A Preprint

Zhen Xu
email: xuzhen@4paradigm.com

Lanning Wei
email: weilanning@4paradigm.com

Huan Zhao
email: zhaohuan@4paradigm.com

Rex Ying
email: rexying@stanford.edu

Quanming Yao

Wei-Wei Tu
email: tuweiwei@4paradigm.com

Isabelle Guyon Chalearn

BRIDGING THE GAP OF AUTOGRAPH BETWEEN ACADEMIA AND INDUSTRY: ANALYSING AUTOGRAPH CHALLENGE AT KDD CUP 2020

Keywords: Graph Neural Networks, Automated Machine Learning, Data Challenge, Node Classification

Graph structured data is ubiquitous in daily life and scientific areas and has attracted increasing attention. Graph Neural Networks (GNNs) have been proved to be effective in modeling graph structured data and many variants of GNN architectures have been proposed. However, much human effort is often needed to tune the architecture depending on different datasets. Researchers naturally adopt Automated Machine Learning on Graph Learning, aiming to reduce the human effort and achieve generally top-performing GNNs, but their methods focus more on the architecture search. To understand GNN practitioners' automated solutions, we organized AutoGraph Challenge at KDD Cup 2020, emphasizing on automated graph neural networks for node classification. We received top solutions especially from industrial tech companies like Meituan, Alibaba and Twitter, which are already open sourced on Github. After detailed comparisons with solutions from academia, we quantify the gaps between academia and industry on modeling scope, effectiveness and efficiency, and show that (1) academia AutoML for Graph solutions focus on GNN architecture search while industrial solutions, especially the winning ones in the KDD Cup, tend to obtain an overall solution (2) by neural architecture search only, academia solutions achieve on average 97.3% accuracy of industrial solutions (3) academia solutions are cheap to obtain with several GPU hours while industrial solutions take a few months' labors. Academic solutions also contain much fewer parameters.

Introduction

Graph structured data has been prominent in our life and various tasks are studied based upon, including recommendation on Social Networks [START_REF] Fan | Graph neural networks for social recommendation[END_REF], traffic forecasting on road networks [START_REF] Li | Diffusion convolutional recurrent neural network: Data-driven traffic forecasting[END_REF], drug discovery on molecule graph [START_REF] Torng | Graph convolutional neural networks for predicting drug-target interactions[END_REF], link prediction on knowledge graph [START_REF] Zhang | Relational graph neural network with hierarchical attention for knowledge graph completion[END_REF], etc. Graph Neural Networks (GNN) [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] have been proved to be effective in modeling graph data and tremendous GNN architectures are proposed every year [START_REF] Hamilton | Inductive representation learning on large graphs[END_REF][START_REF] Xu | How powerful are graph neural networks?[END_REF][START_REF] Wu | Simplifying graph convolutional networks[END_REF][START_REF] Veličković | Graph attention networks[END_REF].

When applying GNN on graph structured data, expertise and domain knowledge is often required and numerous human effort is required to adapt to new datasets. Automated Machine Learning (AutoML) [START_REF] Hutter | Automated machine learning: methods, systems, challenges[END_REF][START_REF] Yao | Taking human out of learning applications: A survey on automated machine learning[END_REF] aims to reduce human efforts in deploying on various applications. AutoML, especially Neural Architecture Search (NAS), has been successfully explored on tremendous applications, including Image Classification [START_REF] Tan | Efficientnet: Rethinking model scaling for convolutional neural networks[END_REF], Object Detection [START_REF] Tan | Efficientdet: Scalable and efficient object detection[END_REF], Semantic Segmentation [START_REF] Nekrasov | Fast neural architecture search of compact semantic segmentation models via auxiliary cells[END_REF], Language Modeling [START_REF] Jiang | Improved differentiable architecture search for language modeling and named entity recognition[END_REF], Time Series Forecasting [START_REF] Chen | Scale-aware neural architecture search for multivariate time series forecasting[END_REF], etc. As a result, researchers start to explore Automated Graph Neural Networks (AutoGraph). AutoGraph researchers focus mainly on automatically designing GNN architectures by NAS. The majority of these methods focus on designing the aggregation functions/layers in GNNs with different search algorithms [START_REF] Gao | Graph neural architecture search[END_REF][START_REF] Zhou | Auto-gnn: Neural architecture search of graph neural networks[END_REF][START_REF] Yoon | Autonomous graph mining algorithm search with best speed/accuracy trade-off[END_REF], Li et al., 2021, Peng et al., 2020]. Other works, SANE [START_REF] Zhao | Search to aggregate neighborhood for graph neural network[END_REF] and AutoGraph [START_REF] Li | Autograph: Automated graph neural network[END_REF], provide the extra layer-wise skip connections design dimension; GNAS [START_REF] Cai | Rethinking graph neural network search from message-passing[END_REF], DeepGNAS [START_REF] Feng | Search for deep graph neural networks[END_REF] and Policy-GNN [START_REF] Lai | Policy-gnn: Aggregation optimization for graph neural networks[END_REF] learn to design the depth of GNNs. DiffMG [START_REF] Ding | Diffmg: Differentiable meta graph search for heterogeneous graph neural networks[END_REF] proposed to use NAS to search data-specific meta-graphs in heterogeneous graph, and PAS [START_REF] Wei | Pooling architecture search for graph classification[END_REF] is proposed to search data-specific pooling architectures for graph classification. The recently proposed F 2 GNN [START_REF] Wei | Designing the topology of graph neural networks: A novel feature fusion perspective[END_REF] method decouples the design of aggregation operations with architecture topology, which is not considered before.

Despite the rich literature from academia, we ask the question of how AutoGraph is used in industrial practitioners. Towards this end, we organize the first AutoGraph challenge at KDD Cup 2020, collaborated with 4Paradigm, ChaLearn and Stanford University. This challenge asks participants to provide AutoGraph solutions for node classification task. The code is executed by the platform on various graph datasets without any human intervention. Through the AutoGraph challenge, we wish to push forward the limit of AutoGraph as well as to understand the gap between industrial solutions and academia ones. In this paper, we first introduce the AutoGraph challenge setting. Then, we present the winning solutions which are open sourced for everyone to use. At last, we experiment further and compare with NAS methods for GNN methods and quantify empirically the gap with respect to top solutions. We conclude three gaps of AutoGraph between academia and industry: Modeling scope, Effectiveness and Efficiency.

2 Challenge background

General statistics

The AutoGraph challenge lasted for two months. We received over 2200 submissions and more than 140 teams from both high-tech companies (Ant Financial, Bytedance, Criteo, Meituan Dianping, Twitter, NTT DOCOMO, etc.) and universities (MIT, UCLA, Tsinghua University, Peking University, Nanyang Technological University, National University of Singapore, IIT Kanpur, George Washington University, etc.), coming from various countries. The top three teams are: aister, PASA_NJU, qqerret. Top 10 winners' information are shown in Table 1. The 1st winner aister comes from Meituan Dianping, a company on location-based shopping and retailing service. This makes the challenge particularly valuable since we can compare academic solutions with industrial best AutoGraph practices.

Table 1: General information about winning teams. Two teams tie on the 6th and 10th place. We list them both.

Place Team Name Institute Place Team

Problem formulation

The task of AutoGraph challenge is node classification under the transductive setting. Formally speaking, consider a graph G = (V, E), where V = {v 1 , • • • , v N } is the set of nodes, i.e. |V| = N and E is the set of edges, which is usually encoded by an adjacency matrix A ∈ [0, 1] N ×N . A ij is positive if there is an edge connecting from node v i to node v j . Additionally, a feature matrix X ∈ R N ×D gives features of each node. Each node v i has a class label y i ∈ L = {1, • • • , c}, resulting in the label vector Y ∈ L N . In the transductive semi-supervised node classification task, part of labels are available during training and the goal is to learn a mapping F : V → L and predict classes of unlabeled nodes.

Protocol

The protocol of AutoGraph challenge is straightforward. Participants should submit a python file containing a Model class with required fit and predict method. We prepare an ingestion program reading dataset and instantiate the class and call fit and predict method until prediction finishes or the running time has reached the budget limit. Ingestion program outputs model's prediction on test data and save to a shared space. Then, another scoring program reads the prediction and ground truth and outputs evaluation scores. The execution of the program is totally on the challenge platform. When developing locally, we provide script to call model.py file methods directly.

Metric

We use Accuracy (Acc) and Balanced Accuracy (BalAcc) as evaluation metrics, defined as

Acc = 1 |Ω| i∈Ω 1 ŷi=yi , BalAcc = 1 |C| i∈C Recall i ,
where Ω is the set of test nodes indexes, y i is the ground truth label for node v i and ŷi is the predicted label, C is the set of classes and Recall i is the recall score for class i. Accuracy (Acc) is used in the challenge to rank participants and Balanced Accuracy (BalAcc) is applied for additional analyses since it takes into account the imbalanced label distribution of datasets.

Datasets

Fifteen graph datasets were used during the competition: 5 public datasets were directly downloadable by the participants so they could develop their solutions offline. Five feedback datasets were made available on the platform during the feedback phase to evaluate AutoGraph algorithms on the public leaderboard. Finally, the AutoGraph algorithms were evaluated with 5 private datasets, without human intervention. These dataset are quite diverse in domains, shapes, density and other graph properties because we expect AutoGraph solutions to have good generalization ability. On the other hands, we intentionally keep the characteristics of 5 feedback datasets and 5 private datasets similar to enable transferability. We summarize dataset statistics in Table 2. The licenses and original sources of these datasets are also provided1 .

Solutions

In this part, we introduce various methods suitable for the AutoGraph challenge, including the provided challenge baseline and solutions from top-3 winners. We conclude the first gap at the end.

Baselines (GCN(L2)). In the provided baselines, there is no feature engineering except for using the raw node features.

For graph without node features, e.g. dataset i,j, one hot encoding is used to unroll the node lists to a dummy feature 1st placed winner. The 1st winner is from team aister. Their code is open source here2 . The authors use four GNN models, two spatial ones: GraphSage [START_REF] Hamilton | Inductive representation learning on large graphs[END_REF] and GAT [START_REF] Veličković | Graph attention networks[END_REF], two spectral ones: GCN [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] and TAGConv [START_REF] Du | Topology adaptive graph convolutional networks[END_REF] to process node features collectively. For each GNN model, a heavy search is applied offline to determining the important hyperparameters as well as the boundaries. In the online stage, they use a smaller search space to determine the hyperparameters. In order to accelerate the search, they do not fully train each configuration but instead early stop in 16 epochs if the validation loss is not satisfactory. Additional features are used: node degrees, distribution of 1-hop and 2-hop neighbor nodes' features, etc.

2nd place winner. The 2nd winner is from team PASA_NJU. Their code is open source here3 . They also split the solution in two stages: offline stage and online stage. In the offline stage, the authors train a decision tree based on public data and other self collected datasets to classify graph type into one of three classes. Then they use GraphNAS [START_REF] Gao | Graph neural architecture search[END_REF] to search massively optimal GNN architectures including aggregation function, activation, number of heads in attention, hidden units, etc. In the online stage, the authors rapidly classify the dataset and fine tune the offline searched model.

3rd place winner. The 3rd winner is from team qqerret. Their code is open source here4 . The core model is a variant of spatial based GNN, which aggregates 2-hop neighbors of a node with additional linear parts for the node itself.

Basically, the new embedding of node i is ĥ(i) = j∈N2(i) a j h(j) + α(wh(i) + b). Additionally, in the GNN output layer, a few features per node are concatenated for final fully connected layer, including number of edges, whether this node connects to a central node who has a lot of edges, label distribution of 1-hop neighbor nodes, and label distribution of 2-hop neighboring nodes.

Results

We conduct additional experiments after the AutoGraph challenge to further analyze the results. We first reproduce winning solutions and then we compare with academia solutions. Three gaps are concluded. The first gap is presented as follows and two other gaps are concluded in Sec 4.2.

Gap #1: Modeling scope is the first gap of AutoGraph between academia and industry. In academia, researchers focus mainly on Neural Architecture Search methods to find better GNN architectures. Their contributions differ in their search space, search strategy and evaluation methods. However, industrial solutions, e.g. 1st solution, focus more on the feature engineering and model ensemble. For GNN architectures, they prefer existing found ones with little modification. In other words, industrial people provide a full pipeline solution including data preprocessing, feature engineering, model architecture, hyperparameter optimization, model ensemble, while academia researchers focus on model architecture part only. The gap is also illustrated in Figure 1. It might be an interesting direction for both groups to merge, i.e. AutoGraph researchers could explore the automated feature engineering, automated ensemble and AutoGraph practitioners could adopt NAS methods for GNN.

Reproducing winning solutions

We reproduce all winning methods on all the datasets and include their results in Table 3. We observe that all three winning solutions are close in performance and all significantly beat the GCN baselines. On the other hand, in the AutoGraph challenge, due to the nature of the competition, we rank methods based on accuracy, we state that this is not sufficient to evaluate comprehensively solutions from the scientific perspective. We add balanced accuracy here just to show that for some methods that show close performance in accuracy, they could diverge a lot in balanced accuracy.

Regarding both accuracy and balanced accuracy, we conclude that 1st solution which comes from Meituan Dianping Company, is indeed best among top winners. Thus, we will later use their solutions for comparing with academia solutions. These winning solutions are already open sourced, which are reproducible and lower the barriers of using AutoGraph. We further adopt NAS methods for GNN and compare with the baseline and 1st solution coming from industry. We choose the recent F 2 GNN [START_REF] Wei | Designing the topology of graph neural networks: A novel feature fusion perspective[END_REF], which searches for data-specific GNN topology, in our experiment. To compare fairly with GCN baselines, we fix the aggregation to GCN and search only the GNN topology, which we call F 2 GCN. Since F 2 GCN requires at least 4 layers, we also run a 4 layer GCN baseline for better comparison. The results are given in Table 4. Gap #2: Effectiveness is the second gap of AutoGraph between academia and industry. We observe from Table 4 and Figure 2 that all baselines and F 2 GCN methods are not as good as 1st winning solution. However, for many datasets, e.g. e, f, g, i, j, F 2 GCN is very close to the best industrial solution. On average, F 2 GCN which focuses only on architecture search, reaches 97.3% of best solution. Note that the 1st solution constructs additional node features and uses multiple GNN architectures for ensemble while F 2 GCN does not use any feature engineering or model ensemble. This shows the effectiveness of winner's engineering practices as well as F 2 GCN's adaptive NAS search. Winning teams also have access to public datasets and public leaderboard to iteratively fine tune their methods. F 2 GCN does not assume any prior knowledge of the datasets, which shows further its effectiveness.

To better understand the solutions, we calculate the number of parameters of baseline, F 2 GCN, and 1st solution, as shown in Table 5.

Gap #3: Efficiency is the third gap of AutoGraph between academia and industry. From Table 5 and Figure 3, F 2 GCN uses significantly fewer parameters than the best industrial solution on most datasets (13 out of 15

Conclusion

We organized the first Automated Graph Learning (AutoGraph) Challenge at KDD Cup 2020. We presented in this paper its settings, dataset, and solutions, which are all open sourced. We ran additional post-challenge experiments to compare the baseline (Graph Convolution Network (GCN)), the winning solution (feature engineering-based ensemble of various Graph Neural Networks), and a recent and efficient Neural Architecture Search (NAS) for GNN method called F 2 GCN. This paper provides results that could bridge the gap between academic research and industry practices, by correcting bias favoring certain approaches. This gap is currently at 3 aspects: Gap #1 modeling scope. (academia focuses more on model-centric approaches, emphasizing NAS; industry emphasizes data centric approaches and feature engineering); Gap #2 effectiveness. (academic solutions are perceived by industry to be less effective than their industry counterpart); Gap #3 efficiency. (academic solutions are perceived to be parsimonious or slower than industry solutions). Our results indicate that the "academic" NAS-based approach that we applied attains performances closely matching those of the winning industry solution, while being both faster and more parsimonious in number of parameters, therefore closing Gap #2 and #3. Moreover, we hope that these results will help reducing Gap #1, by encouraging industry practitioners to apply NAS methods (and particularly F 2 GCN), eventually combining the best of both approaches. We believe the results we obtained are significant, since they involve a benchmark on 15 datasets.

Figure 1 :

 1 Figure 1: Illustraion of AutoGraph scope. Industrial people provide a full pipeline solution that covers from data preprocessing to evaluation. Academic researchers focus mainly on model architecture and hyperparameter optimization.

Figure 2 :

 2 Figure 2: Accuracy improvement with respect to baseline.

Figure 3 :

 3 Figure 3: Comparison on number of parameters of baseline, 1st solution and F 2 GCN.

Table 2 :

 2 Statistics

	Dataset Phase	Domain	#Node	#Edge #Feature #Class Avg Deg Directed? Weighted? Skewness
	a	Public	Citation	2.7K	5.3K	1.4K	7	1.9	F	F	5
	b	Public	Citation	3.3K	4.6K	3.7K	6	1.4	F	F	3
	c	Public	Social	10K	733K	0.6K	41	73.3	F	F	81
	d	Public	News	10K 2,917K	0.3K	20	291.7	T	T	467
	e	Public	Finance	7.5K	7.8K	0	3	1.0	F	F	111
	f	Feedback Sales	10K	194K	0.7K	10	19.4	F	F	18
	g	Feedback Citation	10K	41K	8K	5	4.1	F	F	6
	h	Feedback Medicine	10K 2,461K	0.3K	23	246.1	T	T	1,773
	i	Feedback Finance	15K	16K	0	3	1.1	F	F	213
	j	Feedback Medicine	11K	22K	0	9	2.0	F	F	227
	k	Private	Sales	8K	119K	0.7K	8	14.9	F	F	6
	l	Private	Citation	10K	40K	7K	15	4	F	F	34
	m	Private	News	10K 1,425K	0.3K	8	142.5	T	T	360
	n	Private	Finance	14K	22K	0	10	1.6	F	F	61
	o	Private	Social	12K	19K	0	19	1.6	F	F	62

of all datasets. "Avg Deg" is the average number of edges per node. "Directed" and "Weighted" indicate the two properties of a graph. "Skewness" here is calculated by number of nodes in the largest class divided by number of nodes in the smallest class.

  table. During model training, a MLP is first used to map node features to the same embedding dimension. Then a two layer vanilla GCN is applied for learning node embeddings. Another MLP with softmax outputs the final classification. Dropout is used. All the hyperparameters are fixed by experience. No time management since the model is simple and one full training will not cost more than the allowed time budget.

Table 3 :

 3 Accuracy and Balanced accuracy of top methods on all datasets (%). Baseline is a two layer GCN.

	Dataset Phase	Baseline (GCN(L2))	1st solution	2nd solution	3rd solution
			Acc	BalAcc	Acc BalAcc Acc BalAcc Acc BalAcc
	a	Public	85.7	84.9	88.5	87.8	88.2	87.2	87.2	85.5
	b	Public	71.4	67.8	75.2	71.2	75.8	71.2	75.6	69.0
	c	Public	86.5	72.0	94.3	87.5	94.2	90.9	95.4	91.3
	d	Public	93.7	6.1	96.5	48.7	95.1	28.8	94.6	21.0
	e	Public	59.6	38.8	88.7	92.8	88.5	90.7	88.8	92.8
	f	Feedback 86.6	78.2	92.8	92.1	92.3	92.1	92.4	91.4
	g	Feedback 94.7	92.8	95.3	93.5	95.6	93.8	95.8	94.2
	h	Feedback 90.4	8.8	93.5	26.3	92.2	17.6	92.1	16.6
	i	Feedback 88.2	59.2	88.4	87.5	88.4	92.6	88.5	91.1
	j	Feedback 90.7	68.1	95.9	89.0	96.1	93.7	96.6	93.3
	k	Private	93.5	92.2	95.5	94.4	95.5	94.4	94.8	93.1
	l	Private	90.9	84.5	94.9	92.6	94.7	91.8	94.5	92.6
	m	Private	85.5	24.5	98.1	79.7	95.7	69.0	98.0	79.4
	n	Private	85.6	47.3	99.0	97.3	99.0	98.4	98.9	97.0
	o	Private	49.6	15.6	91.0	84.6	91.3	90.6	91.4	88.5
	4.2 Neural Architecture Search for GNN

Table 4 :

 4 Accuracy comparison of GCN baselines, F 2 GCN and industrial best solution (%). L2, L4 means 2 and 4 layers for the GNN architecture. Numbers in parentheses are relative accuracy w.r.t 1st solution. We regard 1st solution as 100%. Last line is the average percentage.

  ). On average, F 2 GCN consumes 45.1% of the 1st solution in terms of parameter size, which is quite resource efficient. Note that feature engineering and ensemble do not contain additional parameters and basically, F 2 GCN searches one GNN model to compete with ensemble of 4 types of GNN models in 1st solution. As for time devotion, winning solutions come from a team's months of work, which consists of 5 or more members. F 2 GCN only runs for a few GPU hours per dataset, demonstrating its time efficiency compared to industrial solutions.

	Dataset GCN(L2)	F 2 GCN(L4) 1st solution
	a	0.023	0.908 (75.7) 1.199 (100)
	b	0.059	0.700 (44.2) 1.583 (100)
	c	0.011	1.598 (98.0) 1.631 (100)
	d	0.006	0.042 (3.20) 1.296 (100)
	e	0.121	0.354 (31.8) 1.114 (100)
	f	0.013	0.039 (2.30) 1.688 (100)
	g	0.134	0.313 (13.1) 2.389 (100)
	h	0.006	0.271 (20.9) 1.294 (100)
	i	0.241	2.269 (113.0) 2.013 (100)
	j	0.171	0.834 (60.6) 1.376 (100)
	k	0.012	1.478 (108.0) 1.395 (100)
	l	0.108	0.614 (25.6) 2.395 (100)
	m	0.005	0.010 (0.80) 1.278 (100)
	n	0.218	0.488 (27.8) 1.756 (100)
	o	0.192	0.822 (52.5) 1.565 (100)
	Avg		-(45.1)	-(100)

Table 5 :

 5

Number of parameters of baseline, 1st solution and F 2 GCN (Unit: Millions). Numbers in parentheses are relative # parameters w.r.t 1st solution. We regard 1st solution as 100%. Last line is the average percentage.

https://github.com/AutoML-Research/AutoGraph-KDDCup2020

https://github.com/aister2020/KDDCUP_2020_AutoGraph_1st_Place

https://github.com/Unkrible/AutoGraph2020

https://github.com/white-bird/kdd2020_GCN

Acknowledgments

Funding and support have been received by several research grants, including 4Paradigm, Big Data Chair of Excellence FDS Paris-Saclay, Paris Région Ile-de-France, and ANR Chair of Artificial Intelligence HUMANIA ANR-19-CHIA-0022, ChaLearn, Microsoft, Google. We acknowledge the following people for helping organize AutoGraph challenge: Xiawei Guo, Shouxiang Liu. We also appreciate the following people and institutes for open sourcing datasets which are used in our use cases: Andrew McCallum, C. Lee Giles, Ken Lang, Tom Mitchell, William L. Hamilton, Maximilian


William Oçafrain
email: w.ocafrain@hotmail.fr

An ergodic theorem for asymptotically periodic time-inhomogeneous Markov processes, with application to quasi-stationarity with moving boundaries

Keywords: ergodic theorem, law of large numbers, time-inhomogeneous Markov processes, quasi-stationarity, quasi-ergodic distribution, moving boundaries

This paper deals with ergodic theorems for particular time-inhomogeneous Markov processes, whose timeinhomogeneity is asymptotically periodic. Under a Lyapunov/minorization condition, it is shown that, for any measurable bounded function f , the time average 1 t t 0 f (Xs)ds converges in L 2 towards a limiting distribution, starting from any initial distribution for the process (Xt) t≥0 . This convergence can be improved to an almost sure convergence under an additional assumption on the initial measure. This result will be then applied to show the existence of a quasi-ergodic distribution for processes absorbed by an asymptotically periodic moving boundary, satisfying a conditional Doeblin condition.

Notation

Throughout we shall use the following notation:

• N = {1, 2, . . . , } and Z + = {0} ∪ N.

• M 1 (E) denotes the space of the probability measures whose support is included in E.

• B(E) denotes the set of the measurable bounded functions defined on E.

• B 1 (E): denotes the set of the measurable functions f defined on E such that f ∞ ≤ 1.

• For all µ ∈ M 1 (E) and p ∈ N, L p (µ) denotes the set of the measurable functions f : E → R such that

E |f (x)| p µ(dx) < +∞. • For any µ ∈ M 1 (E) and f ∈ L 1 (µ), denote µ(f ) := E f (x)µ(dx).
• For any positive function ψ, M 1 (ψ) := {µ ∈ M 1 (E) : µ(ψ) < +∞}.

• Id denotes the Identity operator.

Introduction

In general, an ergodic theorem for a Markov process (X t ) t≥0 and probability measure π refers to the almost sure convergence 1 t

t 0 f (X s )ds -→ t→∞ π(f ), ∀f ∈ L 1 (π). (1)
In the time-homogeneous setting, such an ergodic theorem holds for positive Harris recurrent Markov processes with the limiting distribution π corresponding to an invariant measure for the underlying Markov process. For time-inhomogeneous Markov processes, such a result does not hold in general (in particular the notion of invariant measure is in general not well-defined), except for specific types of time-inhomogeneity such as periodic timeinhomogeneous Markov processes, defined as time-inhomogeneous Markov processes for which there exists γ > 0 such that, for any s ≤ t, k ∈ Z + and x,

P[X t ∈ •|X s = x] = P[X t+kγ ∈ •|X s+kγ = x]. (2)
In other words, a time-inhomogeneous Markov process is periodic when the transition law between any times s and t remains unchanged when the time interval [s, t] is shifted by a multiple of the period γ. In particular, this implies that, for any s ∈ [0, γ), the Markov chain (X s+nγ ) n∈Z+ is time-homogeneous. This fact allowed Höpfner et al. (in [20,[START_REF] Höpfner | Ergodicity and limit theorems for degenerate diffusions with time periodic drift. application to a stochastic hodgkin-huxley model[END_REF][START_REF] Höpfner | Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input[END_REF]) to show that, if the skeleton Markov chain (X nγ ) n∈Z+ is Harris recurrent, then the chains (X s+nγ ) n∈Z+ , for all s ∈ [0, γ), are also Harris recurrent and

1 t t 0 f (X s )ds -→ t→∞ 1 γ γ 0
π s (f )ds, almost surely, from any initial measure, where π s is the invariant measure for (X s+nγ ) n∈Z+ . This paper aims to prove a similar result for time-inhomogeneous Markov processes said to be asymptotically periodic. Roughly speaking (a precise definition of which will be explicitly given later), an asymptotically periodic Markov process is such that, given a time interval T ≥ 0, its transition law on the interval [s, s+T ] is asymptotically "close to" the one, on the same interval, of a periodic time-inhomogeneous Markov process called an auxiliary Markov process, when s → ∞. This definition is very similar to the notion of asymptotic homogeneization, defined as follows in [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF]Subsection 3.3]: a time-inhomogeneous Markov process (X t ) t≥0 is said to be asymptotically homogeneous if there exists a time-homogeneous Markovian semigroup (Q t ) t≥0 such that, for all s ≥ 0,

lim t→∞ sup x P[X t+s ∈ •|X t = x] -δ x Q s T V = 0, (3)
where, for two positive measures with finite mass µ 1 and µ 2 , µ 1 -µ 2 T V is the total variation distance between µ 1 and µ 2 :

µ 1 -µ 2 T V := sup f ∈B1(E) |µ 1 (f ) -µ 2 (f )|. (4)
In particular, it is well-known (see [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF]Theorem 3.11]) that, under this suitable additional conditions, an asymptotically homogeneous Markov process converges towards a probability measure which is invariant for (Q t ) t≥0 .

It is similarly expected that an asymptotically periodic process has the same asymptotic properties as a periodic Markov process; in particular an ergodic theorem holds for the asymptotically periodic process.

The main result of this paper provides for an asymptotically periodic Markov process to satisfy

1 t t 0 f (X s )ds L 2 (P0,µ) ------→ t→∞ 1 γ γ 0 β s (f )ds, ∀f ∈ B(E), ∀µ ∈ M 1 (E), (5)
whereby P 0,µ is a probability measure under which X 0 ∼ µ, and where β s is the limiting distribution of the skeleton Markov chain (X s+nγ ) n∈Z+ , if it satisfies a Lyapunov-type condition, a local Doeblin condition (defined further in Section 2), and is such that its auxiliary process satisfies a Lyapunov/minorization condition. Furthermore, this convergence result holds almost surely if a Lyapunov function of the process (X t ) t≥0 , denoted by ψ, is integrable with respect to the initial measure:

1 t t 0 f (X s )ds P0,µ-almost surely ----------→ t→∞ 1 γ γ 0 β s (f )ds, ∀µ ∈ M 1 (ψ).
This will be more precisely stated and proved in Section 2.

The main motivation of this paper is then to deal with quasi-stationarity with moving boundaries, that is the study of asymptotic properties for the process X, conditioned not to reach some moving subset of the state space.

In particular, such a study is motivated by models such as those presented in [START_REF] Cattiaux | A stochastic model for cytotoxic T lymphocyte interaction with tumor nodules[END_REF], studying Brownian particles absorbed by cells whose volume may vary over time.

Quasi-stationarity with moving boundaries has been studied in particular in [START_REF] Oçafrain | Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically[END_REF][START_REF] Oçafrain | Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF], where a "conditional ergodic theorem" (see further the definition of a quasi-ergodic distribution) has been shown when the absorbing boundaries move periodically. In this paper, we show that a similar result holds when the boundary is asymptotically periodic, assuming that the process satisfies a conditional Doeblin condition (see Assumption (A')). This will be dealt with in Section 3.

The paper will be concluded by using these results in two examples: an ergodic theorem for an asymptotically periodic Ornstein-Uhlenbeck process, and the existence of a unique quasi-ergodic distribution for a Brownian motion confined between two symmetric asymptotically periodic functions.

Ergodic theorem for asymptotically periodic time-inhomogeneous semigroup.

Asymptotic periodicity: the definition.

Let (E, E) be a measurable space In particular, associated to {(E t , E t ) t≥0 , (P s,t ) s≤t } is a Markov process (X t ) t≥0 and a family of probability measures (P s,x ) s≥0,x∈Es such that, for any s ≤ t, x ∈ E s and A ∈ E t ,

P s,x [X t ∈ A] = P s,t 1 A (x).
We denote P s,µ := Es P s,x µ(dx) for any probability measure µ supported on E s . We also denote by E s,x and E s,µ the expectations associated to P s,x and P s,µ respectively. Finally, the following notation will be used for

µ ∈ M 1 (E s ), s ≤ t and f ∈ B(E t ), µP s,t f := E s,µ [f (X t )], µP s,t := P s,µ [X t ∈ •].
The periodicity of a time-inhomogeneous semigroup is defined as follows. We say a semigroup

{(F t , F t ) t≥0 , (Q s,t ) s≤t } is γ-periodic (for γ > 0), if, for any s ≤ t, (F t , F t ) = (F t+kγ , F t+kγ ), Q s,t = Q s+kγ,t+kγ , ∀k ∈ Z + .
It is now possible to define an asymptotically periodic semigroup.

Definition 1 (Asymptotically periodic semigroups). A time-inhomogeneous semigroup {(E t , E t ) t≥0 , (P s,t ) s≤t } is said to be asymptotically periodic if (for some γ > 0) there exists a γ-periodic semigroup {(F t , F t ) t≥0 , (Q s,t ) s≤t } and two families of functions (ψ s ) s≥0 and ( ψs ) s≥0 such that ψs+γ = ψs for all s ≥ 0, and for any s ∈ [0, γ):

1.

∞ k=0 l≥k E s+lγ ∩ F s = ∅; 2. there exists x s ∈ ∞ k=0 l≥k E s+lγ ∩ F s such that, for any n ∈ Z + , δ xs P s+kγ,s+(k+n)γ [ψ s+(k+n)γ × •] -δ xs Q s,s+nγ [ ψs × •] T V -→ k→∞ 0. ( 6
)
The semigroup {(F t , F t ) t≥0 , (Q s,t ) s≤t } is then called the auxiliary semigroup of (P s,t ) s≤t .

When ψ s = ψs = 1 for all s ≥ 0, we say that the semigroup (P s,t ) s≤t is asymptotically periodic in total variation. By extension, we will say that the process (X t ) t≥0 is asymptotically periodic (in total variation) if the associated semigroup {(E t , E t ) t≥0 , (P s,t ) s≤t } is asymptotically periodic (in total variation).

In what follows, the functions (ψ s ) s≥0 and ( ψs ) s∈[0,γ) will play the role of Lyapunov functions (that is to say satisfying Assumption 1 (ii) below) for the semigroups (P s,t ) s≤t and (Q s,t ) s≤t respectively. The introduction of these functions in the definition of asymptotically periodic semigroups will allow us to establish an ergodic theorem for processes satisfying the Lyapunov/minorization conditions written below.

Lyapunov/minorization conditions.

The main assumption of Theorem 1, which will be provided later, will be that the asymptotically periodic Markov process satisfies the following assumption. Assumption 1. There exist t 1 ≥ 0, n 0 ∈ N, c > 0, θ ∈ (0, 1), a family of measurable sets (K t ) t≥0 such that K t ⊂ E t for all t ≥ 0, a family of probability measures (ν s ) s≥0 on (K s ) s≥0 , and a family of functions (ψ s ) s≥0 , all lower-bounded by 1, such that: (i) For any s ≥ 0, x ∈ K s and n ≥ n 0 , δ x P s,s+nt1 ≥ cν s+nt1 .

(ii) For any s ≥ 0, P s,s+t1 ψ s+t1 ≤ θψ s + C1 Ks .

(iii) For any s ≥ 0 and t ∈ [0, t 1 ), P s,s+t ψ s+t ≤ Cψ s .

When a semigroup (P s,t ) s≤t satisfies Assumption 1 as stated above, we will say that the family of functions (ψ s ) s≥0 are Lyapunov functions for the semigroup (P s,t ) s≤t . In particular, under (ii) and (iii), it is easy to prove that for any s ≤ t,

P s,t ψ t ≤ C 1 + C 1 -θ ψ s . ( 7
)
We remark in particular that Assumption 1 implies an exponential weak ergodicity in ψ t -distance, that is we have the existence of two constants C > 0 and κ > 0 such that, for all s ≤ t and for all probability measures

µ 1 , µ 2 ∈ M 1 (E s ), µ 1 P s,t -µ 2 P s,t ψt ≤ C [µ 1 (ψ s ) + µ 2 (ψ s )]e -κ(t-s) , (8)
whereby, for a given function ψ, µ -ν ψ is the ψ-distance defined to be

µ -ν ψ := sup |f |≤ψ |µ(f ) -ν(f )| , ∀µ, ν ∈ M 1 (ψ).
In particular, when ψ = 1 for all t ≥ 0, the ψ-distance is the total variation distance. If we have weak ergodicity [START_REF] Champagnat | Uniform convergence to the Q-process[END_REF] in the time-homogeneous setting (see in particular [START_REF] Hairer | Yet another look at Harris' ergodic theorem for markov chains[END_REF]), the proof of [15, Theorem 1.3.] can be adapted to a general time-inhomogeneous framework (see for example [6, Subsection 9.5] ).

The main theorem and proof.

The main result of this paper is the following. Then, for any µ ∈ M 1 (E 0 ) such that µ(ψ 0 ) < +∞,

1 t t 0 µP 0,s [ψ s × •]ds - 1 γ γ 0 β γ Q 0,s [ ψs × •]ds T V -→ t→∞ 0, (9)
whereby β γ ∈ M 1 (F 0 ) is the unique invariant probability measure of the skeleton semigroup (Q 0,nγ ) n∈Z+ satisfying β γ ( ψ0 ) < +∞. Moreover, for any f ∈ B(E) we have:

1. For any µ ∈ M 1 (E 0 ), E 0,µ 1 t t 0 f (X s )ds - 1 γ γ 0 β γ Q 0,s f ds 2 -→ t→∞ 0. ( 10
)
2. If moreover µ(ψ 0 ) < +∞, then

1 t t 0 f (X s )ds -→ t→∞ 1 γ γ 0 β γ Q 0,s f ds, P 0,µ -almost surely. ( 11
)
Remark 1. When Assumption 1 hold for K s = E s for any s, then the condition (i) in Assumption 1 implies Doeblin condition.

Doeblin condition.

There exists t 0 ≥ 0, c > 0 and a family of probability measure (ν t ) t≥0 on (E t ) t≥0 such that, for any s ≥ 0 and x ∈ E s ,

δ x P s,s+t0 ≥ cν s+t0 . ( 12
)
In fact, if we assume that Assumption 1 (i) holds for K s = E s , Doeblin condition holds by setting t 0 := n 0 t 1 .

Conversely, the Doeblin condition implies the conditions (i)-(ii)-(iii) with K s = E s and ψ s = 1 Es for all s ≥ 0, so that these conditions are equivalent. In fact, (ii) and (iii) straightforwardly hold true for (K s ) s≥0 = (E s ) s≥0 , (ψ s ) s≥0 = (1 Es ) s≥0 , C = 1, any θ ∈ (0, 1) and any t 1 ≥ 0. Setting t 1 = t 0 and n 0 = 1, the Doeblin condition entails that, for any s ∈ [0, t 1 ),

δ x P s,s+t1 ≥ cν s+t1 , ∀x ∈ E s .
Integrating this inequality over µ ∈ M 1 (E s ), one obtains

µP s,s+t1 ≥ cν s+t1 , ∀s ∈ [0, t 1 ), ∀µ ∈ M 1 (E s ).
Then, by the Markov property, for all s ∈ [0, t 1 ), x ∈ E s and n ∈ N, we have

δ x P s,s+nt1 = (δ x P s,s+(n-1)t1 )P s+(n-1)t1,s+nt1 ≥ cν s+nt1 ,
which is (i). Theorem 1 then entails the following corollary.

Corollary 1. Let (X t ) t≥0 be asymptotically γ-periodic in total variation distance. If (X t ) t≥0 and its auxiliary semigroup satisfy a Doeblin condition, then the convergence (10) is improved to

sup µ∈M1(E0) sup f ∈B1(E) E 0,µ 1 t t 0 f (X s )ds - 1 γ γ 0 β γ Q 0,s f ds 2 -→ t→∞ 0.
Moreover, the almost sure convergence [START_REF] Collet | Quasi-stationary distributions[END_REF] holds for any initial measure µ.

Remark 2. We also note that, if the convergence (6) holds for all x ∈ ∞ k=0 l≥k E s+lγ ∩ F s , then this implies (6) and therefore the pointwise convergence of (ψ s+nγ ) n∈Z+ to ψs (by taking n = 0 in (6)).

Proof of Theorem 1. The proof is divided into five steps.

First step. Since the auxiliary semigroup (Q s,t ) s≤t satisfies Assumption 1 with ( ψs ) s≥0 as Lyapunov functions, the time-homogeneous semigroup (Q 0,nγ ) n∈Z+ satisfies assumptions 1 and 2 of [START_REF] Hairer | Yet another look at Harris' ergodic theorem for markov chains[END_REF], which we now recall (using our notation).

Assumption 2 (Assumption 1, [15]

). There exists V :

F 0 → [0, +∞), n 1 ∈ N and constants K ≥ 0 and κ ∈ (0, 1) such that Q 0,n1γ V ≤ κV + K.

Assumption 3 (Assumption 2, [15]

). There exists a constant α ∈ (0, 1) and a probability measure ν such that

inf x∈C R δ x Q 0,n1γ ≥ αν(•), with C R := {x ∈ F 0 : V (x) ≤ R} for some R > 2K/(1 -κ),
whereby n 1 , K and κ are the constants from Assumption 2.

In fact, since (Q s,t ) s≤t satisfies (ii) and (iii) of Assumption 1, there exists C > 0, θ ∈ (0, 1),

t 1 ≥ 0 and (K s ) s≥0 such that Q s,s+t1 ψs+t1 ≤ θ ψs + C1 Ks , ∀s ≥ 0, (13)
and

Q s,s+t ψs+t ≤ C ψs , ∀s ≥ 0, ∀t ∈ [0, t 1 ).
We let n 2 ∈ N be such that

θ n2 C(1 + C 1-θ ) < 1.
By [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time finite Markov chains[END_REF] and recalling that ψt = ψt+γ for all t ≥ 0, one has for any s ≥ 0 and n ∈ N,

Q s,s+nt1 ψs+nt1 ≤ θ n ψs + C 1 -θ . ( 14
)
Thus, for all

n 1 ≥ n2t1 γ , Q 0,n1γ ψ0 = Q 0,n1γ-n2t1 Q n1γ-n2t1,n1γ ψn1γ ≤ θ n2 Q 0,n1γ-n2t1 ψn1γ-n2t1 + C 1 -θ ≤ θ n2 C(1 + C 1 -θ ) ψ0 + C 1 -θ ,
where we successively used the property of semigroup of (Q s,t ) s≤t , ( 14) and ( 7) applied to (Q s,t ) s≤t . Hence one has Assumption 2 by setting V = ψ0 , κ := θ n2 C(1 + C 1-θ ) and K := C 1-θ . We now prove Assumption 3. To this end, we introduce a Markov process (Y t ) t≥0 and a family of probability measures ( Ps,x ) s≥0,x∈Fs such that

Ps,x (Y t ∈ A) = Q s,t 1 A (x), ∀s ≤ t, x ∈ F s , A ∈ F t .
In what follows, for all s ≥ 0 and x ∈ F s , we will use the notation Ês,x for the expectation associated to Ps,x . Moreover, we define

T K := inf{n ∈ Z + : Y nt1 ∈ K nt1 }.
Then, using [START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time finite Markov chains[END_REF] recursively, for all k ∈ N, R > 0 and x ∈ C R (recalling that C R is defined in the statement of Assumption 3) we have

Ê0,x [ ψkt1 (Y kt1 )1 T K >k ] = Ê0,x [1 T K >k-1 Ê(k-1)t1,Y (k-1)t 1 ( ψkt1 (Y kt1 )1 T K >k )] ≤ θ Ê0,x [ ψ(k-1)t1 (Y (k-1)t1 )1 T K >k-1 ] ≤ θ k ψ0 (x) ≤ Rθ k . Since ψkt1 ≥ 1 for all k ∈ Z + , then for all x ∈ C R , for all k ∈ Z + , P0,x (T k > k) ≤ Rθ k .
In particular, there exists

k 0 ≥ n 0 such that, for all k ≥ k 0 -n 0 , P0,x (T K > k) ≤ 1 2 .
Hence, for all x ∈ C R ,

δ x Q 0,k0t1 = P0,x (Y k0t1 ∈ •) ≥ k0-n0 i=0 Ê0,x (1 T k =i Pit1,Xit 1 (Y k0t1 ∈ •)) ≥ c k0-n0 i=0 Ê0,x (1 T K =i ) × ν k0t1 = c P0,x (T K ≤ k 0 -n 0 )ν k0t1 ≥ c 2 ν k0t1 .
Hence, for all

n 1 ≥ k0t1 γ , for all x ∈ C R , δ x Q 0,k0t1 Q k0t1,n1γ ≥ c 2 ν k0t1 Q k0t1,n1γ .
Thus, Assumption 3 is satisfied taking

n 1 := n2t1 γ ∨ k0t1 γ , α := c 2 and ν(•) := ν k0t1 Q k0t1,n1γ
. Then, by [15, Theorem 1.2], Assumptions 2 and 3 imply that Q 0,n1γ admits a unique invariant probability measure β γ . Furthermore, there exists constants C > 0 and δ ∈ (0, 1) such that, for all µ ∈ M 1 (F 0 ),

µQ 0,nn1γ -β γ ψ0 ≤ Cµ( ψ0 )δ n . ( 15
)
Since β γ is the unique invariant probability measure of Q 0,n1γ and noting that β γ Q 0,γ is invariant for Q 0,n1γ , we deduce that β γ is the unique invariant probability measure for Q 0,γ and, by [START_REF] Hairer | Yet another look at Harris' ergodic theorem for markov chains[END_REF], for all µ such that µ( ψ0 ) < +∞,

µQ 0,nγ -β γ ψ0 -→ n→∞ 0.
Now, for any s ≥ 0, note that δ x Q s, s γ γ ψ0 < +∞ for all x ∈ F s (this is a consequence of (7) applied to the semigroup (Q s,t ) s≤t ), and therefore, taking µ = δ x Q s, s γ γ in the above convergence, [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed onedimensional diffusions[END_REF], we conclude from the above convergence that

δ x Q s,nγ -β γ ψ0 -→ n→∞ 0 for all x ∈ F s . Hence, since Q nγ,nγ+s ψs ≤ C(1 + C 1-θ ) ψnγ by
δ x Q s,s+nγ -β γ Q 0,s ψs ≤ C(1 + C 1 -θ ) δ x Q s,nγ -β γ ψ0 -→ n→∞ 0. (16)
Moreover, β γ ( ψ0 ) < +∞.

Second step. The first part of this step (until the equality ( 20)) is inspired by the proof of [START_REF] Bansaye | Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions[END_REF]Theorem 3.11].

We fix s ∈ [0, γ]. Without loss of generality, we assume that l≥0 E s+lγ ∩ F s = ∅. Then, by Definition 1, there exists x s ∈ l≥0 E s+lγ ∩ F s such that for any n ≥ 0,

δ xs P s+kγ,s+(k+n)γ [ψ s+(k+n)γ × •] -δ xs Q s,s+nγ [ ψs × •] T V -→ k→∞ 0, which implies by (16) that lim n→∞ lim k→∞ δ xs P s+kγ,s+(k+n)γ [ψ s+(k+n)γ × •] -β γ Q 0,s [ ψs × •] T V = 0. ( 17
)
Then, by the Markov property, [START_REF] Champagnat | Uniform convergence to the Q-process[END_REF], and ( 7), one obtains that, for any k, n ∈ N and x ∈ l≥0 E s+lγ ,

δ x P s,s+(k+n)γ -δ x P s+kγ,s+(k+n)γ ψ s+(k+n)γ = (δ x P s,s+kγ ) P s+kγ,s+(k+n)γ -δ x P s+kγ,s+(k+n)γ ψ s+(k+n)γ ≤ C [P s,s+kγ ψ s+kγ (x) + ψ s+kγ (x)]e -κγn ≤ C [ψ s (x) + ψ s+kγ (x)]e -κγn , (18)
whereby

C := C C 1 + C 1-θ ∨ 1 .
Then, for any k, n ∈ N,

δ xs P s,s+(k+n)γ [ψ s+(k+n)γ × •] -β γ Q 0,s [ ψs × •] T V ≤ C [ψ s (x) + ψ s+kγ (x)]e -κγn + δ xs P s+kγ,s+(k+n)γ [ψ s+(k+n)γ × •] -β γ Q 0,s [ ψs × •] T V , ( 19
)
which by [START_REF] He | Some conditional limiting theorems for symmetric Markov processes with tightness property[END_REF] and the pointwise convergence of (ψ s+kγ ) k∈Z+ implies that lim n→∞

δ xs P s,s+nγ [ψ s+nγ ו]-β γ Q 0,s [ ψs ו] T V = lim n→∞ lim sup k→∞ δ xs P s,s+(k+n)γ [ψ s+(k+n)γ ו]-β γ Q 0,s [ ψs ו] T V = 0.
(20) The weak ergodicity [START_REF] Champagnat | Uniform convergence to the Q-process[END_REF] implies therefore that the previous convergence actually holds for any initial distribution 21) and Lebesgue's dominated convergence theorem implies that

µ ∈ M 1 (E 0 ) satisfying µ(ψ 0 ) < +∞, so that µP 0,s+nγ [ψ s+nγ × •] -β γ Q 0,s [ ψs × •] T V -→ n→∞ 0. ( 21
) Since µP 0,s+nγ [ψ s+nγ × •] -β γ Q 0,s [ ψs × •] T V ≤ 2 for all µ ∈ M 1 (E 0 ), s ≥ 0 and n ∈ Z + , (
1 γ γ 0 µP 0,s+nγ [ψ s+nγ × •] -β γ Q 0,s [ ψs × •] T V ds -→ n→∞ 0, which implies that 1 γ γ 0 µP 0,s+nγ [ψ s+nγ × •]ds - 1 γ γ 0 β γ Q 0,s [ ψs × •]ds T V -→ n→∞ 0.
By Cesaro's lemma, this allows us to conclude that, for any µ ∈ M 1 (E 0 ) such that µ(ψ 0 ) < +∞,

1 t t 0 µP 0,s [ψ s × •]ds - 1 γ γ 0 β γ Q 0,s [ ψs × •]ds T V ≤ 1 t γ t γ k=0 1 γ γ 0 µP 0,s+kγ [ψ s+kγ × •]ds - 1 γ γ 0 β γ Q 0,s [ ψs × •]ds T V + 1 t t t γ γ µP 0,s [ψ s × •]ds T V -→ t→∞ 0,
which concludes the proof of (9).

Third step. In the same manner, we now prove that, for any µ ∈ M 1 (E 0 ) such that µ(ψ 0 ) < +∞,

1 t t 0 µP 0,s ds - 1 γ γ 0 β γ Q 0,s ds T V -→ t→∞ 0. ( 22
)
In fact, for any function f bounded by 1 and

µ ∈ M 1 (E 0 ) such that µ(ψ 0 ) < +∞, µP 0,s+nγ ψ s+nγ × f ψ s+nγ -β γ Q 0,s ψs × f ψs ≤ µP 0,s+nγ ψ s+nγ × f ψ s+nγ -β γ Q 0,s ψs × f ψ s+nγ + β γ Q 0,s ψs × f ψ s+nγ -β γ Q 0,s ψs × f ψs ≤ µP 0,s+nγ [ψ s+nγ × •] -β γ Q 0,s [ ψs × •] T V + β γ Q 0,s ψs × f ψ s+nγ -β γ Q 0,s ψs × f ψs .
We now remark that, since ψ s+nγ ≥ 1 for any s and n ∈ Z + , one has that ψs

ψ s+nγ -1 ≤ 1 + ψs .
Since (ψ s+nγ ) n∈Z+ converges pointwisely towards ψs and β γ Q 0,s ψs < +∞, Lebesgue's dominated convergence theorem implies sup f ∈B1(E)

β γ Q 0,s ψs × f ψ s+nγ -β γ Q 0,s ψs × f ψs -→ n→∞ 0.
Then, using [START_REF] Höpfner | Ergodicity and limit theorems for degenerate diffusions with time periodic drift. application to a stochastic hodgkin-huxley model[END_REF], one has µP 0,s+nγ -

β γ Q 0,s T V -→ n→∞ 0,
which allows us to conclude [START_REF] Höpfner | Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input[END_REF], using the same argument as in the first step.

Fourth step. In order to show the L 2 -ergodic theorem, we let f ∈ B(E). For any x ∈ E 0 and t ≥ 0,

E 0,x 1 t t 0 f (X s )ds -E 0,x 1 t t 0 f (X s )ds 2 = 2 t 2 t 0 t s (E 0,x [f (X s )f (X u )] -E 0,x [f (X s )]E 0,x [f (X u )]) duds = 2 t 2 t 0 t s E 0,x [f (X s ) (f (X u ) -E 0,x [f (X u )])] duds = 2 t 2 t 0 t s E 0,x f (X s ) E s,Xs [f (X u )] -E s,δxP0,s [f (X u )] duds,
whereby the Markov property was used on the last line. By (8) (weak ergodicity) and ( 7), one obtains for any s ≤ t,

E s,Xs [f (X t )] -E s,δxP0,s [f (X t )] ≤ C f ∞ [ψ s (X s ) + ψ 0 (x)]e -κ(t-s) , P 0,x -almost surely, (23)
whereby C was defined in the first part. As a result, for any x ∈ E 0 and t ≥ 0,

E 0,x 1 t t 0 f (X s )ds -E 0,x 1 t t 0 f (X s )ds 2 ≤ 2C f ∞ t 2 t 0 t s E 0,x [|f (X s )|(ψ s (X s )+ψ 0 (x))]e -κ(u-s) duds = 2C f ∞ t 2 t 0 E 0,x [|f (X s )|(ψ s (X s ) + ψ 0 (x))]
e κs e -κs -e -κt κ ds

= 2C f ∞ κt ×E 0,x 1 t t 0 |f (X s )|(ψ s (X s ) + ψ 0 (x))ds - 2C f ∞ e -κt κt 2 t 0 e κs E 0,x [|f (X s )|(ψ s (X s )+ψ 0 (x))]ds.
Then, by [START_REF] Champagnat | Uniform convergence of penalized time-inhomogeneous Markov processes[END_REF], there exists a constant C > 0 such that, for any x ∈ E 0 , when t → ∞,

E 0,x 1 t t 0 f (X s )ds -E 0,x 1 t t 0 f (X s )ds 2 ≤ C f ∞ ψ 0 (x) t × 1 γ γ 0 β γ Q 0,s [|f | ψs ]ds + o 1 t . ( 24
)
Since f ∈ B(E) and by definition of the total variation distance, [START_REF] Höpfner | Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input[END_REF] implies that, for all x ∈ E 0 ,

1 t t 0 P 0,s f (x) - 1 γ γ 0 β γ Q 0,s f ds ≤ f ∞ 1 t t 0 δ x P 0,s ds - 1 γ γ 0 β γ Q 0,s ds T V -→ t→∞ 0.
Then, using [START_REF] Höpfner | Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input[END_REF], one deduces that, for any x ∈ E 0 and bounded function f ,

E 0,x 1 t t 0 f (X s )ds - 1 γ γ 0 β γ Q 0,s f ds 2 ≤ 2 E 0,x 1 t t 0 f (X s )ds - 1 t t 0 P 0,s f (x) 2 + 1 t t 0 P 0,s f (x) - 1 γ γ 0 β γ Q 0,s f ds 2 -→ t→∞ 0.
The convergence for any probability measure µ ∈ M 1 (E 0 ) comes from Lebesgue's dominated convergence theorem.

Fifth step. We now fix non-negative f ∈ B(E), and µ ∈ M 1 (E 0 ) satisfying µ(ψ 0 ) < +∞. The following proof is inspired by the proof of [START_REF] Vassiliou | Laws of large numbers for non-homogeneous Markov systems[END_REF]Theorem 12].

Since µ(ψ 0 ) < +∞, the inequality [START_REF] Oçafrain | Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically[END_REF] entails that there exists a finite constant C f,µ ∈ (0, ∞) such that, for t large enough,

E 0,µ 1 t t 0 f (X s )ds -E 0,µ 1 t t 0 f (X s )ds 2 ≤ C f,µ t .
Then, for n large enough,

E 0,µ   1 n 2 n 2 0 f (X s )ds -E 0,µ 1 n 2 n 2 0 f (X s )ds 2   ≤ C f,µ n 2 .
Then, by Chebyshev's inequality and the Borel-Cantelli lemma, this last inequality implies that

1 n 2 n 2 0 f (X s )ds -E 0,µ 1 n 2 n 2 0 f (X s )ds -→ n→∞ 0, P 0,µ -almost surely.
One thereby obtains by the convergence ( 22) that

1 n 2 n 2 0 f (X s )ds -→ n→∞ 1 γ γ 0 β γ Q 0,s f ds, P 0,µ -almost surely. ( 25
)
Since the nonnegativity of f is assumed, this implies that for any t > 0 we have

√ t 2 0 f (X s )ds ≤ t 0 f (X s )ds ≤ √ t 2 0 f (X s )ds.
These inequalities and (25) then give that

1 t t 0 f (X s )ds -→ t→∞ 1 γ γ 0 β γ Q 0,s f ds, P 0,µ -almost surely.
In order to conclude that the result holds for any bounded measurable function f , it is enough to decompose f = f + -f -with f + := f ∨ 0 and f -= (-f ) ∨ 0 and apply the above convergence to f + and f -. This concludes the proof of Theorem 1.

Proof of Corollary 1. We remark as in the previous proof that, if f ∞ ≤ 1 and ψ s = 1, an upper-bound for the inequality (24) can be obtained, which does not depend on f and x. Likewise, the convergence (21) holds uniformly in the initial measure due to [START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF].

Remark 3. The proof of Theorem 1, as written above, does not allow to deal with semigroups satisfying a Doeblin condition with time-dependent constant c s , that is such that there exists t 0 ≥ 0 and a family of probability measure (ν t ) t≥0 on (E t ) t≥0 such that, for all s ≥ 0 and x ∈ E s ,

δ x P s,s+t0 ≥ c s+t0 ν s+t0 .
In fact, under the condition written above, we can show (see for example the proof of the formula (2.7) of [9, Theorem 2.1]) that, for all s ≤ t and µ 1 , µ 2 ∈ M 1 (E s ),

µ 1 P s,t -µ 2 P s,t T V ≤ 2 t-s t 0 k=1 (1 -c t-kt0 ).
Hence, by this last inequality with µ 1 = δ x P s,s+kγ , µ 2 = δ x , replacing s by s + kγ and t by s + (k + n)γ, one obtains

δ x P s,s+(k+n)γ -δ x P s+kγ,s+(k+n)γ T V ≤ 2 nγ t 0 l=1 (1 -c s+(k+n)γ-lt0 ),
replacing therefore the inequality [START_REF] He | On the quasi-ergodic distribution of absorbing Markov processes[END_REF] in the proof of Theorem 1. Plugging therefore this last inequality into the formula [START_REF] Hening | Stochastic Lotka-Volterra food chains[END_REF], one obtains

δ x P s,s+(k+n)γ -β γ Q 0,s T V ≤ 2 nγ t 0 l=1 (1 -c s+(k+n)γ-lt0 ) + δ x P s+kγ,s+(k+n)γ -β γ Q 0,s T V .
Hence, we see that we cannot conclude a similar result when c s -→ 0, as s → +∞ since, for n fixed, lim sup

k→∞ nγ t 0 l=1 (1 -c s+(k+n)γ-lt0 ) = 1.

Application to quasi-stationarity with moving boundaries

In this section, (X t ) t≥0 is assumed as being a time-homogeneous Markov process. We consider a family of measurable subsets (A t ) t≥0 of E, and denote the hitting time

τ A := inf{t ≥ 0 : X t ∈ A t }.
For all s ≤ t, denote by F s,t the σ-field generated by the family (X u ) s≤u≤t and F t := F 0,t . Assume that τ A is a stopping time with respect to the filtration (F t ) t≥0 . Assume also that for any x ∈ A 0 ,

P 0,x [τ A < +∞] = 1 and P 0,x [τ A > t] > 0, ∀t ≥ 0.
We will be interested in a notion of quasi-stationarity with moving boundaries, which studies the asymptotic behavior of the Markov process (X t ) t≥0 conditioned not to hit (A t ) t≥0 up to the time t. For non-moving boundaries (A t = A 0 for any t ≥ 0), the quasi-limiting distribution is defined as a probability measure α such that, for at least one initial measure µ and for all measurable subsets A ⊂ E,

P 0,µ [X t ∈ A|τ A > t] -→ t→∞ α(A).
Such a definition is equivalent (still in the non-moving framework) to the notion of quasi-stationary distribution defined as a probability measure α such that, for any t ≥ 0,

P 0,α [X t ∈ •|τ A > t] = α. ( 26
)
If quasi-limiting and quasi-stationary distributions are in general well-defined for time-homogeneous Markov processes and non-moving boundaries (see [START_REF] Collet | Quasi-stationary distributions[END_REF][START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF] for a general overview on the theory of quasi-stationarity), these notions could be not well-defined for time-inhomogeneous Markov processes or moving boundaries and are not equivalent anymore. In particular, under reasonable assumptions on irreducibility, it was shown in [START_REF] Oçafrain | Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically[END_REF] that the notion of quasi-stationary distribution as defined by ( 26) is not well-defined for time-homogeneous Markov processes absorbed by moving boundaries.

Another asymptotic notion to study is the quasi-ergodic distribution, related to a conditional version of the ergodic theorem and usually defined as follows.

Definition 2. A probability measure β is a quasi-ergodic distribution if, for some initial measure µ ∈ M 1 (E \ A 0 ) and for any bounded continuous function f ,

E 0,µ 1 t t 0 f (X s )ds τ A > t -→ t→∞ β(f ).
In the time-homogeneous setting (in particular for non-moving boundaries), this notion has been extensively studied (see for example [START_REF] Breyer | A quasi-ergodic theorem for evanescent processes[END_REF][START_REF] Champagnat | Uniform convergence to the Q-process[END_REF][START_REF] Chen | A deviation inequality and quasi-ergodicity for absorbing Markov processes[END_REF][START_REF] Colonius | Quasi-ergodic limits for finite absorbing Markov chains[END_REF][START_REF] Darroch | On quasi-stationary distributions in absorbing discrete-time finite Markov chains[END_REF][START_REF] He | A note on the quasi-ergodic distribution of one-dimensional diffusions[END_REF][START_REF] He | Some conditional limiting theorems for symmetric Markov processes with tightness property[END_REF][START_REF] He | On the quasi-ergodic distribution of absorbing Markov processes[END_REF][START_REF] Oçafrain | Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically[END_REF]). In the "moving boundaries" framework, the existence of quasi-ergodic distributions has been dealt with in [START_REF] Oçafrain | Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically[END_REF] for Markov chains on finite state spaces absorbed by periodic boundaries, and in [START_REF] Oçafrain | Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF] for processes satisfying a Champagnat-Villemonais condition (see Assumption (A') set later) absorbed by converging or periodic boundaries. In this last paper, the existence of the quasi-ergodic distribution is dealt with through the following inequality (see [START_REF] Oçafrain | Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF]Theorem 1]), holding for any initial state x, s ≤ t and for some constant C, γ > 0 independent of x, s and t:

P 0,x (X s ∈ •|τ A > t) -Q 0,x (X s ∈ •) T V ≤ Ce -γ(t-s) ,
where the family of probability measure (Q s,x ) s≥0,x∈Es is defined by

Q s,x [Γ] := lim T →∞ P s,x [Γ|τ A > T ], ∀s ≤ t, x ∈ E \ A s , Γ ∈ F s,t .
Moreover, [9, Proposition 3.1], there exists a family of positive bounded functions (η t ) t≥0 defined in such that, for all s ≤ t and x ∈ E s , E s,x (η t (X t )1 τ A >t ) = η s (x). Then, we can show (this is actually shown in [START_REF] Champagnat | Uniform convergence of penalized time-inhomogeneous Markov processes[END_REF]) that

Q s,x (Γ) = E s,x (1 Γ,τ A >t η t (X t ) η s (x) )
and that, for all µ ∈ M 1 (E 0 ),

P 0,µ (X s ∈ •|τ A > t) -Q 0,η0 * µ (X s ∈ •) T V ≤ Ce -γ(t-s) .
where η 0 * µ(dx) := η0(x)µ(dx) µ(η0)

. By triangular inequality, one has

1 t t 0 P 0,µ [X s ∈ •|τ A > t]ds - 1 t t 0 Q 0,η0 * µ [X s ∈ •]ds T V ≤ C γt , ∀t > 0, (27)
In particular, the inequality (27) implies that there exists a quasi-ergodic distribution β for the process (X t ) t≥0 absorbed by (A t ) t≥0 if and only if there exist some probability measures µ ∈ M 1 (E 0 ) such that 1 t t 0 Q 0,η0 * µ [X s ∈ •]ds converges weakly to β, when t goes to infinity. In other terms, under Assumption (A'), the existence of a quasi-ergodic distribution for the absorbed process is equivalent to the law of large number for its Q-process.

Assumption (A') is now set.

Assumption (A').

There exists a family of probability measures (ν t ) t≥0 , defined on E \ A t for each t, such that:

(A'1) There exists t 0 ≥ 0 and c 1 > 0 such that

P s,x [X s+t0 ∈ •|τ A > s + t 0 ] ≥ c 1 ν s+t0 , ∀s ≥ 0, ∀x ∈ E \ A s .
(A'2) There exists c 2 > 0 such that

P s,νs [τ A > t] ≥ c 2 P s,x [τ A > t], ∀s ≤ t, ∀x ∈ E \ A s .
In what follows, we say that the couple {(X t ) t≥0 , (A t ) t≥0 } satisfies Assumption (A') when it holds for the Markov process (X t ) t≥0 considered as absorbed by the moving boundary (A t ) t≥0 .

The condition (A'1) is a conditional version of the Doeblin condition ( 12) and (A'2) is a Harnack-like inequality on the probabilities of surviving, necessary to deal with the conditioning. They are equivalent to the set of conditions presented in [1, Definition 2.2], when the non-conservative semigroup is sub-Markovian. In the timehomogeneous framework, we obtain the Champagnat-Villemonais condition defined in [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF] (see Assumption (A)), shown as being equivalent to the exponential uniform convergence to quasi-stationarity in total variation.

In [START_REF] Oçafrain | Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF], the existence of a unique quasi-ergodic distribution has been only proved for converging or periodic boundaries. However, we can expect such a result of existence (and uniqueness) for other kinds of movement for the boundary. Hence, the aim of this section is to extend the results on the existence of quasi-ergodic distributions obtained in [START_REF] Oçafrain | Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF] for Markov processes absorbed by asymptotically periodic moving boundaries. Now, let us state the following theorem.

Theorem 2. Assume that there exists a γ-periodic sequence of subsets (B t ) t≥0 such that, for any s ∈ [0, γ),

E s := E \ k∈Z+ l≥k A s+lγ ∪ B s = ∅
and there exists x s ∈ E s such that, for any n ≤ N ,

P s+kγ,xs [X s+(k+n)γ ∈ •, τ A > s + (k + N )γ] -P s,xs [X s+nγ ∈ •, τ B > s + N γ] T V -→ k→∞ 0. ( 28
)
Assume also that Assumption (A') is satisfied by the couples {(X t ) t≥0 , (A t ) t≥0 } and {(X t ) t≥0 , (B t ) t≥0 }.

Then there exists a probability measure

β ∈ M 1 (E) such that sup µ∈M1(E\A0) sup f ∈B1(E) E 0,µ 1 t t 0 f (X s )ds -β(f ) 2 τ A > t -→ t→∞ 0. ( 29
)
Remark 4. Remark that the condition (28) implies that, for any n ∈ Z + ,

P s+kγ,xs [τ A > s + (k + n)γ] -→ k→∞ P s,xs [τ B > s + nγ].
Under the additional condition B t ⊂ A t for all t ≥ 0, these two conditions are equivalent, since for all n ≤ N ,

P s+kγ,xs [X s+(k+n)γ ∈ •, τ A > s + (k + N )γ] -P s,xs [X s+nγ ∈ •, τ B > s + N γ] T V = P s+kγ,xs [X s+(k+n)γ ∈ •, τ B ≤ s + (k + N )γ < τ A ] T V ≤ P s+kγ,xs [τ B ≤ s + (k + N )γ < τ A ] = |P s+kγ,xs [τ A > s + (k + N )γ] -P s,xs [τ B > s + N γ]|,
where we used the periodicity of (B t ) t≥0 , writing that P The main idea of this proof is to apply Corollary 1. Since {(X t ) t≥0 , (A t ) t≥0 } and {(X t ) t≥0 , (B t ) t≥0 } satisfy Assumption (A'), [START_REF] Oçafrain | Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF]Theorem 1] implies that there exist two families of probability measures (Q A s,x ) s≥0,x∈E\As and (Q B s,x ) s≥0,x∈E\Bs such that, for any

s ≤ t, x ∈ E \ A s , y ∈ E \ B s and Γ ∈ F s,t , Q A s,x [Γ] = lim T →∞ P s,x [Γ|τ A > T ], and Q B s,y [Γ] = lim T →∞ P s,y [Γ|τ B > T ].
In particular, the quasi-ergodic distribution β is the limit of t → 1

t t 0 Q B 0,µ [X s ∈ •]
ds, when t goes to infinity (see [START_REF] Oçafrain | Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF]Theorem 5]). Also, by [START_REF] Oçafrain | Q-processes and asymptotic properties of Markov processes conditioned not to hit moving boundaries[END_REF]Theorem 1], there exists a constant C > 0 and κ > 0 such that, for any s ≤ t ≤ T , for any

x ∈ E \ A s , Q A s,x [X t ∈ •] -P s,x [X t ∈ •|τ A > T ] T V ≤ Ce -κ(T -t) , and for any x ∈ E \ B s , Q B s,x [X t ∈ •] -P s,x [X t ∈ •|τ B > T ] T V ≤ Ce -κ(T -t) .
Moreover, for any s ≤ t ≤ T and x ∈ E s ,

P s,x [X t ∈ •|τ A > T ] -P s,x [X t ∈ •|τ B > T ] T V = P s,x [X t ∈ •, τ A > T ] P s,x [τ A > T ] - P s,x [X t ∈ •, τ B > T ] P s,x [τ B > T ] T V = P s,x (τ B > T ) P s,x (τ A > T ) P s,x [X t ∈ •, τ A > T ] P s,x [τ B > T ] - P s,x [X t ∈ •, τ B > T ] P s,x [τ B > T ] T V ≤ P s,x (τ B > T ) P s,x (τ A > T ) P s,x [X t ∈ •, τ A > T ] P s,x [τ B > T ] - P s,x [X t ∈ •, τ A > T ] P s,x [τ B > T ] T V + P s,x [X t ∈ •, τ A > T ] P s,x [τ B > T ] - P s,x [X t ∈ •, τ B > T ] P s,x [τ B > T ] T V ≤ |P s,x (τ B > T ) -P s,x (τ A > T )| P s,x (τ B > T ) + P s,x [X t ∈ •, τ A > T ] -P s,x [X t ∈ •, τ B > T ] T V P s,x [τ B > T ] ≤ 2 P s,x [X t ∈ •, τ A > T ] -P s,x [X t ∈ •, τ B > T ] T V P s,x [τ B > T ] , ( 30
) since |P s,x (τ B > T ) -P s,x (τ A > T )| ≤ P s,x [X t ∈ •, τ A > T ] -P s,x [X t ∈ •, τ B > T ] T V .
Then, we obtain for any s ≤ t ≤ T and x ∈ E s ,

Q A s,x [X t ∈ •] -Q B s,x [X t ∈ •] T V ≤ 2Ce -κ(T -t) + 2 P s,x [X t ∈ •, τ A > T ] -P s,x [X t ∈ •, τ B > T ] T V P s,x [τ B > T ] . ( 31
)
The condition (28) implies the existence of x s ∈ E s such that, for any n ≤ N , for all k ∈ Z + ,

lim k→∞ P s+kγ,xs [X s+(k+n)γ ∈ •, τ A > s + (k + N )γ] -P s,xs [X s+nγ ∈ •, τ B > s + N γ] T V = 0,
which implies by (31) that, for any n ≤ N , lim sup

k→∞ Q A s+kγ,xs [X s+(k+n)γ ∈ •] -Q B s+kγ,xs [X s+(k+n)γ ∈ •] T V ≤ 2Ce -κγ(N -n) . Now, letting N → ∞, for any n ∈ Z + lim k→∞ Q A s+kγ,xs [X s+(k+n)γ ∈ •]-Q B s+kγ,xs [X s+(k+n)γ ∈ •] T V = lim k→∞ Q A s+kγ,xs (X s+(k+n)γ ∈ •)-Q B s,xs (X s+nγ ∈ •) T V = 0.
In other words, the semigroup (Q A s,t ) s≤t defined by

Q A s,t f (x) := E Q A s,x (f (X t )), ∀s ≤ t, ∀f ∈ B(E \ A t ), ∀x ∈ E \ A s ,
is asymptotically periodic (according to the Definition 1, with ψ s = ψs = 1 for all s ≥ 0), associated to the auxiliary semigroup (Q B s,t ) s≤t defined by

Q B s,t f (x) := E Q B s,x (f (X t )), ∀s ≤ t, ∀f ∈ B(E \ B t ), ∀x ∈ E \ B s .
Moreover, assumptions (A') satisfied for {(X t ) t≥0 , (A t ) t≥0 } and {(X t ) t≥0 , (B t ) t≥0 } imply that Doeblin condition holds for these two Q-processes. As a matter of fact, by the Markov property, for all s ≤ t ≤ T and x ∈ E \ A s ,

P s,x (X t ∈ •|τ A > T ) = E s,x 1 Xt∈•,τ A >t P t,Xt (τ A > T ) P s,µ (τ A > T ) = E s,x 1 Xt∈•,τ A >t P s,x (τ A > t) P t,Xt (τ A > T ) P t,φt,s(µ) (τ A > T ) = E s,x 1 Xt∈• P t,Xt (τ A > T ) P t,φt,s(δx) (τ A > T ) τ A > t . ( 32
)
By (A 1 ), for all s ≥ 0, T ≥ s + t 0 and x ∈ E \ A s ,

E s,x 1 Xs+t 0 ∈• P s+t0,Xs+t 0 (τ A > T ) P s+t0,φs+t 0 ,s(δx ) (τ A > T ) τ A > s + t 0 ≥ c 1 • ν s+t0 (dy) P s+t0,y (τ A > T ) P s+t0,φs+t 0 ,s (δx) (τ A > T )
, that is to say by (32) that

P s,x (X s+t0 ∈ •|τ A > T ) ≥ c 1 • ν s+t0 (dy) P s+t0,y (τ A > T ) P s+t0,φs+t 0 ,s (δx) (τ A > T ) .
Letting T → ∞ in this last inequality and using [9, Proposition 3.1], for all s ≥ 0 and x ∈ E \ A s ,

Q A s,x (X s+t0 ∈ •) ≥ c 1 • ν s+t0 (dy) η s+t0 (y) φ s+t0,s (δ x )(η s+t0 )
.

The measure • ν s+t0 (dy) ηs+t 0 (y) φs+t 0 ,s(δx )(ηs+t 0 ) is then a positive measure whose the mass is lower-bounded by c 2 by (A 2 ), since for all s ≥ 0 and T ≥ s + t 0 ,

E\As+t 0 ν s+t0 P t,x (τ A > T ) P t,φt,s(δx) (τ A > T ) ≥ c 2 .
This proves therefore a Doeblin condition for the semigroup (Q A s,t ) s≤t . The same reasoning applies to show also a Doeblin condition for the semigroup (Q B s,t ) s≤t . Then, using (27) then Corollary 1, lim t→∞

1 t t 0 P 0,µ [X s ∈ •|τ A > t]ds = lim t→∞ 1 t t 0 Q A 0,η0 * µ (X s ∈ •)ds = lim t→∞ 1 t t 0 Q B 0,η0 * µ [X s ∈ •]ds = β,
where the limits refer to the convergence in total variation and hold uniformly in the initial measure.

For

any µ ∈ M 1 (E \ A 0 ), f ∈ B 1 (E) and t ≥ 0, E 0,µ 1 t t 0 f (X s )ds 2 τ A > t = 2 t 2 t 0 t s E 0,µ [f (X s )f (X u )|τ A > t]duds.
Then, by [25, Theorem 1], for any s ≤ u ≤ t, for any µ ∈ M 1 (E \ A 0 ) and f ∈ B(E),

E 0,µ [f (X s )f (X u )|τ A > t] -E Q A 0,η0 * µ [f (X s )f (X u )] ≤ C f ∞ e -κ(t-u) ,
where the expectation

E Q A 0,η0 * µ is associated to the probability measure Q A 0,η0 * µ . Hence, for any µ ∈ M 1 (E \ A 0 ), f ∈ B 1 (E) and t > 0, E 0,µ 1 t t 0 f (X s )ds -β(f ) 2 τ A > t -E Q A 0,η0 * µ 1 t t 0 f (X s )ds -β(f ) 2 ≤ 4C t 2 t 0 t s e -κ(t-u) duds ≤ 4C κt - 4C(1 -e -κt ) κ 2 t 2 .
Moreover, since (Q A s,t ) s≤t is asymptotically periodic in total variation and satisfies the Doeblin condition, like

(Q B s,t ) s≤t , Corollary 1 implies that sup µ∈M1(E\A0) sup f ∈B1(E) E Q A 0,η0 * µ 1 t t 0 f (X s )ds -β(f ) 2 -→ t→∞ 0. Then sup µ∈M1(E\A0) sup f ∈B1(E) E 0,µ 1 t t 0 f (X s )ds -β(f ) 2 τ A > t -→ t→∞ 0.
Remark 5. It seems that Assumption (A') can be weaken by a conditional version of Assumption 1. In particular, such conditions can be derived from the Assumption (F) presented in [START_REF] Champagnat | General criteria for the study of quasi-stationarity[END_REF], as it will be shown later by the preprint [START_REF] Champagnat | Quasi-stationarity for time-(in)homogeneous Markov processes absorbed by moving boundaries through Lyapunov criteria[END_REF] in preparation.

Examples

Asymptotically periodic Ornstein-Uhlenbeck processes

Let (X t ) t≥0 be a time-inhomogeneous diffusion process on R satisfying the following stochastic differential equation

dX t = dW t -λ(t)X t dt,
where (W t ) t≥0 is a one-dimensional Brownian motion and λ By Itô's lemma, for any s ≤ t,

: [0, ∞) → [0, ∞) is
X t = e - t s λ(u)du X s + t s e u s λ(v)dv dW u .
In particular, denoting (P s,t ) s≤t the semigroup associated to (X t ) t≥0 , for any f ∈ B(R), t ≥ 0 and x ∈ R,

P s,t f (x) = E   f   e - t s λ(u)du x + e - t s λ(u)du t s e 2 u s λ(v)dv du × N (0, 1)     ,
where N (0, 1) denotes a standard Gaussian variable.

Theorem 3. Assume that there exists a γ-periodic function g, bounded on R, such that λ ∼ t→∞ g. Then the assumptions of Theorem 1 hold.

Proof. In our case, the auxiliary semigroup (Q s,t ) s≤t of Definition 1 will be defined as follows: for any f ∈ B(R), t ≥ 0 and x ∈ R,

Q s,t f (x) = E   f   e - t s g(u)du x + e - t s g(u)du t s e 2 u s g(v)dv du × N (0, 1)     .
In particular, the semigroup (Q s,t ) s≤t is associated to the process (Y t ) t≥0 following

dY t = dW t -g(t)Y t dt.
We first remark that the function ψ : x → 1 + x 2 is a Lyapunov function for (P s,t ) s≤t and (Q s,t ) s≤t . In fact, for any s ≥ 0 and x ∈ R, λ(u)du > 0. Taking θ ∈ (e -2γc inf , 1), there exists a compact set K such that, for any s ≥ 0, P s,s+γ ψ(x) ≤ θψ(x) + C1 K (x).

P s,s+γ ψ(x) = 1 + e -2 s+γ s λ(u)du x 2 + e
Moreover, for any s ≥ 0 and t ∈ [0, γ), the function P s,s+t ψ/ψ is upper-bounded uniformly in s and t. It remains therefore to prove Assumption 1 (i) for (P s,t ) s≤t , which is a consequence of the following lemma. The proof of this lemma is postponed after the end of this proof. Since λ ∼ t→∞ g and these two functions are bounded on R + , Lebesgue's dominated convergence theorem implies that, for all s ≤ t, Using [14, Theorem 1.3.], for any x ∈ R,

δ x P s+kγ,t+kγ -δ x Q s+kγ,t+kγ T V -→ k→∞ 0. ( 33
)
To deduce the convergence in ψ-distance, we will inspire from the proof of [START_REF] Hening | Stochastic Lotka-Volterra food chains[END_REF]Lemma 3.1]. Since the variances are uniformly bounded in k (for s ≤ t fixed), there exists H > 0 such that, for any k ∈ N and s ≤ t,

δ x P s+kγ,t+kγ [ψ 2 ] ≤ H and δ x Q s,t [ψ 2 ] ≤ H. ( 34
)
Since lim |x|→∞ ψ(x) ψ 2 (x) = 0, for any > 0, there exists l > 0 such that, for any function f such that |f | ≤ ψ and for any |x| ≥ l ,

|f (x)| ≤ ψ(x) 2 H .
This implies with (34) that, denoting K := [-l , l ], for any k ∈ Z + , f such that |f | ≤ ψ and x ∈ R,

δ x P s+kγ,t+kγ [f 1 K c ] ≤ and δ x Q s,t [f 1 K c ] ≤ .
Then, for any k ∈ Z + and f such that |f | ≤ ψ,

|δ x P s+kγ,t+kγ f -δ x Q s,t f | ≤ 2 + |δ x P s+kγ,t+kγ [f 1 K ] -δ x Q s,t [f 1 K ]| (35) ≤ 2 + (1 + l 2 ) δ x P s+kγ,t+kγ -δ x Q s,t T V (36)
Hence, (33) implies that, for k large enough, for any f bounded by ψ,

|δ x P s+kγ,t+kγ f -δ x Q s,t f | ≤ 3 , ( 37
)
implying that

δ x P s+kγ,t+kγ -δ x Q s,t ψ -→ k→∞ 0.
We now prove Lemma 1.

Proof of Lemma 1. Defining

f ν (x) := e - (x-a) 2 2b - 2 ∧ e - (x+a) 2 2b - 2 ,
we conclude easily that, for any m ∈ [-a, a] and σ ≥ b -, for any x ∈ R,

e -(x-m) 2 2σ 2 ≥ f ν (x).
Imposing moreover that σ ≤ b + , one has

1 √ 2πσ e -(x-m) 2 2σ 2 ≥ 1 √ 2πb + f ν (x),
which concludes the proof.

Quasi-ergodic distribution for Brownian motion absorbed by an asymptotically periodic moving boundary

Let (W t ) t≥0 be a one-dimensional Brownian motion and h be a C 1 -function such that

h min := inf t≥0 h(t) > 0, and
h max := sup t≥0 h(t) < +∞.
We assume also that

-∞ < inf t≥0 h (t) ≤ sup t≥0 h (t) < +∞.
Denote by

τ h := inf{t ≥ 0 : |W t | ≥ h(t)}.
Since h is continuous, the hitting time τ h is a stopping time with respect to the natural filtration of (W t ) t≥0 . Moreover, since sup t≥0 h(t) < +∞ and inf t≥0 h(t) > 0,

P s,x [τ h < +∞] = 1 and P s,x [τ h > t] > 0, ∀s ≤ t, ∀x ∈ [-h(s), h(s)].
The main assumption on the function h is the existence of a γ-periodic function g such that h(t) ≤ g(t), for any t ≥ 0, and such that h ∼ t→∞ g, and h ∼ t→∞ g .

Similarly to τ h , denote

τ g := inf{t ≥ 0 : |W t | = g(t)}.
Finally, let us assume that there exists n 0 ∈ N such that, for any s ≥ 0,

inf{u ≥ s : h(u) = inf t≥s h(t)} -s ≤ n 0 γ. ( 38
)
This condition says that there exists n 0 ∈ N such that, for any time s ≥ 0, the infimum of the function h on the domain [s, +∞) is reached on the subset [s, s + n 0 γ].

We first show the following proposition.

Proposition 1. The Markov process (W t ) t≥0 , considered as absorbed by h or by g, satisfies Assumption (A').

Proof. In what follows, Assumption (A') w.r.t. the absorbing function h will be shown. The following proof could easily be adapted for the function g.

• Proof of (A'1). Denote T := {s ≥ 0 : h(s) = inf t≥s h(t)}. The condition (38) implies that this set contains an infinity of times.

In what follows, the following notation is needed: for any z ∈ R, define τ z as

τ z := inf{t ≥ 0 : |W t | = z}.
Also, let us state that, since the Brownian motion absorbed at {-1, 1} satisfies Assumption (A) of [START_REF] Champagnat | Exponential convergence to quasi-stationary distribution and Q-process[END_REF] at any time (see [START_REF] Champagnat | Uniform convergence of conditional distributions for absorbed onedimensional diffusions[END_REF]), it follows that, for a given t 0 > 0, there exists c > 0 and ν ∈ M 1 ((-1, 1)) such that, for any x ∈ (-1, 1),

P 0,x W t 0 h 2 max ∧t0 ∈ • τ 1 > t 0 h 2 max ∧ t 0 ≥ cν. ( 39
)
Moreover, regarding the proof of [7, Section 5.1], the probability measure can be expressed as

ν = 1 2 (P 0,1-[W t2 ∈ •|τ 1 > t 2 ] + P 0,-1+ [W t2 ∈ •|τ 1 > t 2 ]) , ( 40
)
for some 0 < t 2 < t0 h 2 max ∧ t 0 and ∈ (0, 1).

The following lemma is very important for the following.

Lemma 2. For all z ∈ [h min , h max ],

P 0,x [W u ∈ •|τ z > u] ≥ cν z , ∀x ∈ (-z, z), ∀u ≥ t 0 ,
where t 0 was evoked before, c > 0 is the same constant as in (39) and

ν z (f ) = (-1,1) f (zx)ν(dx),
with ν ∈ M 1 ((-1, 1)) defined in (40).

The proof of this lemma is postponed after the current proof.

Let s ∈ T . Then, for all x ∈ (-h(s), h(s)) and t ≥ 0,

P s,x [W s+t ∈ •|τ h > s + t] ≥ P s,x [τ h(s) > s + t] P s,x [τ h > s + t] P s,x [W s+t ∈ •|τ h(s) > s + t],
By Lemma 2, for all x ∈ (-h(s), h(s)) and t ≥ t 0 ,

P s,x [W s+t ∈ •|τ h(s) > s + t] ≥ cν h(s) ,
which implies therefore that, for any t ∈ [t 0 , t 0 + n 0 γ],

P s,x [W s+t ∈ •|τ h > s + t] ≥ P s,x [τ h(s) > s + t] P s,x [τ h > s + t] cν h(s) ≥ P s,x [τ h(s) > s + t 0 + n 0 γ] P s,x [τ h > s + t 0 ] cν h(s) . ( 41
)
Let us introduce the process X h defined by, for all t ≥ 0,

X h t := W t h(t) .
By Itô's formula, for any t ≥ 0,

X h t = X h 0 + t 0 dW s h(s) - t 0 h (s) h(s) X h s ds.

Denote by (M

= h(t) h(s) exp - 1 2 h (t)h(t)w 2 I h (t) -h (s)h(s)w 2 I h (s) + t s w 2 I h (u) [(h (u)) 2 -[h(u)h (u)] ]du , (42)
Girsanov theorem implies that, for all x ∈ (-h(s), h(s)),

P s,x [τ h > s + t 0 ] = E I h (s), x h(s) E h s,s+t0 (W )1 τ1> s+t 0 0 1 h 2 (u) du . ( 43
)
On the event {τ 1 > s+t0 0 1 h 2 (u) du}, and since h and h are bounded on R + , the random variable E h s,s+t0 (W ) is almost surely bounded by a constant C > 0, uniformly in s, such that for all x ∈ (-h(s), h(s)),

E I h (s), x h(s) E h s,s+t0 (W )1 τ1> s+t 0 0 1 h 2 (u) du ≤ CP 0, x h(s) τ 1 > s+t0 s 1 h 2 (u) du . ( 44
) Since h(t) ≥ h(s) for all t ≥ s (since s ∈ T ), I h (s + t 0 ) -I h (s) ≤ t0 h(s) 2 .
By the scaling property of the Brownian motion and by Markov property, one has for all x ∈ (-h(s), h(s)),

P s,x [τ h(s) > s + t 0 ] = P 0,x [τ h(s) > t 0 ] = P 0, x h(s) τ 1 > t 0 h 2 (s) = E 0, x h(s) 1 τ1> s+t 0 s 1 h 2 (u) du P 0,W s+t 0 s 1 h 2 (u) du τ 1 > t 0 h 2 (s) - s+t0 s 1 h 2 (s) ds = P 0, x h(s) τ 1 > s+t0 s 1 h 2 (u) du P 0,φ I h (s+t 0 )-I h (s) (δx) τ 1 > t 0 h 2 (s) - s+t0 s 1 h 2 (u) du ,
where, for any initial distribution µ and any t ≥ 0,

φ t (µ) := P 0,µ [W t ∈ •|τ 1 > t].
The family (φ t ) t≥0 satisfies the equality φ t • φ s = φ t+s for all s, t ≥ 0. By this property, and using that

I h (s + t 0 ) -I h (s) ≥ t0 h 2 max
for any s ≥ 0, the minorization (39) implies that, for all s ≥ 0 and x ∈ (-1, 1),

φ I h (s+t0)-I h (s) (δ x ) ≥ cν.
Hence, by this minorization, and using that h is upper-bounded and lower-bounded positively on R + , one has for all x ∈ (-1, 1),

P 0,φ I h (s+t 0 )-I h (s) (δx) τ 1 > t 0 h 2 (s) - s+t0 s 1 h 2 (u) du ≥ cP 0,ν τ 1 > inf s≥0 t 0 h 2 (s) - s+t0 s 1 h 2 (u) du ,
that is to say,

P s,x [τ h(s) > s + t 0 ] P 0, x h(s) τ 1 > s+t0 s 1 h 2 (u) du ≥ cP 0,ν τ 1 > inf s≥0 γ h 2 (s) - s+t0 s 1 h 2 (u)
du .

In other words, we just showed that, for all x ∈ (-h(s), h(s)), This concludes the proof of (A'1).

P
• Proof of (A'2 This concludes the proof since, using (40), one has ν([-a, a]) > 0.

We now prove Lemma 2.

Proof of Lemma 2. This result comes from the scaling property of a Brownian motion. In fact, for any z ∈ [h min , h max ], x ∈ (-z, z) and t ≥ 0, and for any measurable bounded function f ,

E 0,x [f (W t )|τ z > t] = E 0,x f z × 1 z W z 2 t z 2 τ z > t = E 0, x z f z × W t z 2 τ 1 > t z 2 .
Then, the minorization (39) implies that, for any x ∈ (-1, 1),

P 0,x W t 0 h 2 max ∈ • τ 1 > t 0 h 2 max ≥ cν.
This inequality holds for any time greater than t0 h 2 max . In particular, for any z ∈ [h min , h max ] and x ∈ (-1, 1),

P 0,x W t 0 z 2 ∈ • τ 1 > t 0 z 2 ≥ cν.
Then, for any z ∈ [a, b], f positive and measurable, and x ∈ (-z, z),

E 0,x [f (W t0 )|τ z > t 0 ] ≥ cν z (f ) ,
where ν z (f ) := E f (z × x)ν(dx). This completes the proof of Lemma 2.

The section is now concluded by stating and proving the following result. which concludes the proof.

  du ≤ e -2γc inf ψ(x) + C, where C ∈ (0, +∞) and c inf := inf t≥0 1 γ t+γ t

Lemma 1 .

 1 For any a, b -, b + > 0, define the subset C(a, b -, b + ) ⊂ M 1 (R) as C(a, b -, b + ) := {N (m, σ) : m ∈ [-a, a], σ ∈ [b -, b + ]}.Then, for any a, b -, b + > 0, there exists a probability measure ν and a constant c > 0 such that, for any µ ∈ C(a, b -, b + ), µ ≥ cν.

  way, for all s ≤ t, )dv du.Hence, for any s ≤ t, e -

  s,x [τ h(s) > s + t 0 ] P s,x [τ h > s + t 0 ]Moreover, by Lemma 2 and the scaling property of the Brownian motion, for all x ∈ (-h(s), h(s)),P s,x [τ h(s) > s + t 0 + n 0 γ] P s,x [τ h(s) > s + t 0 ] = P 0,P0,x[Wt 0 ∈•|τ h(s) >t0] [τ h(s) > n 0 γ] ≥ cP 0,ν h(s) [τ h(s) > n 0 γ]Thus, gathering (41), (45) and (46), for any x ∈ (-h(s), h(s)) and any t ∈ [t 0 , t 0 + n 0 γ],P s,x [W s+t ∈ •|τ h > s + t] ≥ c 1 ν h(s) ,

			≥	c C	P 0,ν τ 1 > inf s≥0	t 0 h 2 (s)	-	s	s+t0	1 h 2 (u)	du	> 0.	(45)
							= c		ν(dy)P h(s)y [τ h(s) > n 0 γ]
							(-1,1)
							≥ cP 0,ν τ 1 >	n 0 γ min h 2		> 0.	(46)
													(47)
	where	c 1 := cP 0,ν τ 1 >	n 0 γ h 2 max	×	c C	P 0,ν τ 1 > inf s≥0	γ h 2 (s)	-	s	s+γ	1 h 2 (u)	du c.

We recall that the Doeblin condition (47) is, for now, only obtained for s ∈ T . Consider now s ∈ T . Then, by the condition (38), there exists s 1 ∈ T such that s < s 1 ≤ s + n 0 γ. Markov property and (47) implies therefore that, for any x ∈ (-h(s), h(s)),

P s,x [W s+t0+n0γ ∈ •|τ h > s + t 0 + n 0 γ] = P s1,φs 1 ,s [W s+t0+n0γ ∈ •|τ h > s + t 0 + n 0 γ] ≥ c 1 ν h(s1) ,

where, for all s ≤ t and µ ∈ M 1 ((-h(s), h(s))), φ t,s (µ) := P s,µ [W t ∈ •|τ h > t].

  ). Since (W t ) t≥0 is a Brownian motion, note that, for any s ≤ t, Thus, by Markov property, and using that the function s → P s,0 [τ g > t] is non-decreasing on [0, t] (for all t ≥ 0), one has for any s ≤ t,P s,a [τ h > t] ≥ E s,a [1 τ0<s+γ<τ h P τ0,0 [τ h > t]] ≥ P s,a [τ 0 < s + γ < τ h ]P s,0 [τ h > t]. (48)Denoting a := hmin hmax , by Lemma 2 and taking s 1 := inf{u ≥ s : u ∈ T }, one obtains that, for all s ≤ t, P s,ν h(s 1 ) [τ h > t] =

	sup
	x∈(-1,1)
	ν(dx)P s,h(s1)x [τ h > t]
	(-1,1)

P s,x [τ h > t] = P s,0 [τ h > t].

Also, for any a ∈ (0, h(s)), inf

[-a,a] P s,x [τ h > t] = P s,a [τ h > t]. ≥ ν([-a, a])P s,h(s1)a [τ h > t] ≥ ν([-a, a])P 0,hmin [τ 0 < γ < τ h ] sup x∈(-h(s),h(s)) P s,x [τ h > t].

  (W ) 1 τ1>I h (t+kγ)-I h (s+kγ) ,where, for any trajectory w = (w u ) u≥0 ,A h s,t,k (w) = h (t + kγ)h(t + kγ)w 2 I h (t+kγ)-I h (s+kγ) -h (s + kγ)h(s + kγ)w 2Moreover, since h ∼ t→∞ g and h ∼ t→∞ g , one has for all trajectories w = (w u ) u≥0 ands ≤ t ∈ [0, γ], I g (u) [(g (u)) 2 -[g(u)g (u)] ]du.Since the random variable exp -1 2 A h s,t,k (W ) 1 τ1>I h (t+kγ)-I h (s+kγ) is bounded almost surely, Lebesgue's dominated convergence theorem implies that P s+kγ,x [τ h > t + kγ] -→

					t
	A h s,t,k (w) -→ k→∞	g (t)g(t)w 2 I g (t)-I g (s) -g (s)g(s)w 2 0 +	s	w 2
		h(t + kγ) h(s + kγ)	E 0,x exp -	1 2	A h s,t,k 0
		t-s
	+ I Since h ∼ t→∞ g, one has for any s, t ∈ [0, γ] 0 w 2
		h(t + kγ) h(s + kγ)	-→ k→∞	g(t) g(s)	.

Theorem 4. For any s ≤ t, n ∈ N and any x ∈ R,

P s+kγ,x [τ h ≤ t + kγ < τ g ] -→ k→∞ 0.

In particular, Corollary 2 holds for (W t ) t≥0 absorbed by h.

Proof. Reminding (42), by Markov property for the Brownian motion, one has for any k, n ∈ N and any x ∈ R,

P s+kγ,x [τ h > t + kγ] = h (u+s+kγ)-I h (s+kγ) [(h (u + s + kγ)) 2 -[h(u + s + kγ)h (u + s + kγ)] ]du.

For the same reasons, and using that the function h is bounded on [s + kγ, t + kγ] for all s ≤ t, Lebesgue's dominated convergence theorem implies that

I h (t + kγ) -I h (s + kγ) -→ k→∞ I g (t) -I g (s)

for all s ≤ t ∈ [0, γ]. k→∞ P s,x [τ g > t],

Acknowledgement

I would like to thank the anonymous reviewers for their valuable and relevant comments and suggestions, as well as Oliver Kelsey Tough for reviewing a part of this paper.