1.1 Conditioned stochastic stability

The notion of conditioned stochastic stability that we propose is based on ideas from the theory of absorbing Markov processes [Reference Collet, Martínez and Schmitt28] and conditioned random dynamical systems [Reference Castro, Chemnitz, Chu, Engel, Lamb and Rasmussen17–Reference Castro, Lamb, Olicón-Méndez and Rasmussen19, Reference Engel, Lamb and Rasmussen39, Reference Zmarrou and Homburg87]. As mentioned above, the statistical behaviour of a Markov process $X_n$ on a state space M conditioned upon remaining outside of a subset $\partial \subset M$ is captured by its quasi-ergodic measure $\nu $ on $M \setminus \partial $ [Reference Breyer and Roberts15, Reference Colonius and Rasmussen29, Reference Darroch and Seneta32, Reference Zhang, Li and Song86]. This object describes the limiting distribution of the conditioned Birkhoff averages of $X_n$ , i.e. given an observable $h: M \to \mathbb {R}$ , it holds that for $\nu $ -almost every (a.e.) $x \in M\setminus \partial $ ,

$$ \begin{align*} \mathbb{E}_x\bigg[\frac{1}{n}\sum_{i= 0}^{n-1}h \circ X_i \, \bigg|\, \tau>n \bigg] := \frac{1}{\mathbb{P}_x[\tau > n]} \mathbb{E}_x\bigg[\mathbb{1}_{\{\tau > n\}}\frac{1}{n}\sum_{i= 0}^{n-1} h \circ X_i\bigg]\xrightarrow[]{n \to \infty} \int h(x) \nu(dx), \end{align*} $$

where conditioning upon $\tau := \min \{i \in {\mathbb N}; X_i \in \partial \}> n$ ensures the process has not been absorbed by time n.

Given a map $T:M \to M$ on a manifold M and a subset $\partial \subset M$ , consider the Markov process $X_n^\varepsilon $ on M generated by $\varepsilon $ -bounded random perturbations of T. Conditioned stochastic stability concerns the quasi-ergodic measures of $X_n^\varepsilon $ on $M\setminus \partial $ and their limit as the amplitude of the perturbation $\varepsilon $ goes to 0. When these quasi-ergodic measures converge to a T-invariant measure $\nu _0$ (in the weak $^*$ topology), we say that the limiting measure is conditioned stochastically stable on $M \setminus \partial $ . Observe that this notion depends on the choice of random perturbation generating $X_n^\varepsilon $ , which is also true for (classical) stochastic stability. As is common in the study of (classical) stochastic stability, we only consider random bounded diffusive perturbations [Reference Alves, Araújo and Vásquez2, Reference Araújo3, Reference Baladi and Viana7, Reference Benedicks and Viana9, Reference Benedicks and Young10] (see §2 for the precise details) and prove that all such random perturbations give rise to the same notion.

In the context of uniformly expanding repellers, it is natural to assume that the repelling set is characterized by

(1.1) $$ \begin{align} \Lambda = \bigcap_{n \geq 0} T^{-n}(M \setminus \partial), \end{align} $$

where $\partial $ could be, for example, the complement of a neighbourhood of the repeller (see §§4 and 2.4.2) or a small open neighbourhood of the attractors of T (see §§5 and 2.4.1). In this paper, we prove the following result (see Theorem 2.13 for a more precise and more general result).

Remark. The measure $\nu _0$ is the equilibrium state associated with the (geometric or natural) potential $- \log |\!\det d T|$ , see §1.2. Moreover, recall that a T-invariant probability measure $\mu $ is said to be mixing if for any measurable sets $A,B$ it holds that

$$ \begin{align*}\lim_{n \to \infty} \mu(T^{-n}(A) \cap B - \mu(A)\mu(B) = 0.\end{align*} $$

Importantly, we highlight that the closure of the set of periodic points of $\Lambda $ may not be topologically transitive (see Lemma 2.6) and so, it is not clear whether perturbation arguments on the spectrum of the transfer operator [Reference Gouëzel and Liverani44, Reference Keller and Liverani52] are applicable. Moreover, this result allows for the study of stochastic stability in open systems as it provides a new perspective based on conditioned random dynamics and circumventing the lack of continuity mentioned in [Reference Gouëzel and Liverani44, §8.1.2].

As mentioned in Theorem A (3), it turns out that $\nu _0$ is a well-known object in the theory of thermodynamic formalism [Reference Climenhaga and Pesin26, Reference Ruelle71] and corresponds to the unique equilibrium state on the set $R~:= ~\overline {\{p \in \Lambda ;\ p \text { is } T\text{-periodic}\}}$ associated with the potential ${-}\! \log |\!\det d T|$ , i.e. $\nu _0$ is the unique T-invariant measure satisfying

$$ \begin{align*}h_{\nu_0}(T) - \int \log|\!\det d T| \, d \nu_0 = \sup_{\mu \in {\mathcal{I}}(T, R)}\bigg(h_{\mu}(T) -\int \log|\!\det d T| \, d\mu\bigg),\end{align*} $$

where $h_\mu $ is the Kolmogorov–Sinai (or metric) entropy [Reference Kolmogorov55, Reference Sinaĭ78] and ${\mathcal {I}} (T, R)$ is the set of T-invariant probability measures on R. This result has its parallel in the (classical) theory of stochastic stability. Indeed, given a uniformly hyperbolic transformation $T:M \to M$ on a compact metric space M, it is well known that stochastically stable invariant measures on attractors correspond to the equilibrium states from the thermodynamic formalism associated with the potential ${-}\! \log |\!\det d T |_{E^u}|$ , where $E^u$ denotes the unstable expanding direction of T [Reference Young85].

In this paper, we uncover a stronger connection between conditioned stochastic stability and the thermodynamic formalism, establishing the approximation of any equilibrium state by quasi-ergodic measures of weighted Markov processes.

1.2 Thermodynamic formalism and weighted Markov processes

The thermodynamic formalism is a powerful framework for the analysis of statistical properties of dynamical systems. Pioneered by Sinai, Ruelle and Bowen [Reference Bowen13, Reference Bowen and Ruelle14, Reference Ruelle70, Reference Ruelle71, Reference Sinaĭ79], and motivated by the field of statistical physics, this theory aims to describe properties of equilibrium states, such as the measure of maximal entropy and other invariant Gibbs measures [Reference Baladi6, Reference Climenhaga and Pesin26].

Given a T-invariant set $\Lambda \subset M$ , an equilibrium state on $\Lambda $ is defined for each given potential $\psi :\Lambda \to \mathbb {R}$ as an invariant measure $\nu ^\psi $ on $\Lambda $ whose metric pressure is equal to the topological pressure $P(T,\psi ,\Lambda )$ of the system on $\Lambda $ , i.e. $\nu ^\psi $ satisfies

(1.2) $$ \begin{align} h_{\nu^\psi}(T) +\int \psi\, d \nu^\psi = \sup_{\mu \in {\mathcal{I}}(T,\Lambda)}\bigg(h_{\mu}(T) +\int \psi\, d\mu\bigg)=: P(T,\psi,\Lambda). \end{align} $$

In particular, observe that when $\psi =0$ , the equilibrium states associated with this potential correspond to the measures of maximal entropy [Reference Viana and Oliveira82, §10.5]. Moreover, a classical result of Ruelle (see [Reference Ruelle72, Lemma 1.4] or Lemma 2.6 below) provides the existence and uniqueness of equilibrium states for Hölder potentials on uniformly expanding repellers [Reference Viana and Oliveira82, Theorem 12.1].

It is natural to ask whether the definition of conditioned stochastic stability can be extended to approximate other equilibrium states of T. This question appears not to have been raised in the literature, even for stochastic stability of equilibrium states on attractors. Here, we show that equilibrium states on uniformly expanding repellers are approximated by quasi-ergodic measures of weighted Markov processes [Reference Champagnat, Strickler and Villemonais21, Reference Champagnat and Villemonais23, Reference Del Moral36, Reference Kim, Tagawa and Velleret54], providing a general notion of conditioned stochastic stability.

Given a Markov process $X_n$ on M, consider a non-positive weight function ${\phi :M \to \mathbb R_{\leq 0}}$ (the weight function is sometimes referred to as a ‘potential’ in the literature [Reference Del Moral36, Reference Zhang, Li and Song86], but here we reserve the term ‘potential’ for the symbol $\psi $ in (1.2)) and define the new process $X_n^\phi $ by

(1.3) $$ \begin{align} X_{n+1}^{\phi} = \begin{cases} X_{n+1} & \text{with probability } e^{\phi(X_n)},\\* \partial & \text{with probability } 1-e^{\phi(X_n)}, \end{cases} \end{align} $$

where $\partial $ is a cemetery state. By abuse of notation, we consider the statements $X_n \in \partial $ and $X_n = \partial $ to be equivalent. If $X_n$ is already an absorbing Markov process killed at $\partial ^\prime $ , we may (and do) set $\partial =\partial ^\prime $ . We refer to the new Markov process $X_n^\phi $ as an $e^\phi $ -weighted Markov process. A quasi-ergodic measure $\nu ^\phi $ provides the statistical behaviour of the process when conditioned upon survival on $M \setminus \partial $ .

Definition 1.1. We say that $\nu ^\phi $ is a quasi-ergodic measure for the $e^\phi $ -weighted Markov process $X^\phi _n$ if for any observable $h:M \to \mathbb R$ ,

$$ \begin{align*} \mathbb{E}_x^\phi \bigg[\frac{1}{n}\sum_{i= 0}^{n-1}h \circ X_i^\phi \, \bigg|\, \tau^\phi> n \bigg] \xrightarrow[]{n \to \infty} \int h(x) \nu^{\phi}(dx) \end{align*} $$

for $\nu ^\phi $ -a.e. $x \in M \setminus \partial $ , where $\tau ^{\phi }:= \min \{n \in {\mathbb N};\, X_n^{\phi } \in \partial \}$ and $\mathbb {E}^\phi _x$ is the expectation with respect to the weighted process $X_n^\phi $ with $X_0^\phi = x$ .

The random variable $\tau ^\phi $ denotes the time at which the process is killed, either by dynamically entering $\partial $ (hard killing) or due to the weight $e^\phi $ (soft killing). When both are present, the conditioned Birkhoff averages simplify to (see §2 for precise details)

(1.4) $$ \begin{align} \mathbb{E}_x^\phi \bigg[\frac{1}{n}\sum_{i= 0}^{n-1}h \circ X_i^\phi \, \bigg|\, \tau^\phi> n \bigg] & {:= \frac{1}{\mathbb{E}_x[e^{S_n\phi}\mathbb{1}_{\{\tau > n\}}]}{\mathbb{E}_x\bigg[ e^{S_n\phi}\mathbb{1}_{\{\tau > n\}} \frac{1}{n}\sum_{i= 0}^{n-1}h \circ X_i \bigg]}}\nonumber\\ &\quad { \xrightarrow[]{n \to \infty} \int h(x) \nu^{\phi}(dx),} \end{align} $$

where $\tau = \min \{n;\, X_n \in \partial \}$ relates to hard killing and $S_n\phi := \sum _{i= 0}^{n-1} \phi \circ X_i$ relates to soft killing. Note that when $\phi = 0$ , we recover the setting introduced in the previous section.

Observe that the right-hand side of (1.4) is also well defined as long as $\phi $ is measurable and bounded, even if it is occasionally positive. Indeed,

$$ \begin{align*}\frac{\mathbb{E}_x[ e^{S_n\phi}\mathbb{1}_{\{\tau> n\}} ({1}/{n})\sum_{i= 0}^{n-1}h \circ X_i ]}{\mathbb{E}_x[e^{S_n{\phi}}\mathbb{1}_{\{\tau > n\}}]} = \frac{\mathbb{E}_x[ e^{S_n\bar{\phi}}\mathbb{1}_{\{\tau > n\}} ({1}/{n})\sum_{i= 0}^{n-1}h \circ X_i ]}{\mathbb{E}_x[e^{S_n\bar{\phi}}\mathbb{1}_{\{\tau > n\}}]}, \end{align*} $$

where $\bar {\phi } = \phi - \sup \phi _+$ , with $\phi _+(x):= \max \{\phi (x),0\}$ . Defining the $e^\phi $ -weighted process $X_n^\phi $ to be equal to $X_n^{\bar {\phi }}$ , we recover the interpretation from (1.3).

Recall that $X_n^{\varepsilon }$ is a Markov process on M generated by $\varepsilon $ -bounded random perturbations of the map T and absorbed on $\partial \subset M$ , and denote by $X_n^{\varepsilon , \phi }$ the appropriate weighting of $X_n^{\varepsilon }$ as in (1.4).

We generalize Theorem A to allow for soft killing and show conditioned $e^\phi $ -weighted stochastic stability with the following result (see Theorem 2.13 for a more precise and more general statement).

In particular, for every repeller $R^1, \ldots , R^k \subset R$ of the dynamical decomposition of T (see Lemma 2.6), Theorem B also holds for the process conditioned upon survival in $R^{i}_\delta =~M \setminus \partial , i =1, \ldots , k$ , where $R^i_\delta $ is a sufficiently small $\delta $ -neighbourhood of the repeller $R^i$ . This gives rise to the following corollary for each $i = 1, \ldots , k$ (see Theorem 2.12 for more precise details).

Observe that the main difference between both results relies on the choice of $\partial $ . On the one hand, Theorem B corresponds to the so-called global problem and requires $\Lambda \subset M \setminus \partial $ . On the other hand, Corollary B1 refers to the local problem, as the process is conditioned upon survival locally around some (i.e. in a small neighbourhood of) $R^i$ . We mention that Corollary B1 as stated above can be derived directly from spectral stability arguments [Reference Keller and Liverani52]. In this paper, however, we first prove Corollary B1 using a Hilbert cone contraction argument (§4) which provides a more precise description of the quasi-ergodic measure later used in the proof of Theorem B (§5). For the latter, we identify a graph structure representing the dynamical behaviour of $X_n^{\varepsilon ,\phi }$ conditioned upon staying on $M\setminus \partial $ . This construction resembles the graphs built via chain recurrence and filtration methods [Reference Crovisier and Potrie30, Reference De Leo and Yorke33, Reference De Leo and Yorke34] and allows us to recover the setting of Corollary B1.

1.3 Context of the results

The results in this paper relate to the theories of open systems and spectral stability.

The theory of conditioned random dynamical systems, which builds on the theory of absorbing Markov processes, essentially addresses transient properties of random dynamics. In particular, the quasi-stationary and quasi-ergodic measures of the associated absorbing Markov process provide relevant statistical properties of the random system conditioned upon remaining in a given region: the former is associated with the rate of escape of the system and the latter provides the Birkhoff averages of the process conditioned upon survival.

This has clear parallels with the theory of deterministic open dynamical systems [Reference Demers and Young38], where conditionally invariant measures are similar to quasi-stationary measures and invariant measures on the survival set resemble quasi-ergodic measures.

It is important to note that a conditioned random dynamical system with diffusive-like noise typically has a single quasi-stationary measure and a single quasi-ergodic measure. This differs from the setting of deterministic open systems, where uncountably many conditionally invariant measures and invariant measures may exist. We would like to emphasize that the quasi-ergodic measure is not a stationary measure of the conditioned random dynamical system. It is the relevant object to study ergodic theory from a conditioned point of view [Reference Breyer and Roberts15, Reference Castro, Chemnitz, Chu, Engel, Lamb and Rasmussen17, Reference Champagnat and Villemonais22, Reference Darroch and Seneta32, Reference Engel, Lamb and Rasmussen39]. Indeed, quasi-ergodic measures are similar to invariant measures of open systems, but their support on the state space usually has non-empty interior, while invariant measures of open systems are often supported on a Cantor-like set.

Taking the limit of noise amplitude to zero, we obtain an alternative perspective to transient dynamics in deterministic systems, which naturally aligns with the theory of open dynamical systems. In addition, this conditioned random dynamics approach provides an elegant, natural means to approximate also other equilibrium states on repellers.

In hyperbolic systems, equilibrium states may be constructed from the combination of right and left eigenfunctions of maximal eigenvalue, associated with a particular transfer operator displaying a spectral gap [Reference Baladi4, Reference Baladi6, Reference Demers, Kiamari and Liverani37, Reference Gouëzel and Liverani44]. Spectral stability of such operators then ensures that properties of the peripheral spectrum are stable under suitable (abstract) perturbations [Reference Gouëzel and Liverani43, Reference Keller and Liverani52]. Exploiting spectral stability of transfer operators has proven to be useful in several settings, including the proof of the existence of absolutely continuous conditioned invariant measures for systems with holes [Reference Demers and Young38], the study of decay of correlations for small random perturbations of hyperbolic systems [Reference Gouëzel and Liverani43] and the study of linear response [Reference Baladi5, Reference Baladi6], to name a few.

In this paper, we consider Birkhoff averages of a canonical weighted Markov process whose explicit dynamics are meaningful and can be well understood. We then study the quasi-ergodic measure of this process, a well-established object from the theory of absorbing Markov processes, which provides relevant dynamical and statistical properties of the random system. For the problem at hand, we show that quasi-ergodic measures can be constructed using a functional analytical approach (see Appendix A). In particular, we use an elementary Hilbert cone technique [Reference Liverani60, Reference Viana81] from which we obtain a detailed description of the quasi-ergodic measure and its constituents. Finally, we show that quasi-ergodic measures converge to equilibrium states.

We mention that there are several results in the literature on the zero-noise limit for absorbing Markov processes, where one studies the limiting behaviour of the quasi-stationary distribution as the noise strength tends to zero (see [Reference Faure and Schreiber40, Reference Hening, Qi, Shen and Yi50, Reference Prodhomme and Strickler66, Reference Schreiber, Huang, Jiang and Wang76] and the references therein). In previous works, the deterministic system is assumed to have an attractor on which the limiting quasi-stationary measures concentrate. In contrast, the present paper focuses on repellers: we consider small random perturbations of the dynamics near uniformly expanding repellers and obtain zero-noise limits supported on these repelling sets. Moreover, we treat both quasi-stationary and quasi-ergodic limits within the same framework, showing that quasi-ergodic measures generalise the role of stationary measures in the study of stochastic stability.

It should be noted that the local (conditioned) stochastic stability results in §4 (Theorem 2.12) align with well-known spectral stability arguments as mentioned above. However, the global results in §5 (Theorem 2.13) require the fine description of the quasi-ergodic measure provided by §4, which is not accessible through standard spectral stability results.

1.4 Outline

This paper is organized as follows. In §2, we introduce the objects of interest from the theory of conditioned random dynamics. We also lay out the required technical conditions (Hypotheses H1 and H2) regarding the deterministic systems considered, their random perturbations and present the two main theorems (Theorems 2.12 and 2.13). Three examples are presented in §2.4. In §3, we explore the direct implications of the hypotheses. In §4, we analyse the local problem (i.e. conditioning the random dynamics on a small neighbourhood of a repeller) and prove Theorem 2.12. In §5, we consider the global picture (i.e. conditioning upon not escaping from a general neighbourhood of the repeller) and prove Theorem 2.13. We provide examples in §2.4 of where these theorems are applicable. Finally, we devote Appendix A to a general proof for the existence of (weighted) quasi-ergodic measures, simplifying previous techniques.