\,:\!=\, U_H$ on the event $\left \lbrace \tau_\partial\wedge U_H < \tau_E^1\right \rbrace$ , where $\tau_E^1\,:\!=\, \inf\{s\ge t_{\barwedge}: Z_s \in E\}$ .

• ⋆ On the event $\left \lbrace \tau_E^1 \le \tau_\partial \wedge U_H \right \rbrace$ and conditionally on $\mathcal{F}_{\tau_E^1}$ , the law of $U_H^\infty - \tau_E^1$ coincides with that of $\widetilde U_H^\infty$ for a realization $\widetilde Z$ of the Markov process $(Z_t, t\ge 0)$ with initial condition $\widetilde Z_0\,:\!=\, Z\big(\tau_E^1\big)$ and independent of Z conditionally on $Z\big(\tau_E^1\big)$ .

The quantity $\rho$ as stated in Assumptions (A2) and $(A3_F)$ is required to be strictly larger than the following survival estimate:

$$ \begin{equation*}\rho_S\,:\!=\, \sup\bigg\{\gamma \ge 0;\,\sup_{L\ge 1} \inf_{t>0} \;e^{\gamma t}\,\textrm{P}_\zeta\big(t < \tau_\partial\wedge T_{\mathcal{D}_L}\big)= 0\bigg\}\vee 0.\end{equation*}$$

We are now in a position to state $\mathbf{(AF)}$ :

• • (A1) holds for some $\zeta \in \mathcal{M}_1(\mathcal{Z})$ and a sequence $(\mathcal{D}_\ell)_\ell$ satisfying (A0). Moreover, there exist $\rho > \rho_S$ and a closed set E such that $E\subset \mathcal{D}_\ell$ for some $\ell\ge 1$ and such that (A2) and $(A3_F)$ hold.

As stated next by gathering the results of Theorems 2.2–2.3 and Corollary 2.2.7 of [Reference Velleret32], $\mathbf{(AF)}$ implies the convergence results that we aim for, noting that the sequence $(\mathcal{D}_\ell)_\ell$ will cover the whole space. Some additional properties of approximations are also obtained, where the process is localized to large $\mathcal{D}_L$ by extinction.

For the proof of Theorem 2.1, the sequence $(\mathcal{D}_\ell)_{\ell\ge 1}$ is defined as follows:

(4.4) $$ \begin{equation}\mathcal{D}_\ell\,:\!=\, \bar{B}(0, \ell) \times [1/\ell, \ell],\end{equation}$$

where $\bar{B}(0, \ell)$ denotes the closed ball of radius $\ell$ for the Euclidean norm.

Forbidding deleterious mutations in the case of unidimensional $\mathcal{X}$ will make our proof a bit more complicated. This case is thus treated later on. The expression ‘with deleterious mutations’ will be used a bit abusively to discuss the model under Assumption [D]. On the other hand, the expression ‘with advantageous mutations’ will refer to the case where Assumption [A] holds.

These criteria are proved to hold true under the assumptions of Theorem 2.1 in Theorems 4.2–4.6 below. We see in Subsection 4.2.1 how these theorems together with Theorem 4.1 imply Theorem 2.1. In the next subsections, we then prove Theorems 4.2–4.6. By first mentioning the mixing estimate, we wish to highlight the constraint on the reachable domain under Assumption [A]. The order of the proofs is different and chosen for clarity of presentation. The mixing estimates are handled similarly under the different sets of assumptions and are directly exploited in the proofs of the harvest properties. The escape estimates are very close to those of previously considered models, so more easily dealt with.

4.2. The whole space is accessible: with deleterious mutations or $d\ge 2$

4.2.1. Mixing property and accessibility

With deleterious mutations, the whole space becomes accessible. In fact, this is also the case with only advantageous mutations, provided $d\ge 2$ .

Theorem 4.2. Suppose that the assumptions [H] hold. For $d = 1$ , suppose Assumption [D] holds. For $d\ge2$ , suppose either Assumption [D] or Assumption [A]. Then, for any $\ell_I, \ell_M\ge 1$ , there exist $L>\ell_I\vee \ell_M$ and $c, t>0$ such that

$$ \begin{equation*} \forall \,(x_I,\, y_I) \in \mathcal{D}_{\ell_I},\quad \textrm{P}_{(x_I,\, y_I)} \left[ {(X,\, Y)}_{t}\in (dx,\,dy);\, t < \tau_\partial \wedge T_{\mathcal{D}_L} \right] \ge c\, \mathbf{1}_{\mathcal{D}_{\ell_M}}(x,\, y) \,dx\, dy. \end{equation*}$$

Remarks.

• ⋆ Equation (4.1) is exploited when defining $ T_{\mathcal{D}_L} \,:\!=\, \inf\left \lbrace t\ge 0;\, (X,\, Y)_t \notin \mathcal{D}_{L} \right \rbrace$ .

• ⋆ Theorem 4.2 implies in particular that the density with respect to the Lebesgue measure of any QSD is uniformly lower-bounded on any $\mathcal{D}_\ell$ .

• ⋆ In the case where Assumption [D] holds, $L\,:\!=\, \ell_I\vee \ell_M + \theta$ can be chosen. The choice of t cannot generally be made arbitrary, at least for $d=1$ , since the lower bound on the density of jump sizes is only valid for jumps of size close to $\theta$ . Under Assumption [A] with $d\ge 2$ , the constraint that jumps must be advantageous makes the convenient choice of L less clear.

4.2.3. Almost perfect harvest

We need some reference set on which our reference measure has positive density. With the constants $\theta$ and $\eta$ involved in [H4], let

(4.5) $$ \begin{equation}\varDelta\,:\!=\, \bar{B}({-}\theta\, \mathbf{e_1}, \eta)\times [1/2, 2].\end{equation}$$

This choice (which is rather arbitrary) is made in such a way that the uniform distribution on $\varDelta$ can be taken as the lower bound in the conclusions of Theorems 4.5 and 4.2.

Including deleterious mutations or with $d\ge 2$ , we will exploit the following theorem for sets E of the form $E\,:\!=\, \mathcal{D}_{\ell_{E}}$ , where $\ell_{E}$ is determined thanks to Theorem 4.3. But the theorem holds generally for any closed subsets E of $\mathbb{R}^d\times \mathbb{R}_+^*$ for which there exists $\ell\ge 1$ such that $E\subset \mathcal{D}_\ell$ , a property that, for brevity, we denote by $E \in \mathbf{D}$ .

Theorem 4.4. Suppose that the assumptions [H] hold. For $d = 1$ , suppose Assumption [D]. For $d\ge2$ , suppose either Assumption [D] or Assumption [A]. Then, for any $\rho > 0$ , $\epsilon\in (0,\, 1)$ , and $E \in \mathbf{D}$ , there exist $t_{\barwedge}, c >0$ which satisfy the following property for any $(x, y) \in E$ and $(x_{\zeta}, y_{\zeta})\in \varDelta$ . There exist a stopping time $U_H$ such that

$$ \begin{equation*} \left \lbrace\tau_\partial \wedge t_{\barwedge} \le U_H \right \rbrace= \left \lbrace U_H = \infty\right \rbrace\quad and \quad \textrm{P}_{(x, y)} (U_H = \infty, \, t_{\barwedge}< \tau_\partial) \le \epsilon\, \exp({-}\rho\, t_{\barwedge}), \end{equation*}$$

and an additional stopping time V such that

(4.6) $$ \begin{multline}\textrm{P}_{(x, y)} \big[(X(U_H),Y(U_H)) \in (dx^{\prime}, dy^{\prime});\, U_H < \tau_\partial \big] \\\le c \,\textrm{P}_{\big(x_{\zeta}, y_{\zeta}\big)} \big[ (X(V),Y(V)) \in (dx^{\prime}, dy^{\prime});\, V < \tau_\partial\big]. \end{multline}$$

Moreover, there exists a stopping time $U_H^{\infty}$ satisfying the following properties:

• • $U_H^{\infty}\,:\!=\, U_H$ on the event $\left \lbrace \tau_\partial\wedge U_H < \tau_{E}^1\right \rbrace$ , where $\tau_{E}^1\,:\!=\, \inf\big\{s\ge t_{\barwedge}\,:\, (X_s, Y_s) \in E\big\}$ .

• • On the event $\left \lbrace \tau_{E}^1 < \tau_\partial\right \rbrace \cap\left \lbrace U_H = \infty \right \rbrace$ , and conditionally on $\mathcal{F}_{\tau_{E}^1}$ , the law of $U_H^{\infty} - \tau_{E}^1$ coincides with that of $\widetilde U_H^{\infty}$ for the solution $\big(\widetilde X, \widetilde Y\big)$ of

  (4.7)

  
  where $r\ge 0$ , and $\widetilde M$ and $\widetilde B$ are independent copies of M and B, respectively.

4.3. No deleterious mutations in the unidimensional case

4.3.1. Mixing property and accessibility

When only advantageous mutations are allowed and $d =1$ , as soon as the size of jumps is bounded, the process can no longer access some portion of the space (there is a limit in the X direction). We could prove that the limit is related to the quantity $L_A\,:\!=\, \sup\left \lbrace M;\; \nu [2\,M, +\infty) >0 \right \rbrace \in (\theta/2,\, \infty].$

The accessible domains with maximal extension would then be of the form $[{-}\ell, L_A - 1/\ell] \times [1/\ell, \ell]$ , for some $\ell\ge 1.$ To simplify the proof, however, the limit $L_A$ will not appear in the statements below. We simply want to point out this potential constraint on the visited domain. In fact, the X component is assumed to be negative in the following definition of the accessibility domains:

(4.8) $$ \begin{equation}\Delta_{E} \,:\!=\, \{[{-}L, 0] \times [1/\ell, \ell];\; L, \ell\ge 1\}.\end{equation}$$

Theorem 4.5. Suppose $d=1$ , and that the assumptions [H] and [A] hold. Then, for any $\ell_I\ge 1$ and $E\in \Delta_{E}$ , there exists $L>\ell_I$ and $c, t>0$ such that the following lower bound holds for any $(x_I,\, y_I) \in \mathcal{D}_{\ell_I}$ :

(4.9) $$ \begin{equation} \textrm{P}_{(x_I,\, y_I)} \left[ {(X_{t},\, Y_{t})}\in (dx,\,dy);\, t < \tau_\partial \wedge T_{\mathcal{D}_{L}} \right] \ge c\, \mathbf{1}_{E}(x,\, y) \,dx\, dy. \end{equation}$$

Remark. Theorem 4.5 implies that the density with respect to the Lebesgue measure of any QSD is uniformly lower-bounded on any E of the form given by (4.8).

Proof. Let $\alpha$ be a QSD, and $E \in \Delta_{E}$ . Since $\mathcal{X} = \cup_{\ell} \mathcal{D}_{\ell}$ and $\alpha(\mathcal{X}) = 1$ , there exists $\ell_I$ such that $\alpha(\mathcal{D}_{\ell_I}) >0$ . Let $\lambda$ be the extinction rate of $\alpha$ . Let L, c, t be such that (4.9) holds for this choice of E and $\ell_I$ . Then

$$ \begin{equation*}\alpha(dx, dy) = e^{\lambda t} \alpha P_t(dx, dy)\ge \big(e^{\lambda t}\cdot \alpha\big(\mathcal{D}_{\ell_I}\big)\cdot c\big)\cdot \mathbf{1}_{E}(x, y)\, dx\, dy. \end{equation*}$$

This concludes the proof of the above remark.

4.3.4. Proof of Theorem 2.1 as a consequence of Theorems 4.5–4.6

The argument being very similar to the one for the case $d\ge 2$ or with Assumption [D], we go through it only briefly:

• • (A1) holds thanks to Theorem 4.5, again with the choice of $\zeta$ uniform on $\Delta$ .

• • Thanks to Theorem 4.6, and similarly as in the proof exploiting Theorem 4.3 in Subsection 4.2.4, we deduce that there exists $E \in \Delta_{E}$ such that (A2) holds with some value $\rho > \rho_S$ .

• • Finally, $(A3_F)$ holds for these choices of $\rho$ and E, thanks to Theorem 4.4.

This concludes the proof of the assumptions $\mathbf{(AF)}$ with $\mathbin{\cup}_{\ell \ge 1} \mathcal{D}_\ell = \mathcal{Z}$ . Exploiting Theorem 4.1, it implies Theorems 2.1 and 2.2 in the case where $d=1$ and the assumptions [H] and [A] hold.

5.1. With deleterious mutations or $d\ge 2$

Theorem 4.3 is a direct consequence of the following proposition, which is given as [Reference Velleret30, Proposition 4.2.2].

Proposition 5.1. Assume that (X, N) is a càdlàg process on $\mathbb{R}^d\times \mathbb{R}_+$ such that N is a solution to

$$dN_t = (r(X_t) - c\ N_t)\ N_t\ dt + \sigma \ \sqrt{N_t}\ dB_t,$$

where B is a Brownian motion. Assume that $\tau_\partial$ is upper-bounded by $\inf\{t\ge 0; N_t =0\}$ . Provided that $\limsup_{\|x\| \rightarrow \infty} r(x) = -\infty$ , it holds that for any $\rho>0$ , there exists $n>0$ such that

$$\textstyle{\sup_{\{x\in \mathcal{X}\} } }\, \;\textrm{E}_{x} \left(\exp\left[\rho\, (\tau_\partial\wedge \tau_{\mathcal{D}_n}) \right] \right) < \infty.$$

The proof developed in the next subsection extends that of this result and is sufficient to illustrate the technique.

5.2. Without deleterious mutations, $d=1$

In this section, we prove Theorem 4.6, i.e., the following statement:

• • Suppose that $d=1$ , and that the assumptions [H] and [A] hold. Then, for any $\rho>0$ , there exists some set $E\in \Delta_{E}$ such that the exponential moment of $\tau_{E}\wedge \tau_\partial$ with parameter $\rho$ is uniformly upper-bounded as follows:

  $$\begin{equation*} \underset{(x, y)\in \mathbb{R}\times \mathbb{R}_+}{\sup} \;\textrm{E}_{(x, \, y)} \left( \exp\left[\rho\, (\tau_{E}\wedge \tau_\partial)\right] \right) < \infty. \end{equation*}$$

5.2.1. Decomposition of the transitory domain

The proof is very similar to that of [Reference Velleret30, Subsection 4.2.4] except that, by Theorem 4.7, the domain E cannot be chosen as large. We thus need to consider another subdomain of $\mathcal{T}$ , which will be treated specifically thanks to Assumption [A].

The complementary $\mathcal{T}$ of E is then made up of four subdomains: ‘ $y \approx \infty$ ’, ‘ $y \approx 0$ ’, ‘ $x>0$ ’, and ‘ $\|x\| \approx \infty$ ’, as shown in Figure 1. Thus, we make the following definitions:

• • $\mathcal{T}^Y_\infty\,:\!=\, \left \lbrace ({-}\infty,\, -L) \cup(0, \infty) \right \rbrace \times (y_{\infty}, \infty)$ $\bigcup \; [{-}\ell, 0]\times [\ell, \infty) $ (‘ $y\approx \infty$ ’),

• • $\mathcal{T}_0\,:\!=\, ({-}L,\, L) \times [0, 1/\ell]$ (‘ $y \approx 0$ ’),

• • $\mathcal{T}_+\,:\!=\, (0, L)\times (1/\ell, y_{\infty}]$ (‘ $x > 0$ ’),

• • $\mathcal{T}^X_\infty\,:\!=\, \left \lbrace \mathbb{R} \setminus ({-}L,\, L) \right \rbrace \times (1/\ell, y_{\infty}]$ (‘ $|x| \approx\infty$ ’).

Figure 1.

Subdomains for (A2). Until the process reaches E or extinction, it is likely to escape any region either from below or from the side into $\mathcal{T}_+$ , the reverse transitions being unlikely. As long as $X_t > 0$ , $\|X_t\|$ must decrease (see Fact 5.2.5 in Subsection 5.2.4). Once the process has escaped $\{x \ge L_A\}$ , there is no way (via allowed jumps and v) for it to reach it afterwards.

With some threshold $t_\vee$ (meant to ensure finiteness, but the effect will vanish as it tends to $\infty$ ), let us first introduce the exponential moments of each area (remember that $\tau_{E}$ is the hitting time of E):

• • $\mathcal{E}^Y_\infty \,:\!=\, \sup_{(x,\,y) \in \mathcal{T}^Y_\infty} \textrm{E}_{(x, y)}[\exp(\rho \; V_{E})]$ ,

• • $\mathcal{E}_0 \,:\!=\, \sup_{(x,\,y) \in \mathcal{T}_0} \textrm{E}_{(x, y)}[\exp(\rho \; V_{E})]$ ,

• • $\mathcal{E}^X_\infty \,:\!=\, \sup_{(x,\,y) \in \mathcal{T}^X_\infty} \textrm{E}_{(x, y)}[\exp(\rho \; V_{E})]$ ,

• • $\mathcal{E}_X \,:\!=\, \sup_{(x,\,y) \in \mathcal{T}_+} \textrm{E}_{(x, y)}[\exp(\rho \; V_{E})]$ ,

where $V_{E}\,:\!=\, \tau_{E} \wedge \tau_\partial\wedge t_\vee.$ Implicitly, $\mathcal{E}^Y_\infty,\, \mathcal{E}^X_\infty,\, \mathcal{E}_X $ , and $\mathcal{E}_0$ are functions of $\rho,\, L,\, \ell, \,y_{\infty}$ that need to be specified.

5.2.2. A set of inequalities

The main ingredients for the following propositions are simple comparison properties that are specific to each of the transitory domain. By focusing on each of the domains separately (with the transitions between them), we can greatly simplify our control on the dependency of the processes.

As in [Reference Velleret30, Subsection 4.2.4], we first state some inequalities between these quantities, summarized in Propositions 5.2, 5.3, 5.4, and 5.5 below. Using these inequalities, we prove in Subsection 5.2.3 that those quantities are bounded. This will complete the proof of Theorem 4.6.

Proposition 5.2. Suppose that the assumptions [H] hold. Then, given any $\rho > 0$ , there exist $y_{\infty} > 0$ and $C_{\infty}^Y \ge 1$ such that for any $\ell> y_{\infty}$ and any $L>0$ ,

(5.1) $$\begin{equation}\mathcal{E}^Y_\infty \le C_{\infty}^Y \cdot \left( 1+ \mathcal{E}^X_\infty + \mathcal{E}_X \right).\end{equation}$$

Proposition 5.3. Suppose that the assumptions [H] hold. Then, given any $\rho>0$ , there exists $C^X_\infty \ge 1$ which satisfies the following property for any $\epsilon^X,\, y_{\infty} > 0$ : there exist $L>0$ and $\ell^X> y_{\infty}$ such that for any $\ell \ge \ell^X$ ,

(5.2) $$\begin{equation}\mathcal{E}^X_\infty\le C^X_\infty \cdot \left( 1 + \mathcal{E}_0 + \mathcal{E}_X\right)+ \epsilon^X \cdot\mathcal{E}^Y_\infty.\end{equation}$$

Proposition 5.4. Suppose that the assumptions [H] and [A] hold. Then, given any $\rho,\, L>0$ , there exists $C_X \ge 1$ which satisfies the following property for any $\epsilon^{+},\, y_{\infty} > 0$ : for any $\ell$ sufficiently large ( $\ell \ge \ell^+> y_{\infty}$ ),

(5.3) $$\begin{equation}\mathcal{E}_X \le C_X \cdot \left( 1+ \mathcal{E}_0 \right) + \epsilon^{+} \cdot \mathcal{E}^Y_\infty.\end{equation}$$

Proposition 5.5. Suppose that the assumptions [H] hold. Then, given any $\rho,\, \epsilon^0,\, y_\infty>0$ , there exists $C_0 \ge 1$ which satisfies the following property for any L and for any $\ell$ sufficiently large ( $\ell \ge \ell^0> y_{\infty}$ ):

(5.4) $$\begin{equation}\mathcal{E}_0 \le C_0 + \epsilon^0 \cdot \left( \mathcal{E}^Y_\infty +\mathcal{E}^X_\infty + \mathcal{E}_X\right).\end{equation}$$

Propositions 5.2 and 5.3 are deduced from the estimates given in the following two lemmas, which are stated as Lemmas 4.2.6 and 4.2.7 in [Reference Velleret31], on autonomous processes of the form

(5.5) $$\begin{equation} N^D_t \,:\!=\, n + \int_{0}^{t} \big(r - c\cdot N^D_s\big)\cdot N^D_s\ ds +\int_{0}^{t} \sigma \ \sqrt{N^D_s}\ dB_s.\end{equation}$$

Proposition 5.2 relies on the following property of descent from infinity, which is valid for any value of r.

Lemma 5.1. Let $N^D$ be the solution of (5.5), for some $r \in \mathbb{R}$ and $c>0$ , with n the initial condition. Then, for any $t, \epsilon > 0$ , there exists $n_{\infty} >0$ such that

$$\begin{equation*} \textstyle \sup_{n>0} \textrm{P}_{n} \big(t < \tau^D_{\downarrow} \big) \le \epsilon \quad \quad { with } \tau^D_{\downarrow} \,:\!=\, \inf \left \lbrace s\ge 0, \; N^D_s \le n_{\infty} \right \rbrace. \end{equation*}$$

Proposition 5.4 relies on the strong negativity on the drift term, stated below.

Lemma 5.2. For any $c, t >0$ , with $\tau_\partial^D\,:\!=\, \inf \left \lbrace t\ge 0,\, N^D_t = 0 \right \rbrace$ ,

$$\begin{equation*} \textstyle \sup_{ n>0} \,\textrm{P}_{n} \left( t < \tau_\partial^D \right) \underset{r\rightarrow -\infty}{\longrightarrow} 0. \end{equation*}$$

Moreover, for any $n, \epsilon>0$ , there exists $n_c$ such that, for any r sufficiently low, with $T^D_{\infty}\,:\!=\, \inf \left \lbrace t\ge 0,\, N^D_t \ge n_c \right \rbrace$ , we have $ \textrm{P}_{n} \left( T^D_{\infty} \le t \right) + \textrm{P}_{n} \left( N^D_{t} \ge n \right) \le \epsilon.$

On the other hand, Proposition 5.5 relies on an upper bound given as a continuous-state branching process, for which the extinction rate is much more explicit. The transition probability can clearly be made arbitrarily small by choosing a sufficiently small initial condition.

The only difference between the proofs in the current paper and those of [Reference Velleret30, Appendices A--D] is that here we distinguish transitions into $\mathcal{T}_+$ , which makes the term $\mathcal{E}_X$ appear with factors $C_{\infty}^Y$ , $C^X_\infty$ , and $\epsilon^0$ , respectively. These proofs are provided in Appendix A for the sake of completeness.

We prove next how to deduce Theorem 4.6 from the above set of four propositions. Then we will prove Proposition 5.4. This proof should convey both the main novelty and the common approach behind the proofs of these four propositions.

5.2.3. Proof that Propositions 5.2–5.4 imply Theorem 4.6

We first prove that the inequalities (5.2), (5.3), and (5.4) given by Propositions 5.2–5.4 imply an upper bound on $\mathcal{E}^Y_\infty \vee \mathcal{E}^X_\infty \vee \mathcal{E}_X\vee \mathcal{E}_0$ for sufficiently small $\epsilon^X$ , $\epsilon^+$ and $\epsilon^0$ .

Assuming first that $\epsilon^X \le \big(2\,C_{\infty}^Y \big)^{-1}$ , we have

$$\begin{equation*} \mathcal{E}^X_\infty \le C^X_\infty \, \left( 3 + 3\, \mathcal{E}_X + 2\, \mathcal{E}_0\right), \hspace{1 cm} \mathcal{E}^Y_\infty \le C_{\infty}^Y \,C^X_\infty \, \left( 4 + 4\, \mathcal{E}_X + 2\, \mathcal{E}_0\right).\end{equation*}$$

Assuming additionally that $\epsilon^+ \le \big(8\, C_{\infty}^Y \,C^X_\infty\big)^{-1}$ , we have

$$\begin{equation*}\mathcal{E}_X \le C_X\, \left( 2 + 3\, \mathcal{E}_0\right), \hspace{1 cm} \mathcal{E}^X_\infty \le C^X_\infty \,C_X \, \left( 9 + 11\, \mathcal{E}_0\right), \hspace{1 cm} \mathcal{E}^Y_\infty \le C_{\infty}^Y \,C^X_\infty \, \left( 12 + 14\, \mathcal{E}_0\right).\end{equation*}$$

Assuming also that $\epsilon^0 \le \big(60\, C_{\infty}^Y \,C^X_\infty\, C_X\big)^{-1} $ (and exploiting the fact that $2\times[14+11+3]\le 60$ ), we have

$$\begin{equation*} \mathcal{E}_0 \le 50\,C_0, \hspace{0.5 cm} \mathcal{E}_X \le 152\, C_X\,C_0, \hspace{1 cm} \mathcal{E}^X_\infty \le 559\, C^X_\infty \,C_X \,C_0, \hspace{1 cm} \mathcal{E}^Y_\infty \le 712\,C_{\infty}^Y \,C^X_\infty \, C_0.\end{equation*}$$

In particular,

$$ \begin{equation*} \underset{(x, y)\in \mathbb{R}\times \mathbb{R}_+}{\sup} \;\textrm{E}_{(x, \, y)} \left(\exp\left[\rho\, (\tau_{E}\wedge \tau_\partial)\right] \right)= \mathcal{E}^Y_\infty \vee \mathcal{E}^X_\infty \vee \mathcal{E}_X\vee \mathcal{E}_0< \infty.\end{equation*}$$

Let us now specify the choice of the various parameters involved. For any given $\rho$ , we obtain from Proposition 5.2 the constant $y_{\infty}$ , and $C_{\infty}^Y $ , which gives us a value $\epsilon^X\,:\!=\, \big(2\,C_{\infty}^Y \big)^{-1}$ . We then deduce, thanks to Proposition 5.3, some value for $C^X_\infty$ , $\ell^X$ , and L. We can then fix $\epsilon^+\,:\!=\, \big(8\, C_{\infty}^Y \,C^X_\infty\big)^{-1}$ , and deduce, according to Proposition 5.4, some value $C_X$ and $\ell^+ > 0$ . Now we fix $\epsilon^0\,:\!=\, \big(60\, C_{\infty}^Y \,C^X_\infty\, C_X\big)^{-1}$ and choose, according to Proposition 5.4, some value $C_0$ and $\ell^0 > 0$ . To make the inequalities (5.2), (5.3), and (5.4) hold, we can just take $\ell\,:\!=\, \ell^X \vee \ell^+ \vee \ell^0$ . With the calculations above, we then conclude Theorem 4.6 with $E\,:\!=\, [{-}L, 0] \times [1/\ell, \ell]$ .

5.2.4. Proof of Proposition 5.4: phenotypic lag pushed towards the negatives

Since the norm of X decreases at rate at least v as long as the process stays in $\widetilde{\mathcal{T}}_+\,:\!=\, [0, L]\times \mathbb{R}_+^*$ , we know that the process cannot stay in this area during a time-interval larger than $t_\vee\,:\!=\, L/v$ . This effect will give us the bound $C_X\,:\!=\, \exp\left( \rho\, L / v\right)$ .

Moreover, we need to ensure that the transitions from $\mathcal{E}_X$ to $\mathcal{E}^Y_\infty$ are exceptional enough. This is done exactly as for [Reference Velleret30, Proposition 4.2.2], by taking $\ell^+$ sufficiently larger than $y_\infty$ . The event of having the process reach $\ell^+$ in the time-interval $[0, t_\vee]$ is then exceptional enough.

More precisely, given L and $\ell > y_\infty\ge 1$ and initial condition $(x, y)\in \mathcal{T}_+$ , let

(5.6) $$\begin{equation} C_X\,:\!=\, \exp\left( \dfrac{\rho\, L}{v}\right), \qquad T\,:\!=\, \inf\left \lbrace t\ge 0;\, X_t \le 0 \right \rbrace \wedge V_{E}.\end{equation}$$

Lemma 5.3. Suppose that the assumptions [H] and [A] hold. Then, for any initial condition $(x, y)\in \mathcal{T}_+$ , we have $(X, Y)_{T} \notin \mathcal{T}^X_\infty$ a.s., and

$$\begin{equation*} \forall \,t< T,\qquad X_t \le x - v\, t \le L - v\, t, \hspace{1 cm} {so\ that }\ T \le t_\vee\,:\!=\, L / v.\end{equation*}$$

Thanks to Assumption [H4], an immediate induction on the number of jumps previous to $T\wedge t$ proves that the jumps of X can only make its value decrease (because it is positive, while the absolute value must necessarily decrease). This proves Lemma 5.3. Consequently,

$$\begin{align*}\textrm{E}_{(x, y)}[\exp(\rho V_{E} )]&= \textrm{E}_{(x, y)}\Big[ \exp(T );\, T = V_{E} \Big] + \mathcal{E}_0\; \textrm{E}_{(x, y)}\Big[ \exp(T );\, (X, Y)_{T} \in \mathcal{T}_0 \Big] \\&\hspace{0.5cm} + \mathcal{E}^Y_{\infty}\; \textrm{E}_{(x, y)}\Big[ \exp(T );\, (X, Y)_{T} \in \mathcal{T}^Y_\infty \Big] \\& \le C_X\, \left( 1 + \mathcal{E}_0\right) + C_X\, \mathcal{E}^Y_{\infty}\; \textrm{P}_{y_\infty}\big(T_{\uparrow} \le t_\vee\big),\end{align*}$$

where $T_{\uparrow}\,:\!=\, \inf\left \lbrace t\ge 0;\, Y^{\uparrow}_t \ge \ell \right \rbrace,$ and $Y^{\uparrow}$ is the solution of

$$\begin{equation*} Y^{\uparrow}_t \,:\!=\, y_{\infty} + \int_{0}^{t} \psi_\vee \left( Y^{\uparrow}_s \right) \;ds + B_t \qquad\bigg(\text{again } \psi_\vee(y)\,:\!=\, - \frac{1}{2\, y} +\frac{r_\vee\, y}{2} - \gamma\, y^3\bigg).\end{equation*}$$

We conclude the proof of Proposition 5.4 by noticing that $\textrm{P}_{y_\infty}\big(T_{\uparrow} \le t_\vee\big) \underset{\ell\rightarrow \infty} {\longrightarrow} 0.$

Given the proofs of Propositions 5.2, 5.3, and 5.5 provided in Appendix A and Subsection 5.2.3, the proof of Theorem 4.6 is now completed. The proof of Theorem 4.3 is sufficiently similar to be deduced without the need to refer to [Reference Velleret30, Subsection 4.2.4].

6.1. Construction of the change of probability under [H4]

The idea of this subsection is to prove that we can think of Y as a Brownian motion up to some stopping time which will bound $U_H$ . If we get a lower bound for the probability of events in this simpler setup, this will prove that we also get a lower bound in the general setup.

The limits of our control. Let $t_G,\, x_\vee >0,$ $0< y_\wedge< y_\vee,$ $N_J \ge 1.$ Our aim is to simplify the law of $(Y_t)_{t\in [0, t_G]}$ as long as Y stays in $[y_\wedge, y_\vee]$ , $\|X\|$ stays in $\bar{B}(0, x_\vee)$ , and at most $N_J$ jumps have occurred. Thus, let

(6.1) $$\begin{align} & T_X\,:\!=\, \inf \left \lbrace t\ge 0;\, \|X_t\| \ge x_\vee \right \rbrace,\quad & & T_Y\,:\!=\, \inf \left \lbrace t\ge 0;\, Y_t \notin [y_\wedge, y_\vee] \right \rbrace,\notag \\& g_{\vee}\,:\!=\, \sup\left \lbrace g(x, w) ;\, \|x\|\le x_\vee, w\in \mathbb{R}^d \right \rbrace, \quad & & f_{\vee} \,:\!=\,\sup\left \lbrace f(y);\, y\in [y_\wedge,\, y_\vee] \right \rbrace, \\& \mathcal{J}\,:\!=\, \left \lbrace (w, u_g, u_f)\in \mathbb{R}^d \times [0, f_{\vee}]\times [0, g_{\vee}]\right \rbrace,\notag\end{align}$$

so that $\nu\otimes du_g\otimes du_f(\mathcal{J}) = \nu\big(\mathbb{R}^d\big)\;g_{\vee}\, f_{\vee} <\infty.$

Our Girsanov transform alters the law of Y until the stopping time

(6.2) $$\begin{equation} T_G\,:\!=\, t_G\wedge T_X \wedge T_Y \wedge U_{N_J}, \end{equation}$$

where

(6.3) $$\begin{equation} U_{N_J}\,:\!=\, \inf\left \lbrace\, t;\, M([0, t] \times \mathcal{J}) \ge N_J + 1\; \right \rbrace. \end{equation}$$

Note that the $(N_J+1)$ th jump of X will then necessarily occur after $T_G$ .

The change of probability. We define

(6.4) $$\begin{equation} L_t\,:\!=\, - \int_{0}^{t\wedge T_G} \psi(X_s, Y_s) dB_s, \quad \text{ and } D_{t}\,:\!=\, \exp\left[ L_{t} - \langle L\rangle_t /2 \right],\end{equation}$$

the exponential local martingale associated with $(L_t)$ .

Theorem 6.1. Suppose that the assumptions [H] hold. Then, for any $t_G,\, x_\vee > 0$ and $y_\vee > y_\wedge >0$ , there exists $C_G > c_G>0$ such that, a.s. and for any $t>0$ , $c_G \le D_{t} \le C_G$ . In particular, $D_t$ is a uniformly integrable martingale and $\beta_t = B_t - \langle B, L\rangle_t$ is a Brownian motion under $\textrm{P}^G_{(x, y)}$ defined as $\textrm{P}^G_{(x, y)} \,:\!=\, D_{\infty} \cdot \textrm{P}_{(x, y)}$ . We deduce the following bounds, valid for any $(x,\, y)\in \mathbb{R}^d \times \mathbb{R}_+$ :

$$\begin{equation*} c_G \cdot \textrm{P}^G_{(x,\, y)} \le \textrm{P}_{(x,\, y)} \le C_G \cdot \textrm{P}^G_{(x,\, y)}. \end{equation*}$$

On the event $\{t\le T_G\}$ , $Y_t = y + \beta_t$ ; i.e. Y has the law of a Brownian motion under $\textrm{P}^G_{(x, y)}$ up to time $T_G$ . This means that we can obtain bounds on the probability of events involving Y as in our model by considering Y as a simple Brownian motion. Meanwhile, the independence between its variations as a Brownian motion and the Poisson process still holds by Proposition 1.1.

Proof in the case where r is $C^1$

Let

(6.5) $$\begin{align} \|r\|_\infty^G\,:\!=\, \sup\left \lbrace\, |r(x)|;\, x\in \bar{B}(0, x_\vee) \right \rbrace, \end{align}$$

(6.6) $$ \begin{align} \|r^{\prime}\|_\infty^G\,:\!=\, \sup\left \lbrace\, |r^{\prime}(x)|;\, x\in \bar{B}(0, x_\vee) \right \rbrace. \end{align}$$

With $\psi_G^\vee$ an upper bound of $\psi$ on $\bar{B}(0,\, x_\vee)\times [y_\wedge,\, y_\vee]$ (deduced from [H3]), and recalling that (X, Y) belongs to this subset until $T_G$ (see (6.2)), we have

(6.7) $$\begin{equation} \langle L\rangle_{\infty} = \int_{0}^{T_G} \psi(X_s, Y_s)^2 ds \le t_G \cdot \big(\psi_G^\vee\big)^2.\end{equation}$$

In the following, we look for bounds on $\int_{0}^{T_G} \psi(X_s, Y_s) dY_s$ , noting that

$$\begin{equation*} L_{T_G} + \int_{0}^{ T_G} \psi(X_s, Y_s) dY_s = \int_{0}^{T_G} \psi(X_s, Y_s)^2 ds \in \big[0, t_G \cdot \big(\psi_G^\vee\big)^2\big],\end{equation*}$$

$$\begin{equation*} \int_{0}^{T_G} \psi(X_s, Y_s) dY_s = \int_{0}^{T_G} \left({-} \dfrac{1}{2 Y_s} + \frac{r(X_s)\; Y_s}{2} - \gamma\cdot (Y_s)^3 \right) dY_s.\end{equation*}$$

Now, thanks to Itô’s formula,

$$\begin{equation*} \ln\big(Y_{T_G}\big) = \ln(y) + \int_{0}^{T_G} \dfrac{1}{ Y_s} dY_s - \frac{1}{2} \int_{0}^{T_G} \dfrac{1}{ (Y_s)^2} ds. \end{equation*}$$

Thus,

(6.8) $$\begin{equation} \left|\int_{0}^{T_G} \dfrac{1}{ Y_s} dY_s \right| \le 2\;( |\ln(y_\wedge)|\vee|\ln(y_\vee)|) + \dfrac{t_G}{2\, (y_\wedge)^2} <\infty.\end{equation}$$

Secondly,

$$\begin{equation*} \big(Y_{T_G}\big)^4 = y^4 + 4\, \int_{0}^{T_G} (Y_s)^3 dY_s + 6\, \int_{0}^{T_G} (Y_s)^2 ds.\end{equation*}$$

Thus,

(6.9) $$\begin{equation} \left|\int_{0}^{T_G} (Y_s)^3 dY_s \right| \le (y_\vee)^4/4 + 3\, t_G\, (y_\vee)^2/2 <\infty.\end{equation}$$

Thirdly,

(6.10) $$\begin{align} r\big(X_{T_G-}\big)\cdot \big(Y_{T_G}\big)^2 &= r(x)\, y^2 + 2\, \int_{0}^{T_G} r(X_s)\, Y_s\, dY_s\notag\\&+ \int_{0}^{T_G} r(X_s)\; ds - v \int_{0}^{T_G} r^{\prime}(X_s)\cdot (Y_s)^2 \, ds \notag\\ &+ \int_{[0,T_G) \times \mathbb{R}^d \times \mathbb{R}_+ } \big(\;r(X_{s-} + w) - r(X_{s-})\;\big) \cdot (Y_s)^2 \notag\\ &\hspace{0.5cm} \times \mathbf{1}_{\left \lbrace u_f\le f(Y_s) \right \rbrace} \, \mathbf{1}_{\left \lbrace u_g \le g(X_{s^-},\,w) \right \rbrace} M\big(ds, dw, du_f, du_g\big). \end{align}$$

Since $\forall s\le T_G,\, Y_s \in [y_\wedge, y_\vee]$ , from [H2] and (6.1) we get

$$\begin{equation*} \forall \,s\le T_G,\; \forall \, w \in \mathbb{R}^d,\quad g(X_{s-}, w) \le g_{\vee},\quad f(Y_s) \le f_{\vee},\quad \text{ and } T_G \le U_{N_J}.\end{equation*}$$

Since moreover $T_G \le T_X$ ,

$$\begin{multline*}\int_{[0,T_G) \times \mathbb{R}^d \times \mathbb{R}_+ }\big(\;r(X_{s-} + w) - r(X_{s-})\;\big) \cdot (Y_s)^2\\ \times \mathbf{1}_{\left \lbrace u_f\le f(Y_s) \right \rbrace} \, \mathbf{1}_{\left \lbrace u_g \le g(X_{s^-},\,w) \right \rbrace}M\big(ds, dw, du_f, du_g\big)\le 2\, N_J\, \|r\|_\infty^G\, (y_\vee)^2,\end{multline*}$$

so that (6.10) leads to

(6.11) $$\begin{equation} 2\; \left|\int_{0}^{T_G} r(X_s)\, Y_s\, dY_s \right| \le \big( 2\,(N_J +1)\, \|r\|_\infty^G + \|r^{\prime}\|_\infty^G\; v\, t_G\big) \cdot (y_\vee)^2 + \|r\|_\infty^G \; t_G< \infty.\end{equation}$$

The inequalities (6.8), (6.9), and (6.11) combined with (6.7) allow us to conclude that $L_\infty$ and $\langle L\rangle_\infty$ are uniformly bounded. This proves the existence of $0<c_G<C_G$ such that a.s. $c_G \le D_\infty \le C_G$ .

This statement is a priori adapted for $t_G$ replaced by $t\wedge t_G$ , yet these bounds are actually the largest for $t=t_G$ . So this implies that $c_G \le D_t \le C_G$ holds uniformly in t. The rest of the proof is simply a classical application of Girsanov’s transform theory.

Extension to the case where r is only Lipschitz continuous

The inequalities (6.8) and (6.9) are still true, so we show that we can find the same bound on $\left|\int_{0}^{T_G} r(X_s)\, Y_s\, dY_s \right|$ where we replace $\|r^{\prime}\|_\infty^G$ by the Lipschitz constant $\|r\|_{Lip}^G$ of r on $\bar{B}(0, x_\vee)$ , by approximating r by $C^1$ functions that are $\|r\|_{Lip}^G$ -Lipschitz continuous.

Lemma 6.1. Suppose r is Lipschitz continuous on $\bar{B}(0,\, x_\vee)$ for some $x_\vee>0$ . Then there exists $r_n\in C^1\big( \bar{B}(0,\, x_\vee), \mathbb{R}\big)$ , $n\ge 1$ , such that

$$\|r_n - r\|_\infty^G \underset{n\rightarrow \infty} {\longrightarrow} 0 \quad and \quad \forall \,n\ge 1,\; \|r^{\prime}_n\|_\infty^G \le \|r\|_{Lip}^G.$$

Proof of Lemma 6.1. We begin by extending r on $\mathbb{R}^d$ with $r_G(x)\,:\!=\, r\circ\Pi_G(x)$ , where $\Pi_G$ is the projection on $\bar{B}(0, x_\vee)$ (it is well known that r can be extended on $\bar{B}(0, x_\vee)$ with the same Lipschitz constant). Note that this extension $r_G$ is still $\|r\|_{Lip}^G$ -Lipschitz continuous. If we now define $r_n\,:\!=\, r_G\ast \phi_n \in C^1,$ where $(\phi_n)$ is an approximation of the identity of class $C^1$ , then

$$\begin{multline*}\forall \,x, y,\; |r_n(x) - r_n(y)| = \left| \int_{\mathbb{R}^d} (r_G(x-z) - r_G(y-z)) \phi_n(z) dz\right|\\ \le \|r\|_{Lip}^G \, \|x - y\|\, \int_{\mathbb{R}^d}\phi_n(z) dz = \|r\|_{Lip}^G \, \|x - y\|.\end{multline*}$$

It follows that

(6.12) $$\begin{equation} \forall \,n \ge 1,\quad \|r^{\prime}_n\|_\infty^G \le \|r\|_{Lip}^G,\quad \|r_n - r_G\|_\infty^G \underset{n\rightarrow \infty} {\longrightarrow} 0.\end{equation}$$

Proof that Lemma 6.1 combined with the case $r\in C^1$ proves Theorem 6.1. We just have to prove (6.11) with $\|r\|_{Lip}^G$ instead of $\|r^{\prime}\|_\infty^G$ . If we apply this formula for $r_n$ and exploit Lemma 6.1, we see that there will be some $C = C(t_G, y_\vee, N_J) > 0$ such that

$$\begin{equation*} 2\; \left|\int_{0}^{T_G} r_n(X_s)\, Y_s\, dY_s \right| \le \big( 2\,(N_J +1)\, \|r\|_\infty^G + \|r\|_{Lip}^G\; v\, t_G\big) \, (y_\vee)^2 + r_\infty \; t_G + C \, \|r - r_n\|_\infty^G.\end{equation*}$$

Thus, it remains to bound

$$\begin{equation*} \left|\int_{0}^{T_G} (r_n(X_s) - r(X_s) )\cdot Y_s\, dY_s \right| \le t_G\, y_\vee\, \psi_G^\vee \, \|r - r_n\|_\infty^G + \left|M_n \right|,\end{equation*}$$

where $M_n\,:\!=\, \int_{0}^{T_G} (r_n(X_s) - r(X_s) )\, Y_s\, dB_s$ has mean 0 and variance

$$\begin{align*}\textrm{E}\big( (M_n)^2 \big)&= \textrm{E}\left( \int_{0}^{T_G} (r_n(X_s) - r(X_s) )^2\, Y_s\, ^2\, ds \right)\\&\hspace{1.5cm}\le t_G\, (y_\vee) ^2\, \big(\|r - r_n\|_\infty^G\big) ^2 \underset{n\rightarrow \infty} {\longrightarrow} 0.\end{align*}$$

Thus, we can extract some subsequence $M_{\phi(n)}$ which converges a.s. towards 0, so that a.s.,

$$\begin{align*} &\left|\int_{0}^{T_G} r(X_s) \, Y_s\, dY_s \right| \le \underset{n\rightarrow \infty}{\liminf} \left \lbrace \left|\int_{0}^{T_G} r_{\phi(n)}(X_s)\, Y_s\, dY_s \right| + t_G\, y_\vee\, \psi_G^\vee \, \|r - r_{\phi(n)}\|_\infty^G + \left|M_{\phi(n)}\right|\right \rbrace\\ &\hspace{3cm} \le \frac{1}{2}\,\Big( 2\,(N_J +1)\, \|r\|_\infty^G + \|r\|_{Lip}^G\; v\, t_G\Big) \cdot (y_\vee)^2 + \frac{1}{2}\, \|r\|_\infty^G \; t_G < \infty.\end{align*}$$

The proof in the case $r\in C^1$ can then be exploited without difficulty.

6.2. Mixing for Y

The proof will rely on Theorem and on the following classical property of Brownian motion.

Lemma 6.2. Consider any constants $b_\vee>0$ , $\epsilon >0$ , and $0<t_0 \le t_1$ . Then there exists $c_B>0$ such that for any $b_I \in [0, b_\vee]$ and $t\in [t_0, t_1]$ ,

$$\begin{equation*} \textrm{P}_{b_I}\Big( B_t \in db;\, \min_{s \le t_1} B_s \ge -\epsilon\,,\, \max_{s \le t_1} B_s \le b_\vee + \epsilon \Big) \ge c_B\cdot \mathbf{1}_{[0,\, b_\vee]}(b) \, db, \end{equation*}$$

where B under $\textrm{P}_{b_I}$ has by definition the law of a Brownian motion with initial condition $b_I$ .

Thanks to this lemma and Theorem 6.1, we will be able to control Y to prove that it indeed diffuses and that it stays in some closed interval $I_Y$ away from 0. We can then control the behavior of X independently of the trajectory of Y by appropriate conditioning of M—the Poisson random measure—so as to ensure the jumps we need (conditionally on its remaining in $I_Y$ ).

Proof. Consider the collection of marginal laws of $B_t$ , with initial condition $b\in ({-}\epsilon, b_\vee +\epsilon)$ , killed when it reaches $-\epsilon$ or $b_\vee + \epsilon$ . It is classical that these laws have a density $u(t; b, b^{\prime})$ , $t>0$ , $b^{\prime}\in [{-}\epsilon, b_\vee +\epsilon]$ , with respect to the Lebesgue measure (cf. e.g. Bass [Reference Bass1, Section 2.4] for more details). We have that u is a solution to the Cauchy problem with Dirichlet boundary conditions

$$\begin{align*} & \partial_t u(t; b_I, b) = \Delta_{b} u(t; b_I, b) \hspace{2 cm} \text{for }t>0, b_I, b\in ({-}\epsilon, b_\vee +\epsilon), \\& u(t; b_I,-\epsilon) = u(t; b_I, b_\vee +\epsilon) = 0 \hspace{1 cm}\text{for }t>0.\end{align*}$$

Thanks to the maximum principle (cf. e.g. Evans [Reference Evans17, Theorem 4, Subsection 2.3.3]), $u>0$ on $\mathbb{R}_+^*\mathbin{\! \times \!} [0, b_\vee] \mathbin{\! \times \!} ({-}\epsilon, b_\vee +\epsilon)$ , and since u is continuous in its three variables, it is lower-bounded by some $c_B$ on the compact subset $[t_0, t_1]\mathbin{\! \times \!} [0, b_\vee] \mathbin{\! \times \!} [0, b_\vee]$ .

6.3. Mixing for X

For clarity, we decompose the ‘migration’ along X into different kinds of elementary steps, as already done in [Reference Velleret32, Subsection 4.3.2]. Let

(6.13) $$\begin{equation} \mathcal{A}\,:\!=\, \bar{B}({-}\theta\, \mathbf{e_1},\,\eta/2), \qquad \tau_\mathcal{A}\,:\!=\, \inf\left \lbrace t\ge 0;\, X_t \in \mathcal{A}\,,\, Y_t \in [2, 3] \right \rbrace, \end{equation}$$

where we assume without loss of generality that $\eta \le \theta/8$ (the interval [Reference Bansaye, Cloez, Gabriel and Marguet2, Reference Bürger and Lynch3] is chosen arbitrarily).

Under any of the three sets of assumptions considered in the following, the proof is achieved in three steps. The first step is to prove that, with a lower-bounded probability for any initial condition in $\mathcal{D}_\ell$ , $\tau_\mathcal{A}$ is upper-bounded by some constant $t_\mathcal{A}$ . In the second step, we prove that the process is sufficiently diffuse and that time shifts are not a problem. In the third step, we specify which sets we can reach from $\mathcal{A}$ .

Recall that for any $\ell\ge 1$ , $T_{\mathcal{D}_{\ell}}\,:\!=\, \inf\left \lbrace t\ge 0;\, (X,\, Y)_t \notin \mathcal{D}_{\ell} \right \rbrace< \tau_\partial$ . For $n\ge 3$ , let us define $T_{(n)}\,:\!=\,T_{\mathcal{D}_{2 n}}$ . For $n\ge 3$ and $t, c>0$ , let

(6.14) $$\begin{multline}\mathcal{R}^{(n)}(t, c)\,:\!=\, \left \lbrace x_F\in \mathbb{R}^d;\, \forall \,(x_0,y_0) \in \mathcal{A} \times [1/n,\,n],\; \right. \\ \left. \textrm{P}_{(x_0,y_0)} \left[ (X, Y)_t\in (dx, dy);\, t <T_{(n)}\right] \ge c\; \mathbf{1}_{B(x_F, \eta/2)}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \, dx\, dy \right \rbrace. \end{multline}$$

We will prove the mixing on a global scale by translating local mixing properties into certain induction properties of the sets $(\mathcal{R}^{(n)}(t, c))_{\{t, c>0\}}$ .

Several local mixing properties require local lower and upper bounds on g, so that they can only be exploited in specific areas of $\mathbb{R}^d$ . In order to provide a general framework for these through Proposition 6.1, let us consider the following increasing sequence of sets, indexed by $n\ge 1$ :

$$\begin{align*}\mathcal{G}_n\,:\!=\, \{x\in \bar{B}(0, n);\,&\forall \,z\in [0, \eta/4],\;\forall \,\delta\in \bar{B}(0, \eta/2),\;\forall \,w\in \bar{B}(\theta\, \mathbf{e_1},\eta),\;\\&\hspace{3.5 cm}g(x - (\theta - z) \mathbf{e_1} + \delta, w) \ge 1/n,\\\text{and } \quad &\forall \,z\in [{-}\theta, \eta/4],\;\forall \,\delta\in \bar{B}(0, \eta/2),\;\forall \,w\in \mathbb{R}^d,\;\\&\hspace{4.8 cm}g(x+z \mathbf{e_1} + \delta, w) \le n\}.\end{align*}$$

These steps are deduced from the following elementary properties.

$u\in [0, u_\vee(x_I)]$ , where $u_\vee(x)\,:\!=\, \sup\{ u\ge 0 ;\, (x-v\, u\, \mathbf{e_1}) \in \bar{B}(0, n)\}$ :

$$\begin{equation*} \textrm{P}_{(x_I,y_I)}\left[ (X_u, Y_u) \in (dx, dy) ;\, u < T_{(n)}\right] \ge c_D\, \delta_{\{x_I -v\, u\, \mathbf{e_1}\}}(dx)\cdot \mathbf{1}_{[1/n, n]}(y)\,dy. \end{equation*}$$

In particular, for any $t, c>0$ , $n\ge3$ , the fact that x belongs to $\mathcal{R}^{(n)}(t, c)$ implies the following inclusion:

$$\begin{equation*} \forall \,u\in [0, u_\vee(x)],\quad x -v\, u\, \mathbf{e_1} \in \mathcal{R}^{(n)}(t+u, c\cdot c_D). \end{equation*}$$

The proof of Lemma 6.3 being easily adapted from that of the next proposition, it is deferred until after the proof of the latter.

Proposition 6.1. For any $n \ge 3$ , there exist $t_P, c_P>0$ such that for any $x_I \in \mathcal{G}_n$ , for any $x_0\in B(x_I, \eta/4)$ and $y_0\in [1/n, n]$ ,

$$\begin{equation*} \textrm{P}_{(x_0, y_0)} \left[ (X, Y)_{t_P} \in (dx, dy);\, t_P <T_{(n)}\right] \ge c_P\; \mathbf{1}_{B(x_I,\, 3\eta/4)}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \; dx\, dy.\end{equation*}$$

A direct application of the Markov property implies the following two results.

Corollary 6.1. For any $n \ge 3$ , there exist $t_P, c_P>0$ such that for any $t, c>0$ , the following inclusion holds:

$$\begin{equation*} \big\{x\in \mathbb{R}^d;\, d(x, \mathcal{R}^{(n)}(t, c) \cap \mathcal{G}_n)\le \eta/4\big\} \subset \mathcal{R}^{(n)}(t+t_P, c\cdot c_P). \end{equation*}$$

Lemma 6.4. There exists $c_B>0$ such that the following inclusion holds for any $t, t^{\prime}, c, c^{\prime}>0$ and any $n\ge 1$ , provided that $-\theta\, \mathbf{e_1} \in \mathcal{R}^{(n)}(t, c)$ :

$$\begin{equation*} \mathcal{R}^{(n)}(t^{\prime}, c^{\prime}) \subset \mathcal{R}^{(n)}(t+t^{\prime}, c_B\cdot c\cdot c^{\prime}). \end{equation*}$$

In the previous lemma, we may choose $c_B = Leb(B(x_I,\, \eta/2))>0$ .

Corollary 6.3. as a consequence of Proposition 6.1. For $n\ge 3$ , let $t_P, c_P>0$ be prescribed by Proposition 6.1. We consider $x_I\in \mathcal{R}^{(n)}(t, c) \cap \mathcal{G}_n$ , $x_F$ such that $\|x_F - x_I\| \le \eta/4$ . Combining through the Markov property the fact that $x_I\in \mathcal{R}^{(n)}(t, c)$ and Proposition 6.1, we deduce that for any $(x_0,y_0) \in \mathcal{A} \times [1/n,\,n]$ ,

$$\begin{align*} & \textrm{P}_{(x_0,y_0)} \left[ (X, Y)_{t+t_P}\in (dx, dy) ;\, t+t_P < T_{(n)}\right] \\&\hspace{1 cm} \ge c \int_{B(x_I, \eta/2)} dx^{\prime}_0 \int_{1/n}^{n} dy^{\prime}_0\, \textrm{P}_{(x^{\prime}_0,y^{\prime}_0)} \left[ (X, Y)_{t_P}\in (dx, dy) ;\, t_P < T_{(n)} \right] \\&\hspace{1 cm} \ge c\cdot Leb(B(x_I, \eta/4))\cdot (n-1/n)\cdot c_P \mathbf{1}_{B(x_I,\, 3\eta/4)}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \; dx\, dy \\&\hspace{1 cm} \ge c \cdot c^{\prime}_P \cdot \mathbf{1}_{B(x_F,\, \eta/2)}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \; dx\, dy,\end{align*}$$

where $c^{\prime}_P \,:\!=\, Leb(B(0, \eta/4))\cdot (n-1/n)\cdot c_P>0$ . This means that $x_F \in \mathcal{R}^{(n)}\big(t+t_P, c\cdot c^{\prime}_P\big)$ . The proof of Corollary 6.1 is thus concluded with these choices of $t_P$ and $c^{\prime}_P$ , which are indeed independent from $x_I$ , $x_F$ .

Proof of Proposition 6.1.

Step 1: description of the random event. For $n\ge 3$ , we set $t_P\,:\!=\, \theta/v$ , $t_J\,\,:\!=\,\,\eta/(4 v)$ , $y_\wedge\,:\!=\, 1/(2\,n)$ , $y_\vee\,:\!=\, 2\, n$ . Also, let

(6.15) $$\begin{align}T^Y\,:\!=\, \inf\left \lbrace t\ge 0;\, Y_t \notin [y_\wedge,\, y_\vee] \right \rbrace,\end{align}$$

(6.16) $$ \begin{align} f_\wedge \,:\!=\, \inf \left \lbrace f(y);\, y \in [y_\wedge,\, y_\vee] \right \rbrace, \qquad f_\vee \,:\!=\, \sup \left \lbrace f(y);\, y \in [y_\wedge,\, y_\vee] \right \rbrace. \end{align}$$

We have that $f_\vee$ is finite by [H1]. Thanks to [H1, we also know that $f_\wedge$ is positive.

On the event $\big\{t_P < T^Y\big\}$ , we shall prove that the values of X on $[0, t_P]$ are prescribed as functions of M restricted to the subset

(6.17) $$\begin{equation} \mathcal{X}^M\,:\!=\, [0, t_P]\times \mathbb{R}^d \times [0, f_\vee]\times [0,n]. \end{equation}$$

Let $x_0\,:\!=\, x_I + \delta_0$ , with $x_I\in \mathcal{G}_n$ and $\delta_0\in B(0, \eta/4)$ , and $y_0 \in [1/n,\, n]$ , which we consider as the initial conditions for the process (X, Y).

To ensure one jump of size around $\theta$ , at time nearly $t_P$ , while ‘deleting’ the contribution of $\delta_0$ , let

(6.18) $$ \begin{equation} \mathcal{J} \,:\!=\, [t_P - t_J, t_P] \times B(\theta\, \mathbf{e_1} - \delta_0,\, \eta/2) \times [0, f_\wedge] \times [0, 1/n]. \end{equation}$$

We partition $\mathcal{X}^M = \mathcal{J} \cup \mathcal{N}$ , where $\mathcal{N}\,:\!=\, \mathcal{X}^M \setminus \mathcal{J}.$ The main event under consideration is the following:

(6.19) $$ \begin{equation} \mathcal{W} = \mathcal{W}^{(x_0, y_0)}\,:\!=\, \left \lbrace t_P < T^Y \right \rbrace \cap \left \lbrace M(\mathcal{J}) = 1\right \rbrace \cap \left \lbrace M(\mathcal{N}) = 0\right \rbrace. \end{equation}$$

Thanks to Theorem 6.1 (with $x_\vee \,:\!=\, n + 2 \theta$ , $t_G =t_P$ , and the same values for $y_\wedge$ and $y_\vee$ ), there exists $c_G >0$ such that

(6.20) $$ \begin{equation} \textrm{P}_{(x_0, y_0)} \left( (X, Y)_{t_P} \in (dx, dy);\, \mathcal{W}\right) \ge c_G\; \textrm{P}^G_{(x_0, y_0)} \left( (X, Y)_{t_P} \in (dx, dy);\, \mathcal{W} \right). \end{equation}$$

Under the law $\textrm{P}^G_{(x_0, y_0)}$ , the condition $\left \lbrace M(\mathcal{J}) = 1\right \rbrace$ is independent of $\left \lbrace M(\mathcal{N}) = 0\right \rbrace$ , of $\left \lbrace t_P < T^Y \right \rbrace$ , and of $Y_{t_P}$ ; cf. Proposition 1.1. Thus, on the event $\mathcal{W}$ , the only ‘jump’ coded in the restriction of M on $\mathcal{J}$ is given as $(T_{J}, \theta\, \mathbf{e_1} - \delta_0 + W, U_f, U_g)$ , where $T_{J}$ , $U_f$ , and $U_g$ are chosen uniformly and independently on $[t_P - t_J, t_P]$ , $[0,f_\wedge]$ , and $[0, 1/n]$ , respectively, while $\theta\, \mathbf{e_1} - \delta_0 + W$ are chosen independently according to the restriction of $\nu$ on $B(\theta\, \mathbf{e_1}- \delta_0,\, 3\eta/4)$ (see notably [Reference Daley and Vere-Jones15, Chapter 2.4]). Thanks to [H4], W has a lower-bounded density $d_W$ on $B(0,\, 3\eta/4)$ .

The following lemma motivates this description.

Step 2: proof of Lemma 6.5.

Step 2.1. We prove that on the event $\mathcal{W}$ defined by (6.19),

(6.21) $$ \begin{equation} \forall \,t< T_{J},\quad X_t\,:\!=\, x_0 - v\, t\, \mathbf{e_1}. \end{equation}$$

Indeed, $t_P< T^Y$ implies that for any $t\le T_{J}$ , $Y_t \in [y_\wedge,\, y_\vee]$ . Thanks to (6.16), any ‘potential jump’ $\big(T^{\prime}_{J}, W^{\prime}, U^{\prime}_f, U^{\prime}_g\big)$ such that $T^{\prime}_{J} \le T_{J}$ and either $U^{\prime}_f > f_\vee$ or $U^{\prime}_g > n$ will be rejected. Thanks to the definition of $T_{J}$ , with (6.17), (6.18), and (6.19), no other jump can occur; thus (6.21) holds.

Note that, in order to prove this rejection very rigorously, we would like to consider the first one of such jumps. This cannot be done for (X, Y) directly, but it is easy to prove for any approximation of M where $u_f$ and $u_g$ are bounded. Since the result does not depend on these bounds and the approximations converge to (X, Y) (and are even equal to it, before $T_{J}$ , for bounds larger than $(f_\vee, n)$ ), (6.21) indeed holds.

Step 2.2. We then prove that the jump at time $T_{J}$ is surely accepted.

Since $x_I\in \mathcal{G}_n$ , by (6.15) and the definition of $(T_{J}, W, U_f, U_g)$ ,

$$\begin{align*} &U_f \le f_\wedge \le f(Y_{T_{J}}),\qquad U_g \le 1/n \le g(x_0 - v\, T_{J}\, \mathbf{e_1}, \theta\, \mathbf{e_1} - \delta_0 + W) \\&\hspace{5.3 cm} = g(X_{T_{J}-}, \theta\, \mathbf{e_1} - \delta_0 + W). \end{align*}$$

Thus,

$$ \begin{align*} X_{T_{J}} &= x_I + \delta_0 - v\, T_{J}\, \mathbf{e_1} + \theta\, \mathbf{e_1} - \delta_0 + W\\ &= x_I + (\theta - v\, T_{J})\, \mathbf{e_1} + W. \end{align*}$$

Step 2.3. We say that no jump can be accepted after $T_{J}$ , which is proved as in Step 2.1. This means that the following equalities hold for any $t \in [T_J, t_P]$ :

$$ X_t = X_{T_{J}} - v\cdot (t-T_{J})\,\mathbf{e_1} = x_I + W.$$

This concludes in particular the proof of Lemma 6.5 with $t=t_P = \theta/v$ .

Step 3. concluding the proof of Proposition 6.1.

Note that under $\textrm{P}^G$ , $\left \lbrace M(\mathcal{N}) = 0\right \rbrace$ is also independent of $\left \lbrace t_P < T^Y \right \rbrace$ and of $Y_{t_P}$ , so that

(6.22) $$\begin{multline} \textrm{P}^G_{(x_0, y_0)} \left[ (X, Y)_{t_P} \in (dx, dy);\, \mathcal{W} \right]\\= \textrm{P}(M(\mathcal{N}) = 0) \cdot \textrm{P}(M(\mathcal{J}) = 1) \cdot \textrm{P}^G_{y_0} \left( Y_{t_P} \in dy;\, t_P < T^Y\right)\\\times d_W\cdot \mathbf{1}_{B(x_I,\, 3\eta/4)}(x)\, dx. \end{multline}$$

Thanks to (6.17) and (6.18),

(6.23) $$\begin{multline} \textrm{P}(M(\mathcal{N}) = 0)\cdot \textrm{P}(M(\mathcal{J}) = 1) \\= (t_J\cdot f_\wedge/n)\cdot \nu\{ B(\theta\, \mathbf{e_1}- \delta_0,\, 3\eta/4)\} \cdot \exp\big[{-} t_P\cdot f_\vee\cdot n\cdot \nu\big(\mathbb{R}^d\big) \big] \ge c_X, \end{multline}$$

where the lower bound $c_X$ is chosen independently of $x_0$ and $y_0$ as follows:

$$\begin{equation*} c_X \,:\!=\, (t_J\cdot f_\wedge\cdot d_W/n)\cdot Leb\{B(0, 3\eta/4)\} \cdot \exp\big[{-} t_P\cdot f_\vee\cdot n\cdot \nu\big(\mathbb{R}^d\big) \big] >0. \end{equation*}$$

Thanks to Lemma 6.2 (recall the definitions of $y_\wedge$ and $y_\vee$ at the beginning of this subsection),

(6.24) $$\begin{equation} \textrm{P}^G_{y_0} \left( Y_{t_P} \in dy;\, t_P < T^Y\right) \ge c_B\; \mathbf{1}_{[1/n,\,n]}(y) \, dy. \end{equation}$$

Again, $c_B$ is independent of $x_0$ and $y_0$ .

Thanks to (6.20), (6.22), (6.23), (6.24) and Lemma 6.5, the following lower bound is valid for any $x_0 \in B(x_I, \eta/4)$ and any $y_0\in [1/n,\,n]$ with the constant value $c_P\,:\!=\, c_G\, c_X\, c_B\;d_W >0$ :

$$\begin{multline*} \textrm{P}_{(x_0, y_0)} \left[ (X, Y)_{t_P} \in (dx, dy);\, t_P <T_{(n)}\right] \ge c_P\; \mathbf{1}_{B(x_I,\, 3\eta/4)}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \; dx\, dy. \end{multline*}$$

This completes the proof of Proposition 6.1.

Proof of Lemma 6.3. The proof of Lemma 6.3 relies on principles similar to those of Proposition 6.1. In this case, $t_P$ is to be replaced by $u\in [0, u_\vee(x_I)]$ , and the event under consideration is simply the following:

$$\begin{equation*} \mathcal{W}^{\prime} \,:\!=\, \left \lbrace u < T^Y \right \rbrace \cap \left \lbrace M([0, u]\times \mathbb{R}^d \times [0, f_\vee]\times [0,n]) = 0\right \rbrace. \end{equation*}$$

The reasoning given for Step 2.1 can be applied to prove that for any $t\le u$ , $X_t\,:\!=\, x_0 - v\, t\, \mathbf{e_1}.$ We also exploit Theorem 6.1 for the independence property between X and Y under $\textrm{P}^G_{(x_I, y_I)}$ , and we use Lemma 6.2 to control the diffusion along the Y coordinate. Note that $c_B$ can be taken independently of $x_I$ , $y_I$ , and t (noting that t is uniformly upper-bounded by 2n). These arguments conclude the proof of the lower bound on the marginal of (X, Y) on the event $\{t<T_{(n)}\}$ .

The implication in terms of the sets $\mathcal{R}^{(n)}(t, c)$ is obtained simply by exploiting the Markov property, similarly to the way in which Corollary 6.1 is deduced as a consequence of Proposition 6.1.

Application to the various sets of assumptions

6.4. Proof of Theorem 4.2 under Assumption [D]

We treat in this subsection the mixing of X when both advantageous and deleterious mutations are occurring. More precisely, each step corresponds to each of the following lemmas.

Lemma 6.7. Suppose that the assumptions [H] and [D] hold. Then there exists $n \ge 3$ which satisfies the following property for any $t_1, t_2>0$ : there exist $t_R > t_1$ and $c_R>0$ such that for any $t\in [t_R, t_R+t_2]$ and $(x_0,y_0) \in \mathcal{A} \times [2, 3]$ ,

$$\begin{equation*}\textrm{P}_{(x_0,y_0)} \left[ (X, Y)_t\in (dx, dy);\, t < T_{(n)}\right]\ge c_R\; \mathbf{1}_{\mathcal{A}}(x)\, \mathbf{1}_{[2, 3]}(y) \, dx\, dy.\end{equation*}$$

Lemma 6.8. Suppose that the assumptions [H] and [D] hold. Then, for any $\ell_I>0$ , there exist $c_I, t_I>0$ and $n \ge \ell_I$ such that

$$\begin{equation*} \forall \,(x, y)\in \mathcal{D}_{\ell_I},\quad \textrm{P}_{(x, y)}\big( \tau_\mathcal{A} \le t_I \wedge T_{(n)}\big) \ge c_I. \end{equation*}$$

In the following subsections, we prove these three lemmas, then explain how Theorem 4.2 is deduced as a consequence of them.

6.4.2. Step 2: proof of Lemma 6.7

We keep $x_I\,:\!=\, -\theta\, \mathbf{e_1}$ and $x_1\,:\!=\, ({-}\theta+\eta/2)\, \mathbf{e_1}$ . Thanks to Lemma 6.6, there exist $n, t_1, c_1>0$ such that

$$\begin{equation*} \left \lbrace\, x_I + u\, \mathbf{e_1};\, u\in [\eta/6, 5\,\eta/6]\right \rbrace \subset \mathcal{R}^{(n)}(t_1, c_1).\end{equation*}$$

There exist $t_2, c_2 >0$ , thanks to Lemma 6.3, such that for all $t\in [t_2, t_2 + 2\,\eta/(3\,v)]$ , we have $x_I \in \mathcal{R}^{(n)}(t, c_2).$ Applying Corollary 6.1 twice, with the knowledge that $B(x_I, \eta/2)$ is a subset of $\mathcal{G}_n$ , we deduce that there exist $t_3, c_3>0$ such that

$$\forall \,t\in [t_3, t_3 + 2\,\eta/(3\,v)],\quad \mathcal{A} \subset \mathcal{R}^{(n)}(t, c_3).$$

Inductively applying Lemma 6.4, we deduce the following for any $k\ge 1$ :

$$\forall \,t\in [k\,t_3, k\,t_3 + 2\,k\,\eta/(3\,v)],\quad \mathcal{A} \subset \mathcal{R}^{(n)}\big(t, c_3\cdot [c_3\cdot c_B] ^{k-1}\big).$$

Let $t_1, t_2>0$ and consider $k\ge 1$ sufficiently large for $k\,t_3>t_1$ and $2\,k\,\eta/(3\,v)>t_2$ to hold. Then Lemma 6.7 is proved with this value of n, $t_R\,:\!=\, k\,t_3$ , and $c_R \,:\!=\, c_3\cdot [c_3\cdot c_B] ^{k-1}$ .

6.4.3. Step 3: proof of Lemma 6.8

As before, we can find $n\ge \ell_I$ such that $ \mathcal{D}_{\ell_I}\subset \mathcal{G}_n$ . We go backwards in time from $\mathcal{A}$ by defining, for $t\ge 0$ , $c>0$ ,

$$\begin{equation*} \mathcal{R}^{\prime}(t, c)\,:\!=\, \left \lbrace (x, y)\in \mathcal{G}_n;\, \textrm{P}_{(x,y)} \left[ \tau_\mathcal{A} \le t \wedge T_{(n)}\right] \ge c \right \rbrace.\end{equation*}$$

It is clear that $\mathcal{A}\subset \mathcal{R}^{\prime}(0, 1)$ . Thanks to Proposition 6.1 and the Markov property, there exist $t_P, c_P>0$ such that, for any $t, c>0$ ,

$$\begin{equation*} \left \lbrace x\in \mathcal{G}_n;\, d(x, \mathcal{R}^{\prime}(t, c))\le \eta/4\right \rbrace \subset \mathcal{R}^{\prime}(t+t_P, c\cdot c_P).\end{equation*}$$

Since $\mathcal{D}_{\ell_I}\subset \mathcal{G}_n$ is bounded, an immediate induction ensures that there exist $t_I, c_I>0$ such that $\mathcal{D}_{\ell_I}\subset \mathcal{R}^{\prime}(t_I, c_I)$ . This concludes the proof of Lemma 6.8.

6.4.4. Theorem 4.2 as a consequence of Lemmas 6.6–6.8

The proof is quite naturally adapted from that of Lemma 3.2.1 in [Reference Velleret32]. Note that for any $n_1\le n_2$ , $T_{(n_1)}\le T_{(n_2)}\le \tau_\partial$ holds a.s.

Let $\ell_I, \ell_M\ge 0$ . According to Lemma 6.8, we can find $c_I, t_I>0$ and $n_1\ge \ell_I\wedge \ell_M$ such that for any $(x_I, y_I)\in \mathcal{D}_{\ell_I}$ ,

(6.25) $$\begin{equation} \textrm{P}_{(x_I, y_I)}\big( \tau_\mathcal{A} \le t_I\wedge T_{(n_1)} \big) \ge c_I.\end{equation}$$

Also, let $n_2 \ge n_1$ , $c_R,t_R>0$ be chosen, according to Lemma 6.7, to satisfy that for any $t\in [t_R, t_R+t_I]$ and $(x_0,y_0) \in \mathcal{A} \times [2, 3]$ ,

(6.26) $$\begin{equation} \textrm{P}_{(x_0,y_0)} \left[ (X, Y)_t\in (dx, dy) ;\, t < T_{(n_2)} \right] \ge c_R\; \mathbf{1}_{\mathcal{A}}(x)\, \mathbf{1}_{[2, 3]}(y) \, dx\, dy.\end{equation}$$

Thanks to Lemma 6.6, since $\mathcal{D}_{\ell_M}$ is a bounded set, we know that there exist $n\ge n_2$ , $c_F$ , and $t_F> 0$ such that for any $(x_0,y_0) \in \mathcal{A} \times [2,\,3]$ ,

(6.27) $$\begin{equation} \textrm{P}_{(x_0,y_0)} \left[ (X, Y)_{t_k}\in (dx, dy) ;\, t_k < T_{(n)} \right] \ge c_F\; \mathbf{1}_{\mathcal{D}_{\ell_M}}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \, dx\, dy.\end{equation}$$

The fact that n is larger than $n_1$ and $n_2$ implies without difficulty that (6.25) and (6.26) hold with $n_1$ and $n_2$ replaced by n, which is how these statements are exploited in the following reasoning.

Let $t_M\,:\!=\, t_I + t_R + t_F$ and $c_M\,:\!=\, c_I \cdot c_R \cdot Leb(\mathcal{A}) \cdot c_F$ . For any $(x_I, y_I)\in \mathcal{D}_{\ell_I}$ , by combining (6.26), (6.27), and the Markov property, we deduce that a.s. on the event $\big\{\tau_\mathcal{A} \le t_I\wedge T_{(n)}\big\}$ ,

$$\begin{align*}& \textrm{P}_{(X, Y)[\tau_\mathcal{A}]} \left[ \big(\widetilde X, \widetilde Y\big)[t_M - \tau_\mathcal{A}] \in (dx, dy) ;\, t_M- \tau_\mathcal{A} < \widetilde{T}_{(n)} \right] \\&\hspace{1 cm}\ge c_F\cdot \textrm{P}_{(X, Y)[\tau_\mathcal{A}]} \left[ \big(\widetilde X, \widetilde Y\big)[t_M - t_F- \tau_\mathcal{A}] \in \mathcal{A}\times [2, 3] ;\, t_M - t_F - \tau_\mathcal{A} < \widetilde T_{(n)} \right] \\&\hspace{1.5 cm} \times \mathbf{1}_{\mathcal{D}_{\ell_M}}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \, dx\, dy \\&\hspace{1 cm}\ge c_R \cdot Leb(\mathcal{A}) \cdot c_F \cdot \mathbf{1}_{\mathcal{D}_{\ell_M}}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \, dx\, dy,\end{align*}$$

where we have exploited the knowledge that $\tau_\mathcal{A}\le t_I$ to deduce that $t_M - t_F - \tau_\mathcal{A} \in [t_R, t_R+t_I]$ . By combining this estimate with (6.25) and again the Markov property, we conclude that

$$\begin{align*} &\textrm{P}_{(x_I, y_I)} \left[ \big(X_{t_M}, Y_{t_M}\big) \in (dx, dy) ;\, t_M< T_{(n)} \right] \\&\hspace{1 cm} \ge \textrm{P}_{(x_I, y_I)}( \tau_\mathcal{A} \le t_I\wedge T_{(n)} ) \cdot c_R \cdot Leb(\mathcal{A}) \cdot c_F \cdot \mathbf{1}_{\mathcal{D}_{\ell_M}}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \, dx\, dy \\&\hspace{1 cm} \ge c_M\, \mathbf{1}_{\mathcal{D}_{\ell_M}}(x)\, \mathbf{1}_{[1/n,\,n]}(y) \, dx\, dy.\end{align*}$$

This completes the proof of Theorem 4.2 with $L=2n$ , $c\,:\!=\, c_M$ , and $t\,:\!=\,t_M$ under Assumption [D].

6.5. Proof of Theorem 4.2 under Assumption [A] and $d\ge 2$

The proof of Theorem 4.2 under Assumption [A] and $d\ge 2$ is handled in the same way as the proof in Subsection 6.4.4. Notably, the lemmas that replace Lemmas 6.7–6.8 have identical implications, as shown below.

Lemma 6.10. Suppose that $d\ge 2$ , and that the assumptions [H] and [A] hold. Then there exists $n \ge 3$ which satisfies the following property for any $t_1, t_2>0$ : there exist $t_R > t_1$ and $c_R>0$ such that, for any $t\in [t_R, t_R+t_2]$ and $(x_0,y_0) \in \mathcal{A} \times [2, 3]$ ,

$$\begin{equation*} \textrm{P}_{(x_0,y_0)} \left[ (X, Y)_t\in (dx, dy) ;\, t < T_{(n)}\right] \ge c_R\; \mathbf{1}_{\mathcal{A}}(x)\, \mathbf{1}_{[2, 3]}(y) \, dx\, dy.\end{equation*}$$

Lemma 6.11. Suppose that $d\ge 2$ , and that the assumptions [H] and [A] hold. Then, for any $\ell_I>0$ , there exist $c_I, t_I>0$ and $n \ge \ell_I$ such that

(6.28) $$\begin{equation} \forall \,(x_0, y_0)\in \mathcal{D}_{\ell_I},\quad \textrm{P}_{(x_0, )}( \tau_\mathcal{A} \le t_\mathcal{A} \wedge T_{(n)} ) \ge c_\mathcal{A}. \end{equation}$$

Since the implications are the same, the proof of Theorem 4.2 under Assumption [A] with $d\ge 2$ as a consequence of Lemmas 6.9–6.11 is mutatis mutandis the same as the proof given in Subsection 6.4.4. However, since deleterious mutations are now forbidden, the proof of Lemma 6.9 is much trickier than that of Lemma 6.6. The first step is given by the following two lemmas. To this end, given any direction $\mathbf{u}$ on the sphere $S^d$ of radius 1, we denote its orthogonal component by

(6.29) $$\begin{equation} x^{(\perp \mathbf{u})}\,:\!=\, x - \langle x, \mathbf{u}\rangle \mathbf{u}, \quad \text{ and specifically for $\mathbf{e_1}$,}\quad x^{(\perp 1)}\,:\!=\, x - \langle x, \mathbf{e_1}\rangle \mathbf{e_1}.\end{equation}$$

Lemma 6.12. Suppose that $d\ge2$ , and that the assumptions [H] and [A] hold. Then, for any $x_\vee> 0$ , there exists $\epsilon \le \eta/8$ which satisfies the following property for any $n \ge 3\vee (2\, \theta)$ , $x\in B(0, n)$ , and $\mathbf{u}\in S^d$ such that both $\langle x,\mathbf{u}\rangle \ge \theta$ and $\|x^{(\perp \mathbf{u})}\|\le x_\vee$ : there exist $t_P, c_P>0$ such that for any $t, c>0$ ,

$$ \begin{equation*} x \in \mathcal{R}^{(n)}(t, c) \Rightarrow \bar{B}(x - \theta\, \mathbf{u}, \epsilon)\subset \mathcal{R}^{(n)}(t+t_P, c\cdot c_P). \end{equation*}$$

Lemma 6.13. Suppose that $d\ge2$ , and that the assumptions [H] and [A] hold. Then, for any $m \ge 3\vee (2\, \theta)$ , there exists $\epsilon \le \eta/8$ which satisfies the following property for any $x\in B(0, m)$ with $\langle x, \mathbf{e_1} \rangle\le 0$ : there exist $t_P, c_P>0$ such that

$$ \begin{equation*} \forall \,t, c>0,\quad x \in \mathcal{R}^{(L)}(t, c) \Rightarrow \bar{B}(x, \epsilon)\subset \mathcal{R}^{(L)}(t+t_P, c\cdot c_P). \end{equation*}$$

Lemma 6.13 is actually directly implied by Lemma 6.3 (first applied for a time-interval $[0, \theta/v]$ ), then Lemma 6.12 with $\mathbf{u} \,:\!=\,\mathbf{e_1}$ , combined with the Markov property. Subsection 6.5.1 is dedicated to the proof of Lemma 6.12.

6.5.1. Step 1: proof of Lemma 6.12

Fix $x_\vee>0$ . Consider $\epsilon>0$ ; this will be fixed later, but assume already that $\epsilon \le \theta/8$ . We recall that $\eta\le \theta/8$ is assumed without loss of generality. Let $n \ge 3\vee(2\theta)$ , $x_0\in B(0, n)$ , and $\mathbf{u} \in S^d$ be such that both $\langle x_0,\mathbf{u}\rangle \ge \theta$ and $\|x_0^{(\perp \mathbf{u})}\|\le x_\vee$ hold.

Compared to Proposition 6.3, the first main difference is that the jump is now almost instantaneous. The second is that, in order to have $g_\wedge >0$ , we have much less choice in the value of w when $\|x^{(\perp \mathbf{u})}\|$ is large. In particular, the variability of any particular jump will not be sufficient to wipe out the initial diffusion around x deduced from $x \in \mathcal{R}^{(n)}(t, c)$ , but rather will make it even more diffuse.

To fix $\epsilon>0$ , let us first compute, for $\delta\in B(0, \eta)$ and $w\in B({-}\theta\, \mathbf{u}, \epsilon)$ ,

$$ \begin{align*} &\|x_0+\delta\|^2 - \|x_0+\delta + w\|^2 = 2\, \langle x_0 + \delta, w\rangle - \|w\|^2 \\&\hspace{1 cm} \ge \left(\frac{7}{4} - \frac{9}{8} \mathbin{\! \times \!} \left(\frac{1}{4}+\frac{9}{8}\right)\right)\, \theta^2- 2\, \epsilon \; x_\vee,\end{align*}$$

where we have exploited that $\langle \mathbf{u}, w\rangle \ge 7\, \theta/8$ . We note that

$$\begin{equation*} c\,:\!=\, \frac{7}{4} - \frac{9}{8} \mathbin{\! \times \!} \left(\frac{1}{4}+\frac{9}{8}\right) = \frac{13}{64}>0.\end{equation*}$$

By taking $\epsilon\,:\!=\, \{c\,\theta^2/(4\,x_\vee)\}\wedge \eta$ (recall that $\eta \le \theta/8$ ), we thus ensure that $\|x_0+\delta\|^2> \|x_0+\delta+w\|^2 $ . Note that $\epsilon$ does not depend on the specific choice of $x_0$ .

Let $t_P\,:\!=\,\epsilon/(2\,v)$ . The initial condition for X, Y is taken as $x_I\in B(x, \eta/2)$ and $y_I\in [1/n,\,n]$ . Let

$$\begin{align*} &g_\wedge \,:\!=\, \inf\left \lbrace\, g(x, w);\, x \in \bar{B}(x_0, \eta)\,,\, w\in \bar{B}({-}\theta\,\mathbf{u}, \epsilon)\right \rbrace >0, \\& \mathcal{X}^M\,:\!=\, [0, t_P]\times \mathbb{R}^d \times [0, f_\vee]\times [0,n], \\& \mathcal{J} \,:\!=\, [0, t_P] \times B({-}\theta\, \mathbf{u} +(\epsilon/2)\, \mathbf{e_1}, \epsilon/2) \times [0, f_\wedge] \times [0, g_\wedge].\end{align*}$$

With the same reasoning as in the proof of Proposition 6.1, we obtain a change of probability $\textrm{P}^G_{(x_I, y_I)}$ and an event $\mathcal{W}$ on which the random variable W is uniquely defined from M under $\textrm{P}^G_{(x_I, y_I)}$ , and such that it satisfies, a.s.,

$$ \begin{equation*} X_{t_P} = x_I - (\epsilon/2)\, \mathbf{e_1} - \theta\, \mathbf{u} + (\epsilon/2)\, \mathbf{e_1} + W= x_I - \theta\, \mathbf{u} + W,\end{equation*}$$

where the density of W is lower-bounded by $d_W$ on $B(0, \epsilon/2)$ , uniformly over $x_I$ (given x) and $y_I$ . We thus similarly obtain some constants $c_P, c^{\prime}_P >0$ independent of $x_0$ such that for any such $x_0$ ,

$$ \begin{align*}& \int_{B(x_0, \eta/2)}\,dx_I \int_{[1/n,\,n]}\, dy_I\, \textrm{P}_{(x_I, y_I)} \left[ (X, Y)_{t_P} \in (dx, dy);\, t_P <T_{(n)}\right]\\&\hspace{1 cm}\ge c_P\, \int_{B(x_0, \eta/2)}\,dx_I\mathbf{1}_{B(x_I - \theta\, \mathbf{u},\, \epsilon/2)}(x)\cdot\mathbf{1}_{[1/n,\,n]}(y) \; dx\, dy\\&\hspace{1 cm}\ge c^{\prime}_P\, \mathbf{1}_{B(x_0 - \theta\, \mathbf{u},\,\eta/2 + \epsilon/3)}(x)\mathbf{1}_{[1/n,\,n]}(y) \; dx\, dy.\end{align*}$$

We then reason similarly as in the proof of Corollary 6.1 as a consequence of Proposition 6.1. Assuming further that $x_0 \in \mathcal{R}^{(n)}(t, c)$ for some $t,c >0$ , we can deduce that

$$B(x- \theta\,\mathbf{u}, \epsilon /3) \subset \mathcal{R}^{(n)}(t+t_P, c\cdot c_P).$$

This is exactly the implication of Lemma 6.12, stated in terms of $\epsilon/3$ instead of $\epsilon$ .

6.5.2. Step 2: Lemma 6.9 as a consequence of Lemmas 6.13 and 6.12

Step 2.1: $x_I\in \mathcal{R}^{(n_0)}(t_0, c_0)$ . Let $x_I\,:\!=\, -\theta \mathbf{e_1}$ . We check that there exists $n_1\ge 1$ such that $B(x_I, \eta/2)$ is a subset of $\mathcal{G}_{n_1}$ . Since g is continuous, and thanks to Assumption [A], it is sufficient to prove that $\|x_I-z \mathbf{e_1} + \delta\| > \|x_I-z \mathbf{e_1} + \delta +w\| $ holds for any $z\in [0, \theta]$ , $\delta\in \bar{B}(0, \eta)$ , and $w\in \bar{B}(\theta\, \mathbf{e_1},\eta)$ :

$$\begin{align*} &\|x_I-z \mathbf{e_1} + \delta\|^2 - \|x_I-z \mathbf{e_1} + \delta + w\|^2 = 2 \langle (\theta +z) \mathbf{e_1} - \delta, w\rangle - \|w\|^2 \\& \ge 2\, [\theta\cdot (\theta - \eta) - \eta\cdot(\theta+\eta)] - (\theta+\eta)^2 \\&\hspace{1 cm} = \theta^2 -6\, \theta\,\eta - 3\, \eta^2 \ge \dfrac{13 \theta^2}{64} > 0,\end{align*}$$

since $\eta \le \theta/8$ , as assumed above, just after (6.13). Applying Proposition 6.1 twice, we conclude that there exist $n_0\ge 1$ , $t_0, c_0>0$ such that $x_I\in \mathcal{R}^{(n_0)}(t_0, c_0)$ .

Step 2.2: under the condition that $\langle x_F, \mathbf{e_1}\rangle\,:\!=\, -\theta$ . The purpose of this step is to prove the following lemma, in which we employ the notation $\pi_1:x\mapsto \langle x, \mathbf{e_1}\rangle$ .

Lemma 6.14. For any $n\ge 1$ sufficiently large, there exist $t, c>0$ such that $\pi_1^{-1}({-}\theta)\cap B(0,n)$ is a subset of $\mathcal{R}^{(n)}(t, c)$ .

Let $x_F \in \pi_1^{-1}({-}\theta)\cap B(0,n)$ , where we assume that n is larger than $n_0$ , 3, and $2\theta$ . First, we define $\mathbf{u}$ as $\mathbf{e_1}$ if $x_F^{(\perp 1)}=0$ and as $\mathbf{u}\,:\!=\, x_F^{(\perp 1)}/\big\|x_F^{(\perp 1)}\big\|$ otherwise. Note that $\big\|x_F^{(\perp 1)}\big\|\le n$ . We consider the value of $\epsilon$ given by Lemma 6.13 for $x_\vee \,:\!=\, n$ and make the following definitions:

$$ \begin{align*}K\,:\!=\, \left \lfloor n \epsilon\right \rfloor +1,\quad \text{ and for } 0\le k\le K, \quad x_k\,:\!=\, -\theta\, \mathbf{e_1} + \frac{k\, \big\|x_F^{(\perp 1)}\big\|}{K}\; \mathbf{u}.\end{align*}$$

This choice ensures that for any $k\in [\![0, K-1]\!]$ , $x_{k+1}\in B(x_k, \epsilon)$ , while $x_k\in B(0, n)$ , $\langle x_{k}\, \big| \, \mathbf{e_1}\rangle \le 0$ , and $x_K = x_F$ . Thanks to Step 2.1, $x_0\in \mathcal{R}^{(n)}(t_0, c_0)$ . Thus, by induction over $k\le K$ with Lemma 6.13, $x_k \in \mathcal{R}^{(n)}\big(t_0 + k\, t_P,\, c_0\, [c_P]^k\big)$ . In particular, there exist $t, c>0$ such that $x\in \mathcal{R}^{(n)}(t, c)$ , which concludes Step 2.2.

Step 2.3: the general case. Assume solely that $x \in \mathcal{B}(0, m)$ . We consider the value of $\epsilon$ given by Lemma 6.12 for $x_\vee\,:\!=\, m$ . The choice of $\mathbf{u}$ is as in Step 2.2.

Let

(6.30) $$ \begin{equation}K\,:\!=\, \left \lfloor\frac{m + \theta}{\epsilon}\right \rfloor +1,\quad \text{so that }\frac{\langle x, \mathbf{e_1}\rangle + \theta}{K} \le \epsilon,\end{equation}$$

and for $0\le k \le K$ , let

$$\begin{equation*}x_k\,:\!=\, ({-}\theta+ (k/K)\cdot (\langle x, \mathbf{e_1}\rangle + \theta)) \, \mathbf{e_1}+(K- k)\,\theta\, \mathbf{u}+ x^{(\perp 1)}.\end{equation*}$$

In particular $\langle x_0,\mathbf{e_1}\rangle = -\theta$ , and $x_K = x_F$ , while for any $k\le K-1$ , $x_{k+1}\in B(x_k, \epsilon)$ , $x_k \in B(0, m+K\, \theta)$ , and $\langle x_k, \mathbf{u}\rangle \le \theta\vee \langle x, \mathbf{e_1}\rangle\le m=x_\vee$ .

Since $\langle x_0,\mathbf{e_1}\rangle = -\theta$ , we can exploit Lemma 6.14 to prove that there exist $n\ge 1$ and $t_0, c_0>0$ independent of $x_F$ such that $x_0\in \mathcal{R}^{(n)}(t_0, c_0)$ . Thanks to Lemma 6.12 and induction on k, we deduce that there exist $t_P, c_P>0$ such that $x_k\in \mathcal{R}^{(n)}(t_0 + k\, t_P,\, c_0\, [c_P]^k)$ . In particular, there exist $t,c>0$ such that $x_F \in \mathcal{R}^{(n)}(t, c)$ .