João Saffran
email: joao.saffran@sga.pucminas.br

Gabriel Garcia
email: gabriel.garcia@sga.pucminas.br

Matheus A Souza

Pedro H Penna
email: pedro.penna@posgrad.ufsc.br

Márcio Castro
email: marcio.castro@ufsc.brlfwgoes

Luís F W Góes

Henrique C Freitas

A low-cost energy-efficient Raspberry Pi cluster for data mining algorithms

Keywords: Raspberry Pi cluster, Intel Xeon Phi, Apriori, K-Means

Data mining algorithms are essential tools to extract information from the increasing number of large datasets, also called Big Data. However, these algorithms demand huge amounts of computing power to achieve reliable results. Although conventional High Performance Computing (HPC) platforms can deliver such performance, they are commonly expensive and power-hungry. This paper presents a study of an unconventional low-cost energy-efficient HPC cluster composed of Raspberry Pi nodes. The performance, power and energy efficiency obtained from this unconventional platform is compared with a well-known coprocessor used in HPC (Intel Xeon Phi) for two data mining algorithms: Apriori and K-Means. The experimental results showed that the Raspberry Pi cluster can consume up to 88.35% and 85.17% less power than Intel Xeon Phi when running Apriori and K-Means, respectively, and up to 45.51% less energy when running Apriori.

Introduction

Petaflop computing relied on the advent of massively parallel architectures, such as vector processing units and manycore processors. For instance, Graphics Processing Units (GPUs) and the Intel Xeon Phi have been used extensively in current High Performance Computing (HPC) platforms, since they have proven to significantly increase the overall processing power of these platforms. However, several challenges still have to be overcome to reach the exascale computing era [START_REF] Lim | Technological forecasting of supercomputer development: The march to exascale computing[END_REF], [START_REF] Simon | Barriers to exascale computing[END_REF]. First, current cutting-edge parallel machines are power-hungry. This characteristic has motivated the community to seek for strategies to reduce energy consumption while delivering high performance [START_REF] Ashby | The opportunities and challenges of exascale computing[END_REF], [START_REF] Trefethen | Energy-aware software: Challenges, opportunities and strategies[END_REF]. Second, HPC platforms are expensive to obtain, which may be unaffordable for small-and medium-size institutes. In this context, the use of such architectures might be unworkable, but the demand for performance must not be disregarded.

One of the research domains that are gaining attention in the HPC community is Big Data, which is a term applied to data sets whose size or type is beyond the ability of relational databases to capture, manage, and process the data with low-latency. Big Data comes from sensors, devices, video/audio, networks, log files, transactional applications, web, and social media (much of it generated in real time and in a very large scale). To extract insights from unstructured data in a feasible time, Big Data applications usually exploit HPC platforms.

The already mentioned massively parallel architectures (e.g., GPUs and the Intel Xeon Phi) have been used to process Big Data applications with high performance. However, their expensive financial cost and recent issues related to energy consumption leads the scientific community to consider the use of low-cost and low-power architectures to build scalable machines. This way, high performance may be achieved with lower financial cost [START_REF] Cox | Iridis-pi: a low-cost, compact demonstration cluster[END_REF], [START_REF] Pfalzgraf | A low-cost computer cluster for high-performance computing education[END_REF].

In this paper, we study the use of an unconventional low-cost energy-efficient HPC cluster composed of eight Raspberry Pi boards, interconnected by a network switch, to verify whether it can be used as an alternative for HPC. Although these boards are not optimized for high performance, they can be considered as a good candidate to scalable systems with very low energy consumption.

Our main goal is to evaluate the performance, power and energy consumption of the Raspberry Pi cluster for two well-known data mining algorithms commonly used in Big Data applications: Apriori and K-Means. The results obtained with the Raspberry Pi cluster are compared with a well-known coprocessor used extensively in HPC (Intel Xeon Phi). Our results show that the Raspberry Pi cluster can achieve better energy efficiency than Intel Xeon Phi, consuming up to 45.51% less energy than Intel Xeon Phi when running Apriori and K-Means kernels.

The remainder of this paper is organized as follows. Section 2 presents an overview of the related work. In Section 3 we briefly present the architectures used in this work. In Section 4 we describe the applications we have chosen and explain their implementations. Section 5 presents our experimental methodology and the evaluation of the observed results. Finally, in Section 6, we present our conclusions and suggestions for future research.

Motivation and Related Work

As aforementioned, the energy efficiency of parallel computers is an obstacle to be overcome on the way to the exascale era. Future HPC computers will be limited to about 20 MW of power consumption in the coming decade [START_REF] Ashby | The opportunities and challenges of exascale computing[END_REF], [START_REF] Villa | Scaling the power wall: a path to exascale[END_REF]. Thus, energy efficiency is a relevant aspect to be addressed.

Recently, many high performance architectures have been designed driven by low-power consumption constraints. Kruger et al. [START_REF] Kruger | Building a Parallella board cluster[END_REF] proposed a cluster of lowcost Parallella boards, each one featuring a 16-core Epiphany RISC System-on-Chip (SoC) and a dual-core ARM Cortex-A9. They concluded that the cluster achieved better performance than an Intel i5-3570. However, Parallella lacks hardware for complex arithmetic operations, which can degrade the performance of few specific applications. Although they mention the importance of energy efficiency, the work did not present a rigorous power consumption analysis.

Similarly to our work, d'Amore et al. [START_REF] Amore | A practical approach to big data in tourism: A low cost raspberry pi cluster[END_REF] proposed the use of Raspberry Pi boards. They built a cluster of six Raspberry Pi boards to evaluate the performance of Big Data applications. The authors concluded that the cluster is an affordable solution to their problem, but the work only focused on how the data can be retrieved and used. However, the authors did not conduct any power/energy or quantitative performance evaluation.

In order to analyze the power efficiency of different processors, Aroca et al.

[2] compared Intel, AMD and ARM processors. They used five different processors in their analysis, running web servers, database servers and serial applications from the Linpack benchmark. They concluded that devices with low-power characteristics (e.g., the Intel Atom and ARM ones) are good candidates to compose the infrastructure of data centers. On the other hand, aspects of HPC systems were not considered, such as parallel applications that are mainly CPU-bound.

Other initiatives are also worried about these issues related to energy consumption. ARM processors attempt to be a suitable alternative for low-power consumption in HPC. This is the proposal of the Mont-Blanc and Mont-Blanc2 projects [START_REF] Rajovic | Supercomputing with commodity cpus: Are mobile socs ready for hpc?[END_REF], which intend to design a new computer architecture to establish HPC standards based on energy-efficient platforms. Similarly, the Glasgow Raspberry Pi Cloud (PiCloud) [START_REF] Tso | The glasgow raspberry pi cloud: A scale model for cloud computing infrastructures[END_REF] is a cluster of Raspberry Pi devices designed to be a scale model of a cloud computing platform, in order to address cloud simulations and applications.

Both projects proved to be energy efficient in their proposals, which reinforces the applicability of our work, that evaluates the behavior of Big Data application kernels in parallel architectures, as well as in [START_REF] Amore | A practical approach to big data in tourism: A low cost raspberry pi cluster[END_REF]. We compare the use of a cluster of low-cost and low-power platforms with a coprocessor designed specifically for HPC (Intel Xeon Phi). Furthermore, we measure the power and energy consumption of both architectures when running two data mining algorithms used in the Big Data research domain.

Experimental Platforms

In this section we detail the platforms that we considered in this work, highlighting their main features.

Intel Xeon Phi

The Intel Xeon Phi is a 64-bit coprocessor that primarily targets HPC workloads. This coprocessor features Intel's Many Integrated Core (MIC) architecture and provides full compatibility with the x86-64 Instruction Set Architecture (ISA)3 . An overview of the Intel Xeon Phi is presented in Figure 1. It features 61 dual-issue, out-of-order processing cores with 4-way simultaneous multithreading, enabling up to 244 working threads. Cores are interconnected by a bidirectional ring topology. Each core has 32 kB instruction and 32 kB data L1 caches and 256 kB of L2 cache, which is the last level of cache in the architecture. All caches are private and coherent. With regard to power consumption, this coprocessor has a Thermal Design Power (TDP) of 300 W . The Intel Xeon Phi coprocessor is usually connected to a host machine through a Peripheral Component Interconnect Express (PCIe) bus. A single Intel Xeon Phi board may have up to 16 GB of memory.

Raspberry Pi Cluster

The Raspberry Pi is a low-cost general purpose SoC. It has a quad-core 64-bit processor based on the ARMv7 architecture. In our work, we used a Raspberry Pi 2, which features a quad-core Cortex-A7 CPU running at 900 MHz. Each of the four cores has 64 kB instruction and 64 kB data L1 caches and 512 kB of L2 cache shared among all cores. The L2 is the last level cache in Raspberry Pi 2. The total amount of memory is 1 GB.

This type of SoC is not usually used in HPC due to its low performance. However, some characteristics make it much more energy efficient, e.g., the absence of peripheral controllers and hardware support for complex arithmetic operations. Hence, given the possibility of scaling this type of architectures, the overall energy efficiency of a larger system could be improved while achieving decent performance. The Raspberry has all components that conventional computers have, thus, it can be connected to a local area network using an Ethernet interface. Many Raspberry units can be clustered, conceiving a low-power and low-cost system with high potential to achieve high performance [START_REF] Cox | Iridis-pi: a low-cost, compact demonstration cluster[END_REF], [START_REF] Pfalzgraf | A low-cost computer cluster for high-performance computing education[END_REF].

A schematic diagram of our Raspberry Pi cluster is depicted in Figure 2. In our work, we opted to group eight Raspberry Pi 2 devices in order to conceive the cluster. The devices were interconnected by the means of a layer 3 switch, constituting a local network.

Data Mining Algorithms

To conduct our work, we chose two data mining algorithms used in the Big Data domain. These algorithms play an important role in different fields, including pattern recognition, image analysis and bioinformatics [START_REF] Xu | Survey of clustering algorithms[END_REF]. In this section, we detail these algorithms and their parallel versions.

Association Rule Learning

Association rule learning is a very common method used to discover relations between variables in large databases. Apriori is a state-of-the-art association rule machine-learning technique used for frequent itemset mining [START_REF] Agrawal | Fast algorithms for mining association rules in large databases[END_REF]. Given a list of itemsets, it identifies association rules between those items based on their frequency. These rules reveal subsets of items that frequently occur together in the same itemsets. The algorithm is driven by the following rule: all nonempty frequent itemsets must be also frequent. This rule allows the algorithm to eliminate all itemsets that are not composed of frequent item subsets, reducing significantly the search space. The Apriori algorithm works as follows. For each association rule A → B, where A and B are subsets of items of a frequent itemset, the Apriori algorithm calculates its confidence, presented in Equation 1. High confidence levels mean that most of the time an itemset A is present in a frequent itemset, the itemset B is also there.

Conf(A, B) = support(A ∧ B) support(A) (1)
In this paper, the multi-threaded version of the algorithm follows a Map-Reduce parallel pattern. After identifying the itemsets of size K = 1, items from S are distributed among the threads (Map stage). Each thread then counts the occurrences of its subset of items e i ∈ S. With all frequencies calculated, the subsets are regrouped (Reduce stage). Thus, the itemsets that do not meet the minimum support (i.e. a threshold used to eliminate infrequent itemsets) are removed, and then the confidence is calculated to form the association rules. This steps are repeated incrementing K until the subsets of size K are empty.

In the distributed version, the itemsets are assigned first to the nodes instead of threads. Each node then runs a multi-threaded version of the algorithm, exactly as described before.

K-Means Clustering

The K-Means clustering is a clustering approach widely used and studied [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF]. Formally, the K-Means clustering problem can be defined as follows. Given a set of n points in a real d-dimensional space, the problem is to partition these n points into k partitions, so as to minimize the mean squared distance from each point to the center of the partition it belongs to. Several heuristics have been proposed to address the K-Means clustering problem [START_REF] Jain | Algorithms for Clustering Data[END_REF], [START_REF] Kaufman | Finding groups in data: an introduction to cluster analysis[END_REF]. We opted to use the Lloyd's algorithm [START_REF] Kanungo | An efficient k-means clustering algorithm: analysis and implementation[END_REF], which is based on an iterative strategy that finds a locally minimum solution for the problem. The minimum Euclidean distance between partitions and centroids is used to cluster the data points. The algorithm takes as input the set of data points, the number of partitions k, and the minimum accepted distance between each point and the centroids.

The K-Means algorithm works as follows. First, data points are evenly and randomly distributed among the k partitions, and the initial centroids are computed. Then, data points are re-clustered into partitions taking into account the minimum Euclidean distance between them and the centroids. The centroid of each partition is recalculated taking the mean of all points in the partition. The whole procedure is repeated until no centroid is changed and every point is farther than the minimum accepted distance.

The multi-threaded version of this algorithm takes an additional parameter t, that is the total number of threads. An unique range of points and partitions is assigned to each thread, with two processing phases, A and B. In phase A, each thread re-clusters its own range of points into the k partitions. In phase B, each thread works in its own range of partitions, in order to recalculate centroids.

The distributed K-Means algorithm takes the number of available nodes that, by themselves, spawn working threads. The strategy employed in this algorithm is to first distribute the data points and replicate the data centroids among the available nodes, and then to loop over a two-phase iteration. In the first phase, partitions are populated, as in the multi-threaded algorithm, and in the second phase, data centroids are recalculated. For this recalculation, first each node uses its local data points to compute partial centroids, i.e., a partial sum of data points and population within a partition. Next, nodes exchange partial centroids so that each peer ends up with the partial centroids of the same partitions. Finally, nodes compute their local centroids and broadcast them.

In this section, we first describe the experimental setup and discuss important implementation details for each platform. Then, we present the results obtained with the data mining algorithms on both platforms.

Setup and Implementation Details

We used the multi-threaded implementations of both algorithms to carry out the experiments on the Intel Xeon Phi, since the memory is shared among all cores in this processor. More precisely, we parallelized the algorithms using the OpenMP programming model and we compiled them with Intel C/C++ Compiler (icc) version 16.0.1.

The Intel's MIC System Management and Configuration (MICSMC) tool was used to monitor the processor's power consumption. Power measurements obtained from MICSMC are very accurate as shown in [START_REF] Lawson | Energy evaluation for applications with different thread affinities on the Intel Xeon Phi[END_REF]. Intel Xeon Phi can be either used in offload or native modes. In the former mode, the main application code is executed on the host and performance-critical sections of the application are offloaded to the coprocessor. In the latter mode, the entire application is executed on the coprocessor. In this paper, we chose the latter since we intend to compare the performance and energy consumption of the coprocessor only.

For the Raspberry Pi cluster, on the other hand, we adopted a hybrid programming model: we used the OpenMPI library to implement the distributed version of the applications (i.e., to make use of all nodes available in the Raspberry Pi cluster) and OpenMP inside each node to exploit four cores available in it. We compiled both applications with mpicc version 1.4.5 with GNU C Compiler (GCC) version 4.6.3. To measure the energy consumption of this cluster, we used a watt-meter instrument connected before the power supply of the devices, except the network switch, taking the total consumption in kilowatts-hour (kWh) of the whole system. The Equation 2 was used to convert our results to Joules (J).

E (J) = 1000 × 3600 × E (kW h) (2)
The power consumption was calculated for the Raspberry Pi cluster, as well as the energy consumption, for the Intel Xeon Phi. To calculate these values, we used Equation 3.

E (J) = P (W ) × t (s) (3)
We ran each application varying the number of available nodes or threads, depending on the target platform. For the Raspberry Pi cluster, we used four threads per cluster and varied the number of nodes in 4 (12.5% of the total number of cores), 8 (25% of the total number of cores), 16 (50% of the total number of cores) and 32 (all available cores). Similarly, we varied the number of threads proportionally to the number of cores available in the Intel Xeon Phi, #threads 0 100 200 300 400 500 600 700 Execution Time (s) i.e., 30, 60, 120 and 240 threads. This allows us to compare both platforms when the same percentages of the overall resources available in each platform are used. Three workload sizes were defined for each algorithm, to evaluate the behavior of both systems when the workload increases. For the Apriori algorithm, the increase on the workload size can be achieved by reducing the minimum support. We used the following minimum support values: 70 for the standard workload, 60 for large and 50 for huge. For K-Means, we increased the number of data points to be clustered. We used the following number of data points: 2 14 for the standard workload, 2 15 for large and 2 16 for the huge one.

Finally, we performed 10 runs for each configuration (number of threads/nodes and workload sizes) and computed the average execution time and power. The maximum standard deviation observed in Raspberry Pi cluster executions was 11.04%, while the Intel Xeon Phi presented at most 7.07%.

Evaluation

We used three metrics to compare the platforms: execution time, power consumption, and energy consumption. Figure 3 presents the execution time of the algorithms when executed on both architectures. As it can be observed, Apriori proved to be more scalable than K-Means. It was possible to reduce the execution time by 82.05% with the Apriori when changing from 1 node to 8 nodes in the Raspberry Pi cluster, while in K-Means with same configurations the reduction was about 74.97%. This is due to the fact that the Apriori algorithm has more independent work units than the K-Means, which leads to less synchronization when parallel work finishes.

Comparing the Raspberry Pi cluster and Intel Xeon Phi execution time, the Intel Xeon Phi presented better results. It is an architecture with more processing power, featuring a larger thread support (up to 240), thus it was an expected behavior. However, the Intel Xeon Phi presented poorer scalability than the Raspberry Pi cluster for both algorithms. For instance, the maximum execution time reduction, starting from 30 threads, was 73.85% when running K-Means with the full architecture (240 threads). The communication in the Raspberry Pi cluster, although done by the means of a local area network, is not surpassed by the synchronization time spent by the 240 threads on the Intel Xeon Phi, in the case of Apriori. Considering K-Means, it presents a similar behavior when increasing the resources of the architectures. On the other hand, with this application, the increase in the workload has run better in the Intel Xeon Phi. It is worth noting that the multi-threaded version of the codes are the same for both architectures, without specific optimizations. Thus, despite the Intel Xeon Phi code could be improved, the Raspberry Pi cluster presents better efficiency with a lower number of threads (32 at full use). Figure 4 presents the observed power consumption in our experiment. The applications are naturally unbalanced, thus, the power consumption varies depending on the execution time spent by each work unit, which are irregular (the cores and nodes present different times to solve their own work unit). With the Apriori algorithm, changing the workload size results in less variation in power consumption than when running the K-Means algorithm. Usually, with Apriori, the power consumption reduces when increasing the workload size. With respect to K-Means running in the Raspberry Pi cluster, the power consumption reduces when the workload size is increased, but with the Intel Xeon Phi the opposite occurs. In the Raspberry Pi cluster, each node has its own slave process and, when it finishes its computation, the power consumption of the entire node is drastically reduced while other nodes are still active.

We noticed a much more significant variation in power consumption on the Raspberry Pi cluster than on the Intel Xeon Phi. This is due to the fact that when less cores are used on the Intel Xeon Phi, the coprocessor idle cores keep consuming a portion of the energy, since they are in the same device. On the Raspberry Pi cluster, there is less impact from this fact, since the devices are completely independent. Overall, the Raspberry Pi cluster is superior when compared to Intel Xeon Phi. power consumption with Apriori when using the Raspberry Pi cluster over the Intel Xeon Phi. In the same way, for K-Means, the biggest reduction was 85.17%. Figure 5 presents the energy consumption results, which were obtained by multiplying the average power by the execution time. As it can be observed, the energy consumption increases as we increase the workload size. This is due the increase in the execution time that the algorithms spent to reach a solution.

Another observation concerns the energy consumed when varying the number of threads/nodes. We observed a significant reduction in energy consumption when more threads are used. This can be explained by the fact that if more resources are used, the power consumption increases, but the time to solution tends to decrease, due to the increase in the computational power. Thus, since the power consumption is less determinant than the execution time in our experiment, the energy consumption decreases when more resources are used.

In summary, the Raspberry Pi cluster proved to be more energy efficient than the Intel Xeon Phi for Apriori, although the opposite occurs with K-Means. This is due to the higher execution time difference for K-Means, since as more time is spent running an application more energy is consumed during this time. The Apriori algorithm was less energy-efficient in the Raspberry Pi cluster when using a single node in comparison with 30 threads from the Intel Xeon Phi, however, when more resources are employed proportionally, the Raspberry Pi cluster starts to be more energy-efficient, consuming up to 45.51% less energy than the Intel Xeon Phi. With respect to the financial costs, as mentioned before, the Raspberry Pi cluster is about ten times cheaper than the Intel Xeon Phi, thus presenting better price-performance ratio (i.e., cost-benefit).

Concluding Remarks

In this paper we evaluated the performance, power and energy consumption of an unconventional low-cost energy-efficient HPC cluster composed of Raspberry Pi nodes when running two well-known data mining algorithms used in Big Data (Apriori and K-Means). The results obtained on this cluster were compared to a coprocessor widely adopted in the HPC domain (Intel Xeon Phi). Our results showed that the Raspberry Pi cluster achieved a better tradeoff between execution time and power consumption for the Apriori kernel. On the other hand, the Intel Xeon Phi presented better performance on K-Means.

As future work, we propose to apply load balancing strategies on both applications to improve their performances. Moreover, we intend to implement parallel versions of these applications for Graphical Processor Units (GPUs), and use more HPC devices, for instance, a cluster of Xeon Phi boards. This would allow us to compare this architecture with the ones used in our work. Finally, we also intend to study the impacts on the energy efficiency and performance of the Raspberry Pi cluster when running application kernels from other domains, such as image processing and computational fluid dynamics.

Acknowledgement

This work was partially supported by FAPEMIG, FAPESC, CAPES, CNPq and STIC-AmSud and was developed in the context of EnergySFE and ExaSE cooperation projects, also supported by FAPERGS and INRIA.

Fig. 1 .

 1 Fig. 1. Overview of the Intel Xeon Phi coprocessor.

Fig. 2 .

 2 Fig. 2. The Raspberry Pi cluster (left) and a overview of the ARM Cortex-A7 (right).

Fig. 3 .

 3 Fig. 3. Execution time: (a) Raspberry Pi cluster -Apriori, (b) Raspberry Pi cluster -K-Means, (c) Intel Xeon Phi -Apriori (d), Intel Xeon Phi -K-Means.

Fig. 4 .

 4 Fig. 4. Power consumption: (a) Raspberry Pi cluster -Apriori, (b) Raspberry Pi cluster -K-Means, (c) Intel Xeon Phi -Apriori (d), Intel Xeon Phi -K-Means.

Fig. 5 .

 5 Fig. 5. Energy consumption: (a) Raspberry Pi cluster -Apriori, (b) Raspberry Pi cluster -K-Means, (c) Intel Xeon Phi -Apriori (d), Intel Xeon Phi -K-Means

Intel markets this coprocessor with several different specifications. In this work we refer as Intel Xeon Phi to the one codenamed Knight's Corner.


Henri Elad Altman
email: henri.eladaltman@upmc.fr

Bismut-Elworthy-Li formulae for Bessel processes

In this article we are interested in the differentiability property of the Markovian semigroup corresponding to the Bessel processes of nonnegative dimension. More precisely, for all δ ≥ 0 and T > 0, we compute the derivative of the function x → P δ T F (x), where (P δ t ) t≥0 is the transition semi-group associated to the δ -dimensional Bessel process, and F is any bounded Borel function on R + . The obtained expression shows a nice interplay between the transition semi-groups of the δ -and the (δ + 2)-dimensional Bessel processes. As a consequence, we deduce that the Bessel processes satisfy the strong Feller property, with a continuity modulus which is independent of the dimension. Moreover, we provide a probabilistic interpretation of this expression as a Bismut-Elworthy-Li formula.

Introduction

Bessel processes are a one-parameter family of nonnegative diffusion processes with a singular drift, which present a reflecting behavior when they hit the origin. The smaller the parameter (called dimension), the more intense the reflection. Hence, studying the dynamics of these processes is a non-trivial problem, especially when the dimension is small. Despite these apparent difficulties, Bessel processes have remarkably nice properties. Therefore they provide an instructive insight in the study of stochastic differential equations (SDEs) with a singular drift, as well as the study of reflected SDEs.

For all x ≥ 0 and δ ≥ 0, the squared Bessel process of dimension δ started at x 2 is the unique strong solution of the equation:

X t = x 2 + 2 t 0 X s dB s + δt. (1)
Such a process X is nonnegative, and the law of its square-root ρ = √ X is, by definition, the δ-dimensional Bessel process started at x (see [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], section XI, or Chapter 3 of [START_REF] Zambotti | Random Obstacle Problems, École d' Été de Probabilités de Saint-Flour XLV -2015[END_REF] for

d dx P δ T F (x) = x T P δ+2 T F (x) -P δ T F (x) . (2)
As a consequence, the Bessel processes satisfy the strong Feller property uniformly in δ. In a second part, we interpret the above result probabilistically as a Bismut-Elworthy-Li formula. More precisely, given a realization ρ of the Bessel process through the SDE (1), we introduce the derivative η t of ρ t with respect to the initial condition x, and show that when δ > 0, the stochastic integral t 0 η s dB s is well-defined as an L p martingale, for some p > 1 depending on δ. Moreover, it turns out that T 0 η s dB s is (up to a constant) the Radon-Nikodym derivative of the (δ + 2)-dimensional Bessel process over the interval [0, T ] w.r.t. the δ-dimensional one. As a consequence, we deduce that the above equation can be rewritten:

d dx P δ T F (x) = 1 T E F (ρ t (x)) t 0 η s (x)dB s (3)
which is an apparition, in an unexpected context, of the well-known Bismut-Elworthy-Li formula (see [START_REF] Elworthy | Formulae for the derivatives of heat semi-groups[END_REF] for a precise statement and proof of the Bismut-Elworthy-Li formula in the case of diffusions with smooth coefficients). One surprising feature is that, while (2) is very easy to prove whatever the value of δ ≥ 0, on the other hand, the process ( t 0 η s (x)dB s ) t≥0 has less and less finite moments as δ decreases, which makes the proof of (3) more involved for small δ. In particular, this process is not in L 2 for δ < 2( √ 2 -1), and when δ = 0, we do not even know whether the stochastic integral t 0 η s (x)dB s is well-defined as a local martingale. This article was originally motivated by the hope to prove the strong Feller property for some singular reflected SDEs or SPDEs. Recently, several works have brought about new techniques to prove the strong Feller property for singular SPDEs. Thus, in [START_REF] Tsatsoulis | Spectral Gap for the Stochastic Quantization Equation on the 2-dimensional Torus[END_REF], the authors established this property for the P (Φ) 2 equation, and in [START_REF] Hairer | The strong Feller property for singular stochastic PDEs[END_REF], the authors established it for a large class of singular semilinear SPDEs. The fact, mentioned above, that blowup of η does not affect the strong Feller property of Bessel processes is reminiscent of the latter article, where the setting used to prove the strong Feller property allows blowup in finite time of the solution. Also, we hope that the techniques used in the present article might give inspiration to treat more general cases. Note that, even in the present context, where many computations can be performed explicitly, we still have an open problem concerning the Strong Feller bounds for Bessel processes of dimension δ ≤ 2( √ 2 -1) (see Remark 14 below).

The plan of our paper is as follows. In Section 2 we recall the classical Bismut-Elworthy-Li formula for diffusions in R with a dissipative drift, and show how this implies the strong Feller property. In Section 3 we recall the definition of Bessel processes and their basic properties. In Section 4 we compute the derivative of the Bessel semi-group. In section 5 we establish the differentiability of the Bessel flow at any given point in R * + , and we give an expression for (some modification of) the derivative. In Section 6, we show that this derivative is not bounded in time when δ < 1. We prove, however, that it is linked to an interesting martingale corresponding to the family of Radon-Nikodym derivatives of the (δ + 2)-dimensional Bessel process w.r.t.the δ-dimensional one. In Section 7 we prove the Bismut-Elworthy-Li formula for the Bessel processes of dimension δ > 0 .

Classical Bismut-Elworthy-Li formula for onedimensional diffusions

In this section we recall very briefly the Bismut-Elworthy-Li fomula in the case of onedimensional diffusions, and the way this formula implies the strong Feller property.

Consider an SDE on R of the form :

dX t = b(X t )dt + dB t , X 0 = x (4)
where b : R → R is smooth and satisfies:

|b(x) -b(y)| ≤ C|x -y|, x, y ∈ R b ′ (x) ≤ L, x ∈ R (5)
where C > 0, L ∈ R are some constants. By the classical theory of SDEs, for all x ∈ R, there exists a unique continuous, square-integrable process (X t (x)) t≥0 satisfying (4). Actually, by the Lipschitz assumption on b, there even exists a bi-continuous process (X t (x)) t≥0,x∈R such that, for all x ∈ R, (X t (x)) t≥0 solves (4). Let x ∈ R. Consider the solution (η t (x)) t≥0 to the variation equation obtained by formally differentiating (4) with respect to x:

dη t (x) = b ′ (X t )η t (x)dt, η 0 (x) = 1
Note that this is a (random) linear ODE with explicit solution given by:

η t (x) = exp t 0 b ′ (X s )ds
It is easy to prove that, for all t ≥ 0 and x ∈ R, the map y → X t (y) is a.s. differentiable at x and:

dX t dx a.s. = η t (x) (6)
Remark 1. Note that η t (x) > 0 for all t ≥ 0 and x ∈ R. This reflects the fact that, for all x ≤ y, by a comparison theorem for SDEs (see Theorem 3.7 in Chapter IX in [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]), one has X t (x) ≤ X t (y) .

Recall that a Markovian semi-group (P t ) t≥0 on a Polish space E is said to satisfy the strong Feller property if, for all t > 0 and ϕ : E → R bounded and Borel, the function P t ϕ : E → R defined by:

P t ϕ(x) = ϕ(y)P t (x, dy), x ∈ R is continuous.
The strong Feller property is very useful in the study of SDEs and SPDEs, namely for the proof of ergodicity (see, e.g., the monographs [START_REF] Cerrai | Second Order PDE's in Finite and Infinite Dimensions[END_REF], [START_REF] Da Prato | Ergodicity for Infinite-Dimensional Systems[END_REF] and [START_REF] Zambotti | Random Obstacle Problems, École d' Été de Probabilités de Saint-Flour XLV -2015[END_REF], as well as the recent articles [START_REF] Hairer | The strong Feller property for singular stochastic PDEs[END_REF] and [START_REF] Tsatsoulis | Spectral Gap for the Stochastic Quantization Equation on the 2-dimensional Torus[END_REF], for applications of the strong Feller property in the context of SPDEs).

Let (P t ) t≥0 be the Markovian semi-group associated to the SDE [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]. We are interested in proving the strong Feller property for (P t ) t≥0 . Note that, by assumption (5), η t (x) ≤ e Lt for all t ≥ 0 and x ∈ R. Therefore, by [START_REF] Deuschel | Zambotti : Bismut -Elworthy 's formula and random walk representation for SDEs with reflection[END_REF] and the dominated convergence theorem, for all ϕ : R → R differentiable with a bounded derivative, one has:

d dx (P t ϕ) (x) = d dx E [ϕ(X t (x))] = E [ϕ(X t (x))η t (x)]
As a consequence, for all t ≥ 0, P t preserves the space C 1 b (R) of bounded, continuously differentiable functions on R with a bounded derivative. It turns out that, actually, for all t > 0, P t maps the space C b (R) of bounded and continuous functions into C 1 b (R). This is a consequence of the following, nowadays well-known, result: Theorem 1 (Bismut-Elworthy-Li formula). For all T > 0 and ϕ ∈ C b (R), the function P T ϕ is differentiable and we have:

d dx P T ϕ(x) = 1 T E ϕ(X T (x)) T 0 η s (x)dB s (7)
Proof. See [START_REF] Elworthy | Formulae for the derivatives of heat semi-groups[END_REF], Theorem 2.1, or [START_REF] Zambotti | Random Obstacle Problems, École d' Été de Probabilités de Saint-Flour XLV -2015[END_REF], Lemma 5.17 for a proof.

Corollary 1. The semi-group (P t ) t≥0 satisfies the strong Feller property and, for all T > 0 and ϕ : R → R bounded and Borel, one has:

∀x, y ∈ R, |P T ϕ(x) -P T ϕ(y)| ≤ e L ||ϕ|| ∞ √ T ∧ 1 |x -y|, (8)
where || • || ∞ denotes the supremum norm.

The following remark is crucial.

Remark 2. Inequality (8) involves only the dissipativity constant L, not the Lipschitz constant C. This makes the Bismut-Elworthy-Li formula very useful in the study of SPDEs with a dissipative drift.

Proof of Corollary 1. By approximation, it suffices to prove (8) for ϕ ∈ C b (R). For such a ϕ and for all T > 0, by the Bismut-Elworthy-Li formula, one has:

d dx P T ϕ(x) ≤ ||ϕ|| ∞ T E T 0 η s (x)dB s
Remark that the process (η t (x)) t≥0 is locally bounded since it is dominated by (e Lt ) t≥0 , so that the stochastic integral

t 0 η s (x)dB s t≥0
is an L 2 martingale. Hence using Jensen's inequality as well as Itô's isometry formula, we obtain:

E | T 0 η s (x)dB s | ≤ E T 0 η s (x) 2 ds| ≤ T 0 e 2Ls ds
and the last quantity is bounded by √ e 2L T = e L √ T for all T ∈ (0, 1]. Therefore, we deduce that:

∀x ∈ R, d dx P T ϕ(x) ≤ e L ||ϕ|| ∞ √ T so that: ∀x, y ∈ R, |P T ϕ(x) -P T ϕ(y)| ≤ e L ||ϕ|| ∞ √ T |x -y|
for all ϕ ∈ C b (R) and T ∈ (0, 1]. The case T > 1 follows at once by using the semi-group property of (P t ) t≥0 :

|P T ϕ(x) -P T ϕ(y)| = |P 1 (P T -1 ϕ) (x) -P 1 (P T -1 ϕ) (y)| ≤ e L ||P T -1 ϕ|| ∞ √ 1 |x -y| ≤ e L ||ϕ|| ∞ |x -y|
The claim follows.

Remark 3 (A brief history of the Bismut-Elworthy-Li formula). A particular form of this formula had originally been derived by J.M. Bismut in [START_REF] Bismut | Large deviations and the Malliavin Calculus[END_REF] using Malliavin calculus in the framework of the study of the logarithmic derivative of the fundamental solution of the heat equation on a compact manifold. In [START_REF] Elworthy | Formulae for the derivatives of heat semi-groups[END_REF], K.D. Elworthy and X.-M. Li generalized this formula to a large class of diffusion processes on manifolds with smooth coefficients, and gave also variants of this formula to higher-order derivatives.

The key property allowing the analysis performed in this section is the dissipativity property [START_REF] Da Prato | Ergodicity for Infinite-Dimensional Systems[END_REF]. Without this property being true, one would not even expect the Bismut-Elworthy-Li formula to hold. However, in the sequel, we shall prove that results such as Theorem 1 and Corollary 1 above can also be obtained for a family of diffusions with a non-dissipative drift (informally L = +∞) , namely for the Bessel processes of dimension smaller than 1.

Bessel processes: notations and basic facts

In the sequel, for any subinterval I of R + , C(I) will denote the set of continuous functions I → R. We shall consider this set endowed with the topology of uniform convergence on compact sets, and will denote by B(C(I)) the corresponding Borel σ-algebra.

Consider the canonical measurable space (C(R + ), B(C(I))) endowed with the canonical filtration (F t ) t≥0 . Let (B t ) t≥0 be a standard linear (F t ) t≥0 -Brownian motion. For all x ≥ 0 and δ ≥ 0, there exists a unique continuous, predictable, nonnegative process (X δ t (x)) t≥0 satisfying:

X t = x 2 + 2 t 0 X s dB s + δt. (9)
(X δ t (x)) t≥0 is a squared Bessel process of dimension δ started at x 2 , and the process ρ δ t (x) := X δ t (x) is a δ-dimensional Bessel process started at x. In the sequel, we will also write the latter process as (ρ t (x)) t≥0 , or ρ, when there is no risk of ambiguity.

We recall the following monotonicity property of the family of Bessel processes:

Lemma 1. For all couples (δ, δ ′ ), (x, x ′ ) ∈ R + such that δ ≤ δ ′ and x ≤ x ′ , we have, a.s.:

∀t ≥ 0, ρ δ t (x) ≤ ρ δ ′ t (x ′ ). Proof. By Theorem (3.7) in [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Section IX, applied to the equation ( 9), the following property holds a.s.:

∀t ≥ 0, X δ t (x) ≤ X δ ′ t (x ′ ). Taking the square root on both sides above, we deduce the result.

For all a ≥ 0, let T a (x) denote the (F t ) t≥0 stopping time defined by: T a (x) := inf{t > 0, ρ t (x) ≤ a} (we shall also write T a ). We recall the following fact, (see e.g. Proposition 3.6 of [START_REF] Zambotti | Random Obstacle Problems, École d' Été de Probabilités de Saint-Flour XLV -2015[END_REF]): Proposition 1. The following dichotomy holds:

• T 0 (x) = +∞ a.s., if δ ≥ 2, • T 0 (x) < +∞ a.s., if 0 ≤ δ < 2.
Applying Itô's lemma to ρ t = X δ t (x), we see that ρ satisfies the following relation on the interval [0, T 0 ):

∀t ∈ [0, T 0 ), ρ t = x + δ -1 2 t 0 ds ρ s + B t . ( 10
)
4 Derivative in space of the Bessel semi-group

Let δ ≥ 0. We denote by P δ x the law, on (C(R + ), B(C(R + ))), of the δ-dimensional Bessel process started at x, and we write E δ

x for the corresponding expectation operator. We also denote by (P δ t ) t≥0 the family of transition kernels associated with the δ-dimensional Bessel process, defined by P δ t F (x) := E δ x (F (ρ t )) for all t ≥ 0 and all F : R + → R bounded and Borel. The aim of this section is to prove the following: Theorem 2. For all T > 0 and all F : R + → R bounded and Borel, the function x → P δ t F (x) is differentiable on R + , and for all x ≥ 0:

d dx P δ T F (x) = x T P δ+2 T F (x) -P δ T F (x) . ( 11
)
In particular, the function x → P δ t F (x) satisfies the Neumann boundary condition at 0:

d dx P δ T F (x) x=0 = 0.
Remark 4. By Theorem 2, the derivative of the function x → P δ T F (x) is a smooth function of P δ+2 T F (x) and P δ T F (x). Hence, reasoning by induction, we deduce that the function x → P δ T F (x) is actually smooth on R + . Proof. The proof we propose here relies on the explicit formula for the transition semigroup of the Bessel processes. We first treat the case δ > 0.

Given δ > 0, let ν := δ 2 -1, and denote by I ν the modified Bessel function of index ν. We have (see, e.g., Chap. XI.1 in [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]) :

P δ t F (x) = ∞ 0 p δ T (x, y)F (y)dy
where, for all y ≥ 0:

p δ t (x, y) = 1 T y x ν y exp - x 2 + y 2 2T I ν xy T , if x > 0, p δ t (0, y) = 2 -ν T -(ν+1) Γ(ν + 1) y 2ν+1 exp - y 2 2T
where Γ denotes the gamma function. By the power series expansion of the function I ν we have, for all x, y ≥ 0:

p δ T (x, y) = 1 T exp - x 2 + y 2 2T pδ T (x, y) (12)
with:

pδ T (x, y) := ∞ k=0 y 2k+2ν+1 x 2k (1/2T ) 2k+ν k! Γ(k + ν + 1) .
Note that pδ T (x, y) is the sum of a series with infinite radius of convergence in x, hence we can compute its derivative by differentiating under the sum. We have:

∂ ∂x pδ T (x, y) = ∂ ∂x ∞ k=0 x 2k y 2k+2ν+1 (1/2T ) 2k+ν k! Γ(k + ν + 1) = ∞ k=0 2k x 2k-1 y 2k+2ν+1 (1/2T ) 2k+ν k! Γ(k + ν + 1) = x T ∞ k=1 x 2k-2 y 2k+2ν+1 (1/2T ) 2k+ν-1 (k -1)! Γ(k + ν + 1) .
Hence, performing the change of variable j = k -1, and remarking that ν + 1 = δ+2 2 -1, we obtain:

∂ ∂x pδ T (x, y) = x T ∞ j=0 x 2(j+1)-2 y 2(j+1)+2ν+1 (1/2T ) 2(j+1)+ν-1 j! Γ((j + 1) + ν + 1) = x T ∞ j=0 x 2j y 2j+2(ν+1)+1 (1/2T ) 2j+(ν+1) j! Γ(j + (ν + 1) + 1) = x T pδ+2 T (x, y).
As a consequence, differentiating equality [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF] with respect to x, we obtain:

∂ ∂x p δ T (x, y) = - x T pδ T (x, y) + ∂ ∂x pδ T (x, y) 1 T exp - x 2 + y 2 2T = x T -p δ T (x, y) + p δ+2 T (x, y) .
Hence, we deduce that the function x → P δ T F (x) is differentiable, with a derivative given by [START_REF] Vostrikova | On regularity properties of Bessel flow[END_REF]. Now suppose that δ = 0. We have, for all x ≥ 0:

P 0 T F (x) = exp - x 2 2T F (0) + ∞ 0 p T (x, y)F (y)dy (13)
where, for all y ≥ 0:

p T (x, y) = 1 T exp - x 2 + y 2 2T pT (x, y) with: pT (x, y) := x I 1 xy T = ∞ k=0 x 2k+2 (y/2T ) 2k+1 k!(k + 1)! .
Here again, we can differentiate the sum term by term, so that, for all x, y ≥ 0:

∂ ∂x pT (x, y) = x T ∞ k=0 x 2k y 2k+1 (1/2T ) 2k k! 2 = x T p2 T (x, y).
Therefore, for all x, y ≥ 0, we have:

∂ ∂x p T (x, y) = - x T pT (x, y) + ∂ ∂x pT (x, y) 1 T exp - x 2 + y 2 2T = x T -p T (x, y) + p2 T (x, y) 1 T exp - x 2 + y 2 2T = x T -p T (x, y) + p 2 T (x, y)
Hence, differentiating (13) with respect to x, and using the dominated convergence theorem to differentiate inside the integral, we obtain:

∂ ∂x P 0 T F (x) = - x T exp(- x 2 2T )F (0) + x T ∞ 0 -p T (x, y) + p 2 T (x, y) F (y)dy = x T -P 0 T F (x) + P 2 T F (x) ,
which yields the claim.

Remark 5. Formula (11) can also be derived using the Laplace transform of the onedimensional marginals of the squared Bessel processes. Indeed, denote by (Q δ t ) t≥0 the family of transition kernels of the δ-dimensional squared Bessel process. Then for all δ ≥ 0, x ≥ 0, T > 0, and all function f of the form f (x) = exp(-λx) with λ ≥ 0, one has:

Q δ T f (x) = exp - λx 1 + 2λT (1 + 2λT ) -δ/2
(see [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Chapter XI, Cor. (1.3)). For such test functions f , we check at once that the following equality holds:

d dx Q δ T f (x) = 1 2T Q δ+2 T f (x) -Q δ T f (x) .
By linearity and by the Stone-Weierstrass theorem, we deduce that this equality holds for all bounded, continuous functions f . Then an approximation argument enables to deduce the equality for all functions f : R + → R Borel and bounded. Finally, remarking that for all bounded Borel function F on R + we have

P δ T F (x) = Q δ T f (x 2 )
with f (x) := F ( √ x), we deduce that:

d dx P δ T F (x) = 2x d dx (Q δ T f )(x 2 ) = x T Q δ+2 T f (x 2 ) -Q δ T f (x 2 ) = x T P δ+2 T F (x) -P δ T F (x)
which yields the equality [START_REF] Vostrikova | On regularity properties of Bessel flow[END_REF].

Corollary 2. The semi-group (P δ t ) t≥0 has the strong Feller property. More precisely, for all T > 0, R > 0, x, y ∈ [0, R] and F : R + → R bounded and Borel, we have:

|P δ T F (x) -P δ T F (y)| ≤ 2R||F || ∞ T |y -x|. ( 14
)
Proof. By Theorem 2, for all x, y ∈ [0, R] such that x ≤ y, we have:

|P δ t F (x) -P δ t F (y)| = y x u T P δ+2 T F (u) -P δ T F (u) du ≤ 2||F || ∞ T y x u du ≤ 2R||F || ∞ T |y -x|.
Remark 6. The bound ( 14) is in 1/T , which is not very satisfactory for T small. However, in the sequel, we will improve this bound by getting a better exponent on T , at least for δ ≥ 2( √ 2 -1) (see inequality (28) below).

Differentiability of the flow

In the following, we are interested in finding a probabilistic interpretation of Thm 2, in terms of the Bismut-Elworthy-Li formula. To do so we study, for all δ ≥ 0, and all couple (t, x) ∈ R + × R * + , the differentiability at x of the function:

ρ t : R + → R + y → ρ δ t (y).
In this endeavour, we first need to choose an appropriate modification of the process (ρ t (x)) t≥0,x>0 . We have the following result:

Proposition 2. Let δ ≥ 0 be fixed. There exists a modification (ρ δ t (x)) x,t≥0 of the process (ρ δ t (x)) x,t≥0 such that, a.s., for all x, x ′ ∈ R + with x ≤ x ′ , we have:

∀t ≥ 0, ρδ t (x) ≤ ρδ t (x ′ ). ( 15
)
Proof. For all q, q ′ ∈ Q + , such that q ≤ q ′ , by Lemma 1, the following property holds a.s.:

∀t ≥ 0, ρ δ t (q) ≤ ρ δ t (q ′ ).
For all x ∈ R + , we define the process ρδ (x) by:

∀t ≥ 0, ρδ t (x) := inf q∈Q + ,q≥x ρ δ t (q).
Then (ρ δ t (x)) x,t≥0 yields a modification of the process (ρ δ t (x)) x,t≥0 with the requested property.

Remark 7. We may not have, almost-surely, joint continuity of all the functions t → ρt (x), x ≥ 0. Note however that, by definition, for all x ≥ 0, x ∈ Q, we have a.s.:

∀t ≥ 0, ρt (x) = ρ t (x),
so that, a.s., t → ρt (x) is continuous and satisfies:

∀t ∈ [0, T 0 (x)), ρt (x) = x + δ -1 2 t 0 ds ρs + B t .
As a consequence, by countability of Q, there exists an almost sure event A ∈ F on which, for all x ∈ Q + , the function t → ρt (x) is continuous and satisfies:

∀t ∈ [0, T 0 (x)), ρt (x) = x + δ -1 2 t 0 ds ρs + B t .
In this section, as well as the Appendix, we always work with the modification ρ. Similarly, we work with : T0 (x) := inf{t > 0, ρδ t (x) = 0} instead of T 0 (x), for all δ, x ≥ 0. We will write again ρ and T 0 instead of ρ and T0 . Note that, a.s., the function x → T 0 (x) is non-decreasing on R + . Proposition 3. Let δ ≥ 0, t > 0 and x > 0. Then, a.s., the function ρ t is differentiable at x, and its derivative there is given by:

dρ t (y) dy y=x a.s. = η t (x) := 1 t<T 0 (x) exp 1 -δ 2 t 0 ds ρ s (x) 2 . ( 16
)
The proof of this proposition is quite technical. Since, moreover, the result will not be necessary in the sequel, we prefer to postpone the proof to the Appendix of the article. Remark 8. In particular, when δ = 1, the above formula reduces to:

dρ t (y) dy y=x a.s. = 1 t<T 0 (x) (17)
a formula which was already well-known (see e.g. [START_REF] Arnaudon | Reflected brownian motion: selection, approximation and linearization[END_REF], Lemma A.1).

Remark 9. Note that the indicator function 1 t<T 0 (x) in the right-hand side of ( 16) is related to the behavior of the Bessel process at the boundary 0. It is reminiscent of Theorem 1 in [START_REF] Deuschel | Zambotti : Bismut -Elworthy 's formula and random walk representation for SDEs with reflection[END_REF] , where a similar indicator function appears in the expression of the spatial derivative of the flow of vector-valued solutions to SDEs with reflection.

Remark 10. Proposition 3 shows that, for all t, x > 0, the function ρ t is almost-surely differentiable at x. We may, however, ask if, a.s., the function ρ t is differentiable on the whole of R * + . The case where δ > 1 was treated in detail in [START_REF] Vostrikova | On regularity properties of Bessel flow[END_REF], where it was shown that, a.s., for all t ≥ 0 the function x → ρ t (x) is differentiable on R * + , and that the derivative

dρt(x) dx is continuous in (t, x) ∈ R + × R * + .
However, as δ gets smaller than 1, the regularity of the process (ρ t (x)) t≥0,x>0 becomes much worse. Note that δ = 1 corresponds to the case of the flow of reflected Brownian motion on the half-line; in that case the flow is no longer continuously differentiable as suggested by (17). Many works have been carried out on the study of the flow of reflected Brownian motion on domains in higher dimension (see e.g. [START_REF] Burdzy | Differentiability of stochastic flow of reflected Brownian motions[END_REF] and [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF]) or on manifolds with boundary (see e.g. [START_REF] Arnaudon | Reflected brownian motion: selection, approximation and linearization[END_REF]). By contrast, the regularity of Bessel flows of dimension δ < 1 seems to be a very open problem.

In the remainder of the article, however, we shall not need any regularity results on the Bessel flow. Instead, for all fixed x > 0, we shall study the process (η t (x)) t≥0 defined above as a process in itself.

Properties of η

In the sequel, for all x ≥ 0, we shall consider the process (η t (x)) t≥0 defined as above:

η t (x) := 1 t<T 0 (x) exp 1 -δ 2 t 0 ds ρ s (x) 2 . ( 18
)
When there is no ambiguity we shall drop the x from our notation and denote this process by η.

Regularity of the sample paths of η

We are interested in the continuity property of the process η. It turns out that, as δ decreases, η becomes more and more singular, as shown by the following result.

Proposition 4. If δ > 1, then a.s. η is bounded and continuous on R + . If δ = 1, then a.s. η is constant on [0, T 0 ) and [T 0 , +∞), but has a discontinuity at T 0 . If δ ∈ [0, 1), then a.s. η is continuous away from T 0 , but it diverges to +∞ as t ↑ T 0 .

Proof. When δ ≥ 2, T 0 = ∞ almost-surely, so that, by (18), the following equality of processes holds:

η t = exp 1 -δ 2 t 0 ds ρ s (x) 2 .
Hence, a.s., η takes values in [0, 1] and is continuous on R + . To treat the case δ < 2 we need a lemma: Lemma 2. Let δ < 2 and x > 0. Then the integral:

T 0 0 ds (ρ s (x)) 2
is infinite a.s.

We admit this result for the moment. Then, when δ ∈ (1, 2), η takes values in [0, 1], is continuous away from T 0 and, almost-surely, as t ↑ T 0 :

η t = exp 1 -δ 2 t 0 ds ρ s (x) 2 -→ 0.
Since, η t = 0 for all t ≥ T 0 , η is continuous and the claim follows. When δ = 1,

η t (x) := 1 t<T 0 (x)
so the claim follows at once. Finally, if δ ∈ [0, 1), then η is continuous away from T 0 , but by the above lemma, a.s., as t ↑ T 0 :

η t = exp 1 -δ 2 t 0 ds ρ s (x) 2 -→ +∞ so the claim follows.
We now prove Lemma 2

Proof of Lemma 2. The proof is in two steps. In a first step we prove the lemma when ρ is replaced with a Brownian motion started at some positive point, and in a second step we invoke a representation theorem of Bessel processes as time-changes of some power of the Brownian motion to conclude.

First step: Let (β t ) be a Brownian motion started from some y > 0, and let T 0 denote its hitting time of the origin. Then the integral :

T 0 0 ds (β s (y)) 2
is a.s. infinite. Indeed, denote by h : [0, ∞) → R + the function given by:

h(t) := t| log(1/t)|, if t > 0, 0, if t = 0.
Let A > 0. By Levy's modulus of continuity (see Theorem (2.7), Chapter I, in [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]), there exists a κ > 0, such that the event

M := { ∀s, t ∈ [0, 1], |β t -β s | ≤ κ h(|s -t|) }
has probability one. Therefore, by scale invariance of Brownian motion, setting κ A := √ Aκ, one deduces that the event

M A := { ∀s, t ∈ [0, A], |β t -β s | ≤ κ A h(|s -t|) }
also has probability one. Moreover, under the event {T 0 < A} ∩ M A , we have, for small h > 0.

β 2 T 0 -h = |β T 0 -h -β T 0 | 2 ≤ κ A 2 h log(1/h).
Since

1 h log(1/h)
is not integrable as h → 0 + , we deduce that, under the event {T 0 < A} ∩ M A , we have T 0 0 ds (βs) 2 = +∞. Therefore:

P[T 0 < A] = P[{T 0 < A} ∩ M A ] ≤ P T 0 0 ds (β s ) 2 = +∞ .
Since T 0 < +∞ a.s., we have lim A→∞ P[T 0 < A] = 1. Hence, letting A → ∞ in the above, we deduce that:

P T 0 0 ds (β s ) 2 = +∞ = 1 as claimed.
Second step: Now consider the original Bessel process (ρ t (x)) t≥0 . Suppose that δ ∈ (0, 2). Then, by Thm 3.5 in [START_REF] Zambotti | Random Obstacle Problems, École d' Été de Probabilités de Saint-Flour XLV -2015[END_REF], the process (ρ

t (x)) t≥0 is equal in law to (|β γ(t) | 1 2-δ ) t≥0 ,
where β is a Brownian motion started from y := x 2-δ , and γ : R + → R + is the inverse of the increasing function A : R + → R + given by:

∀u ≥ 0, A(u) = 1 (2 -δ) 2 u 0 |β s | 2(δ-1) 2-δ ds.
Therefore, denoting by T β 0 the hitting time of 0 by the Brownian motion β, we have:

T 0 0 ds (ρ s (x)) 2 (d) = A(T β 0 ) 0 ds |β γ(s) | 2 2-δ = T β 0 0 1 |β u | 2 2-δ 1 (2 -δ) 2 |β u | 2(δ-1) 2-δ du = 1 (2 -δ) 2 T β 0 0 du β u 2
where we have used the change of variable u = γ(s) to get from the first line to the second one. By the first step, the last integral is infinite a.s., so the claim follows.

There still remains to treat the case δ = 0. By Thm 3.5 in [START_REF] Zambotti | Random Obstacle Problems, École d' Été de Probabilités de Saint-Flour XLV -2015[END_REF], in that case, the process (ρ t (x)) t≥0 is equal in law to

β γ(t)∧T β 0 1/2 t≥0
, where β is a Brownian motion started from y := x 2 , T β 0 is its hitting time of 0 and γ : R + → R + is the inverse of the increasing function A : R + → R + given by:

∀u ≥ 0, A(u) = 1 4 u∧T β 0 0 β -1 s ds.
Then, the same computations as above yield the equality in law:

T 0 0 ds (ρ s (x)) 2 (d) = 1 4
T β 0 0 du β u 2
so the result follows as well.

Study of a martingale related to η

Let δ ∈ [0, 2) and x > 0 be fixed. In the previous section, we have shown that, a.s. :

t 0 ds ρ s (x) 2 -→ t→T 0 (x)

+∞

As a consequence, for δ ∈ [0, 1), a.s., the modification η t of the derivative at x of the stochastic flow ρ t diverges at T 0 (x):

η t (x) = 1 t<T 0 (x) exp 1 -δ 2 t 0 ds ρ s (x) 2 -→ t↑T 0 (x)
+∞.

However, since ρ t (x) -→ 0 as t → T 0 (x), this does not exclude the possibility that the product ρ t (x)η t (x) converges as t → T 0 (x). This motivates to study the process :

D t := ρ t (x)η t (x) = 1 t<T 0 (x) ρ t (x) exp 1 -δ 2 t 0 ds ρ s (x) 2 . ( 19
)
As a matter of fact, we will show that (D t ) t≥0 is an L p continuous martingale for some p ≥ 1.

Remark 11. The process (D t ) t≥0 appears as (one half times) the derivative of the stochastic flow associated with the squared Bessel process X t (x) = (ρ t (x)) 2 . Indeed, by applying formally the chain rule, we have, for all t ≥ 0 and x > 0:

dX t (x) dx = 2ρ t (x)η t (x).

Continuity of (D t ) t≥0

In this subsection we show that the process (D t ) t≥0 has a.s. continuous sample paths. By the expression (19), continuity holds as soon as T 0 (x) = ∞ a.s., i.e. as soon as δ ≥ 2. On the other hand, if δ ∈ [0, 2) it suffices to prove that, a.s., D t → 0 as t ↑ T 0 (x). This is the content of the following proposition.

Proposition 5. For all δ ∈ [0, 2) and x > 0, with probability one:

ρ t (x) exp 1 -δ 2 t 0 ds ρ s (x) 2 -→ t→T 0 (x) 0.
Proof. If δ ∈ [1, 2), then exp 1-δ 2 t 0 ds ρs(x) 2 ≤ 1 for all t ≥ 0. Since ρ t -→ 0 as t → T 0 (x), the claim follows at once.

If δ ∈ [0, 1), on the other hand, exp 1-δ

2 t 0 ds ρs(x) 2 ----→ t↑T 0 (x)
+∞ whereas ρ t ----→ t→T 0 (x) 0 so a finer analysis is needed. We have:

log ρ t x exp 1 -δ 2 t 0 ds ρ s (x) 2 = log ρ t x + 1 -δ 2 t 0 ds ρ 2 s
Now, recall that a.s., for all t < T 0 , we have:

ρ t = x + δ -1 2 t 0 ds ρ s + B t
Hence, defining for all integer n ≥ 1 the (F t ) t≥0 -stopping time τ n as:

τ n := inf{t > 0, ρ t ≤ 1/n} ∧ n,
we have:

ρ t∧τn = x + δ -1 2 t∧τn 0 ds ρ s + B t∧τn .
Hence, by Itô's lemma, we deduce that:

Consider now the random time change:

A : [0, T 0 ) → R + t → A t := t 0 ds ρ 2 s .
Note that A is differentiable with strictly positive derivative. Moreover, since A t -→ t→T 0 +∞ a.s. by Lemma 2, we deduce that A is a.s. onto. Hence, a.s., A is a diffeomorphism [0, T 0 ) → R + , the inverse of which we denote by

C : R + → [0, T 0 ) u → C u .
Let β u := Cu 0 dBr ρr , u ≥ 0. Then β is a local martingale started at 0 with quadratic variation β, β u = u, so by Lévy's theorem it is a Brownian motion. The equality (20) can now be rewritten:

log ρ t∧τn x + 1 -δ 2 t∧τn 0 ds ρ 2 s = β At∧τ n - 1 2 A t∧τn .
Letting n → ∞, we obtain, for all t < T 0 :

log ρ t x + 1 -δ 2 t 0 ds ρ 2 s = β At - 1 2 A t .
By the asymptotic properties of Brownian motion (see Corollary (1.12), Chapter II in [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF]), we know that, a.s.:

lim sup s→+∞ β s h(s) = 1
where h(s) := √ 2s log log s. In particular, a.s., there exists T > 0 such that, for all t ≥ T , we have β t ≤ 2h(t). Since, a.s., A t -→ t→T 0 +∞, we deduce that:

lim sup t→+∞ β At - 1 2 A t ≤ lim sup t→+∞ 2h(A t ) - 1 2 A t = -∞.
Hence, a.s. : 2 -→ t↑T 0 (x) 0 as claimed.

log ρ t x exp 1 -δ 2 t 0 ds ρ s (x) 2 -→ t↑T 0 (x) -∞ i.e. ρ t exp 1 -δ 2 t 0 ds ρ s (x)

Martingale property of (D t ) t≥0

Let δ ≥ 0 and x > 0 be fixed. We show in this section that (D t ) t≥0 is an (F t ) t≥0 martingale which, up to a positive constant, corresponds to a Girsanov-type change of probability measure.

Recall that, by definition:

D t = 1 t<T 0 (x) ρ t (x) exp - δ -1 2 t 0 ds ρ 2 s . ( 21
)
Notation 1. For all a ≥ 0 and t ≥ 0, we denote by P a x Ft the image of the probability measure P a

x under the restriction map:

(C(R + ), B(C(R + )) → (C([0, t]), F t ) w → w| [0,t]
Proposition 6. Let δ ≥ 0 and x > 0. Then, for all t ≥ 0, the law P δ+2

x |Ft is absolutely continuous w.r.t. the law P δ

x |Ft , and the corresponding Radon-Nikodym derivative is given by: dP δ+2

x dP δ

x Ft (ρ)

a.s. = 1 t<T 0 (x) ρ t (x) x exp - δ -1 2 t 0 ds ρ 2 s .
Proof. Fix ǫ > 0. Under P δ x |Ft , the canonical process ρ stopped at T ǫ satisfies the following SDE on [0, t]:

ρ s∧Tǫ = x + δ -1 2 s∧Tǫ 0 ds ρ s + B s∧Tǫ .
Consider the process M ǫ defined on [0, t] by:

M ǫ s := s∧Tǫ 0 dB u ρ u M ǫ is an L 2 martingale on [0, t].
The exponential local martingale thereto associated is:

E(M ǫ ) s = exp s∧Tǫ 0 dB u ρ u - 1 2 s∧Tǫ 0 du ρ 2 u .
Since, by Itô's lemma:

log ρ s∧Tǫ x = s∧Tǫ 0 dB u ρ u + δ 2 -1 s∧Tǫ 0 du ρ 2 u ,
we have:

E(M ǫ ) s = exp log ρ s∧Tǫ x - δ -1 2 s∧Tǫ 0 du ρ 2 u = ρ s∧Tǫ x exp - δ -1 2 s∧Tǫ 0 du ρ 2 u .
Note that

E exp 1 2 M ǫ , M ǫ t ≤ exp t 2ǫ < ∞
so that, by Novikov's criterion, E(M ǫ ) is a uniformly integrable martingale on [0, t]. So we may consider the probability measure E(M ǫ )P δ

x Ft . Note also that:

M ǫ , B t = s∧Tǫ 0 du ρ u .
Hence, by Girsanov's theorem, under the probability measure E(M ǫ )P δ x Ft , the process :

ρ s∧Tǫ -x - δ + 1 2 s∧Tǫ 0 du ρ u
is a local martingale, with quadratic variation given by s ∧ T ǫ . Therefore, by Theorem (1.7) in Chapter V of [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], there exists, on some enlarged probability space, a Brownian motion β such that, a.s.:

∀s ∈ [0, t], ρ s∧Tǫ = x + δ + 1 2 s∧Tǫ 0 du ρ u + β s∧Tǫ .
Denote by ρ the unique strong solution on [0, t] of the SDE:

ρs = x + δ + 1 2 s 0 du ρu + β s .
Then, by strong uniqueness of the solution to this SDE, we deduce that, under E(M ǫ )P δ x Ft , a.s.:

∀s ∈ [0, t], s < T ǫ =⇒ ρ s = ρs .

Since ρ has the law of a δ + 2-dimensional Bessel process started at x, we deduce that, for all F : C([0, T ], R + ) → R + Borel, we have:

E δ x [E(M ǫ )F (ρ)1 t<Tǫ ] = E δ+2 x [F (ρ)1 t<Tǫ ]
i.e.:

E δ x ρ t x exp - δ -1 2 t 0 ds ρ 2 s F (ρ)1 t<Tǫ = E δ+2 x [F (ρ)1 t<Tǫ ].
Letting ǫ → 0, by the monotone convergence theorem, we obtain:

E δ x ρ t x exp - δ -1 2 t 0 ds ρ 2 s F (ρ)1 t<T 0 = E δ+2 x [F (ρ)1 t<T 0 ].
But, since P δ+2

x [T 0 < +∞] = 0, this yields:

E δ x ρ t x exp - δ -1 2 t 0 ds ρ 2 s F (ρ)1 t<T 0 = E δ+2 x [F (ρ)]
as stated.

Remark 12. Proposition 6 is actually a particular case of a more general result. Indeed, for all x > 0, t ≥ 0, and δ ′ ≥ δ ≥ 0, such that δ ′ ≥ 2, P δ ′ x |Ft is absolutely continuous w.r.t. the law P δ

x |Ft , and the corresponding Radon-Nikodym derivative is given by:

dP δ ′ x dP δ x Ft (ρ) a.s. = 1 t<T 0 (x) ρ t (x) x δ ′ -δ 2 exp - δ ′ -δ 2 δ ′ + δ 4 -1 t 0 ds ρ 2 s . ( 22
)
The proof of this fact is in all respect similar to that of Proposition 6 above.

Corollary 3. (D t ) t≥0 is an (F t ) t≥0 continuous martingale
Proof. The process (D t ) t≥0 is continuous. Moreover, for all t ≥ 0, 1 x D t is the Radon-Nikodym derivative of P δ+2 x |Ft w.r.t. P δ x |Ft . Therefore ( 1 x D t ) t≥0 is an (F t ) t≥0 martingale, so (D t ) t≥0 is a martingale as well, and the claim follows.

Moment estimates for the martingale (D t ) t≥0

In this section, we prove that the martingale (D t ) t≥0 is actually in L p for some p ≥ 1. We first recall the following fact: Lemma 3. For all a ≥ 0, t ≥ 0, and m ≥ 0, we have:

E a x (ρ m t ) < ∞.
Proof. Denote by d any integer such that d ≥ a. By Lemma 1, we have:

E a x (ρ m t ) ≤ E d x (ρ m t )
Since P d x is the law of (||B s ||) s≥0 , where (B s ) s≥0 is a d-dimensional Brownian motion and || • || is the Euclidean norm in R d (see [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Chapter 11), this inequality can be rewritten as:

E a x (ρ m t ) ≤ E (||B t || m ) Since B t is a Gaussian random variable, E (||B t || m )
is finite, and the result follows.

Proposition 7. (D t ) t≥0 is an L p martingale for all finite positive number p such that p ≤ p(δ), where p(δ) ∈ [1, +∞] is given by:

p(δ) := (2-δ) 2 4(1-δ) if δ < 1, +∞ if δ ≥ 1. ( 23
)
Moreover the above statement is sharp: for δ < 1 and t > 0, the random variable D t is not in L p for p > p(δ).

Remark 13. We emphasize that p is finite in the above result. Indeed D t is never in L ∞ even if δ ≥ 1; for example, when δ = 1, D t = ρ t 1 t<T 0 (x) which is clearly not bounded a.s. .

Proof of Prop 7. If δ ≥ 1, then, for all t ≥ 0, D t ≤ ρ t . Hence, for all p ∈ (0, +∞):

E (D p t ) ≤ E δ x (ρ t p )
which is finite by Lemma 3.

On the other hand, if δ ∈ [0, 1), then, for all t > 0 and p > 0, we have:

E (D p t ) = E δ x 1 t<T 0 ρ t p exp -p δ -1 2 t 0 ds ρ 2 s .
By the absolute continuity relation ( 22) applied with δ ′ := 2, the latter equals:

E 2 x      x 2-δ 2 ρ t p+ δ-2 2 exp      -p δ -1 2 - (δ -2) 2 8 := A(p) t 0 ds ρ 2 s          
.

For p = p(δ), A(p) = 0, so that:

E D p(δ) t = E 2 x x 2-δ 2 ρ t p(δ)+ δ-2 2 = x 1-δ 2 E 2 x ρ t p(δ)+ δ 2 -1 .
Since δ 2 + p(δ) -1 ≥ 0, by Lemma 3, the last quantity is finite. Hence D t is indeed in L p (δ) .

Suppose now that p = p(δ) + r for some r > 0. We show that D t / ∈ L p . We have:

E [D p t ] = E 2 x x 2-δ 2 ρ t p+ δ-2 2 exp -p δ -1 2 - (δ -2) 2 8 t 0 ds ρ s 2 = x 1-δ 2 E 2 x ρ t p+ δ 2 -1 exp 1 -δ 2 r t 0 ds ρ s 2
We claim that the last quantity is infinite. Indeed, first note that by Jensen's inequality and Fubini, for any C > 0 we have:

E 2 x exp C t 0 ds ρ s 2 ≥ exp C t 0 E 2 x ρ -2 s ds
and the right-hand side is infinite since, for all s > 0, E 2 x ρ -2 s = +∞ (indeed, by formula (12), the transition density p 2 s (x, y) does not integrate y -2 as y → 0). Therefore:

E 2 x exp C t 0 ds ρ s 2 = +∞ (24)
Consider now any c > 0 and a, b > 0 such that 1 a + 1 b = 1. By (24) and Hölder's inequality, we have:

+∞ = E 2 x exp 1 -δ 2a r t 0 ds ρ s 2 ≤ E 2 x ρ t ac exp 1 -δ 2 r t 0 ds ρ s 2 1/a E 2 x ρ -bc t 1/b Set c = δ 2 +p-1 δ 2 +p , a = δ 2 + p, and b = δ 2 +p δ 2 +p-1 . Remark that δ 2 + p -1 > 0 since p > p(δ) ≥ 1
, so that this choice for c, a, and b makes sense. We obtain:

E 2 x ρ t δ 2 +p-1 exp 1 -δ 2 r t 0 ds ρ s 2 1 δ 2 +p E 2 x ρ -1 t δ 2 +p-1 δ 2 +p = +∞
By the comparison lemma 1 and the expression [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF] for the transition density of the Bessel process, we have

E 2 x ρ -1 t ≤ E 2 0 ρ -1 t = ∞ 0 1 t exp - y 2 2t dy so that E 2 x ρ -1 t < +∞.
Therefore, we deduce that :

E 2 x ρ t δ 2 +p-1 exp 1 -δ 2 r t 0 ds ρ s 2 = +∞
as claimed. Hence D t / ∈ L p for p > p(δ).

A Bismut-Elworthy-Li formula for the Bessel processes

We are now in position to provide a probabilistic interpretation of the right-hand-side of equation ( 11) in Theorem 2. Let δ > 0, and x > 0. As we saw in the previous section, the process (η t (x)) t≥0 may blow up at time T 0 , so that the stochastic integral t 0 η s (x)dB s is a priori ill-defined, at least for δ ∈ (0, 1). However, it turns out that we can define the latter process rigorously as a local martingale. Proposition 8. Suppose that δ > 0. Then the stochastic integral process t 0 η s dB s is well-defined as a local martingale and is indistinguishable from the continuous martingale D t -x.

Proof. We first treat the case δ ≥ 2, which is much easier to handle. In that case, η t ∈ [0, 1] for all t ≥ 0, so that the stochastic integral t 0 η s dB s is clearly well-defined as an L 2 martingale. Moreover, since T 0 = +∞ a.s., by Itô's lemma we have:

D t = ρ t η t = x + t 0 η s dρ s + t 0 ρ s dη s = x + t 0 η s δ -1 2 ds ρ s + dB s - t 0 ρ s δ -1 2 η s ρ s 2 ds = x + t 0 η s dB s
so the claim follows. Now suppose that δ ∈ (0, 2) and fix an ǫ > 0. Recall that T ǫ (x) := inf{t ≥ 0, ρ t (x) ≤ ǫ} and note that, since T ǫ < T 0 , the stopped process η Tǫ is continuous on R + , so that the stochastic integral t∧Tǫ(x) 0 η s (x)dB s is well-defined as a local martingale. Using as above Itô's lemma, but this time with the stopped processes ρ Tǫ and η Tǫ , we have:

t∧Tǫ 0 η s dB s = D t∧Tǫ -x. (25)
Our aim would be to pass to the limit ǫ → 0 in this equality. By continuity of D, as ǫ → 0, D t∧Tǫ converges to D t∧T 0 = D t almost-surely. So the right-hand side of (25) converges to D t -x almost-surely.

The convergence of the left-hand side to a stochastic integral is more involved, since we first have to prove that the stochastic integral t 0 η s dB s is indeed well-defined as a local martingale. For this, it suffices to prove that, almost-surely:

∀t ≥ 0, t 0 η 2 s ds < ∞.
We actually prove the following stronger fact. For all t ≥ 0

E t 0 η 2 s ds p/2 < ∞ (26)
for all finite positive number p such that p ∈ (1, p(δ)]. Indeed, applying successively the Burkholder-Davis-Gundy (BDG) inequality and Doob's inequality to the martingale Tǫ∧• 0 η s dB s , we have:

E t∧Tǫ 0 η 2 s ds p/2 ≤ C p E sup s≤t∧Tǫ s 0 η u dB u p = C p E sup s≤t∧Tǫ |D s -x| p ≤ C p p p -1 p E [|D t∧Tǫ -x| p ]
where C p is a constant depending only on p. Now, since (D t -x) t≥0 is a continuous martingale, by the optional stopping theorem and Jensen's inequality, we have:

E [|D t∧Tǫ -x| p ] ≤ E(|D t -x| p )
and the right-hand side is finite because D t is in L p . Hence, letting ǫ → 0 in the above, by the monotone convergence theorem we deduce that :

E t∧T 0 0 η 2 s ds p/2 < ∞
But since η t = 0 for all t ≥ T 0 , this implies the bound (26), and hence the stochastic integral t 0 η s dB s is well-defined as a local martingale. Moreover, for all t ≥ 0, by the BDG inequality, we have:

E t 0 η s dB s - t∧Tǫ 0 η s dB s p = E t∧T 0 t∧Tǫ η s dB s p ≤ c p E t∧T 0 t∧Tǫ η 2 s ds p/2
where c p is some constant depending only on p. Now, by the dominated convergence theorem, the last quantity above goes to 0 as ǫ → 0 , and hence: Using the above proposition, Theorem 2 can now be interpreted probabilistically as a Bismut-Elworthy-Li formula.

Theorem 3 (Bismut-Elworthy-Li formula). Let δ > 0. Then, for all T > 0, and all F : R + → R bounded and Borel, the function x → P δ t F (x) is differentiable on R + , and for all x > 0:

d dx P δ T F (x) = 1 T E F (ρ t (x)) T 0 η s (x)dB s . (27)
Proof. By Theorem 2, the differentiablity property holds, and we have:

d dx P δ T F (x) = x T P δ+2 T F (x) -P δ T F (x) .
Moreover, by Proposition 6, for all x > 0:

P δ+2 T F (x) -P δ T F (x) = E δ x F (ρ T ) D T x - 1
and, by Proposition 8, we have:

E δ x F (ρ T ) D T x -1 = 1 x E F (ρ T (x)) T 0 η s (x)dB s so equality (27) follows.
Using the Bismut-Elworthy-Li formula, we are now able to sharpen the Strong Feller estimate obtained in equation ( 14) above.

Corollary 4. Let T > 0 and δ ≥ 2( √ 2 -1). Then, for all R > 0, there exists a constant C > 0 such that, for all x, y ∈ [0, R] and F : R + → R bounded and Borel, we have:

|P δ T F (x) -P δ T F (y)| ≤ C||F || ∞ T α(δ) |y -x| (28)
where the exponent α(δ) ∈ [ 1 2 , 1) is given by:

α(δ) := 1 2 + 1-δ 2-δ if δ ∈ [2( √ 2 -1), 1], 1/2 if δ ≥ 1.
Proof. Let x > 0. By Theorem 3, we have :

d dx P δ T F (x) = 1 T E F (ρ t (x)) T 0 η s (x)dB s .
so that:

d dx P δ T F (x) ≤ ||F || ∞ T E T 0 η s (x)dB s .
We now bound the quantity E T 0 η s (x)dB s . If δ ≥ 1, then the process (η s (x)) s≥0 takes values in [0, 1], so that, using the Cauchy-Schwarz inequality and Itô's isometry formula, we have :

E T 0 η s (x)dB s ≤ E T 0 η s (x) 2 ds ≤ √ T .
Therefore:

d dx P δ T F (x) ≤ ||F || ∞ √
T and the claim follows with C = 1.

Suppose now that δ ∈ [2( √ 2 -1), 1). Letting p := p(δ) as in (23), we have, by Jensen's inequality:

E T 0 η s (x)dB s ≤ E T 0 η s (x)dB s p 1/p
Now, applying successively the BDG inequality, Jensen's inequality and the absolute continuity relation (22) between P 2

x and P δ x , we have, for some constant c p depending only on p:

E T 0 η s (x)dB s p ≤ c p E T 0 η s (x) 2 ds p/2 ≤ c p T p/2-1 E T 0 η s (x) p ds ≤ c p T p/2-1 T 0 E δ x (η p s ) ds = c p T p/2-1 T 0 E 2 x ρ s x δ-2 2 exp 1 -δ 2 p - (2 -δ) 2 8 s 0 du ρ 2 u ds = c p T p/2-1 T 0 E 2 x ρ s x δ-2 2
ds where the last equality follows from the fact that 1-δ 2 p -(2-δ) 2 8 = 0 for p = p(δ). Now, since δ-2 2 ≤ 0, by the comparison lemma 1, as well as the scaling property of the Bessel processes (see, e.g., Remark 3.7 in [START_REF] Zambotti | Random Obstacle Problems, École d' Été de Probabilités de Saint-Flour XLV -2015[END_REF]), for all s ∈ [0, T ], we have:

E 2 x ρ δ-2 2 s ≤ E 2 0 ρ δ-2 2 s = s δ-2 4 E 2 0 ρ δ-2 2 1 . Let c := E 2 0 ρ δ-2 2
1

. Using formula [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF], we have:

c = ∞ 0 y δ/2 exp - y 2 2 dy < ∞.
Hence:

T 0 E 2 x ρ s x δ 2 -1 ds ≤ c x 1-δ 2 T 0 s δ-2 4 ds ≤ 4 c δ + 2 x 1-δ 2 T δ+2 4 .
Therefore, we obtain:

E T 0 η s (x)dB s p ≤ K x 1-δ 2 T p 2 -1 T δ+2 4 ≤ K x 1-δ 2 T p 2 + δ-2 4
where K is a constant depending only on δ. Hence

E T 0 η s (x)dB s ≤ K 1/p x 1 p (1-δ 2 ) T 1 2 + δ-2 4p .
Note that, since p = p(δ), we have 1 p (1 -δ 2 ) = 2(1-δ) 2-δ , and δ-2 4p = -1-δ 2-δ . Therefore, we obtain:

d dx P δ T F (x) ≤ K 1/p x 2 1-δ 2-δ ||F || ∞ T -1 2 -1-δ 2-δ .
Therefore, given R > 0, one has for all x ∈ [0, R]:

d dx P δ T F (x) ≤ C ||F || ∞ T α(δ)
with C := K 1/p R 2 1-δ 2-δ . This yields the claim.

Remark 14. In the above proposition, the value 2( √ 2 -1) that appears is the smallest value of δ for which η is in L 2 . For δ < 2( √ 2 -1), η is no longer in L 2 but only in L p for p = p(δ) < 2, so that we cannot apply Jensen's inequality to bound the quantity

E T 0 η s (x) 2 ds p/2
anymore. It seems reasonable to expect that the bound (28) holds also for δ < 2( √ 2 -1), although we do not have a proof of this fact.

Appendix

In this Appendix, we prove Proposition 3. Recall that we still denote by (ρ t (x)) t,x≥0 the process (ρ δ t (x)) t,x≥0 constructed in Proposition 2.

Lemma 4. For all rational numbers ǫ, γ > 0, let:

U ǫ γ := [0, T ǫ (γ)) × (γ, +∞)
and set:

U := ǫ,γ∈Q * + U ǫ γ .
Then, a.s., the function (t, x) → ρ t (x) is continuous on the open set U .

Proof. By patching, it suffices to prove that, a.s., the function (t, x) → ρ t (x) is continuous on each U ǫ γ , where ǫ, γ ∈ Q * + . Fix ǫ, γ ∈ Q * + , and let x, y ∈ (γ, +∞) ∩ Q. We proceed to show that, a.s., for all t ≤ s < T ǫ (γ) the following inequality holds:

|ρ t (x) -ρ s (y)| ≤ |x -y| exp |δ -1| 2ǫ 2 t + |δ -1| 2ǫ |s -t| + |B s -B t |. (29)
Since T ǫ (γ) < T 0 (γ), a.s., for all t ≤ s ≤ T ǫ (γ), we have:

∀τ ∈ [0, t], ρ τ (x) = x + δ -1 2 τ 0 du ρ u (x) + B τ as well as ∀τ ∈ [0, s], ρ τ (y) = y + δ -1 2 τ 0 du ρ u (y) + B τ and hence : ∀τ ∈ [0, t], |ρ τ (x) -ρ τ (y)| ≤ |x -y| + |δ -1| 2 τ 0 |ρ u (x) -ρ u (y)| ρ u (x)ρ u (y) du.
By the monotonicity property of ρ, we have, a.s., for all t, s as above and u ∈ [0, s]:

ρ u (x) ∧ ρ u (y) ≥ ρ u (γ) ≥ ǫ (30) so that: ∀τ ∈ [0, t], |ρ τ (x) -ρ τ (y)| ≤ |x -y| + |δ -1| 2 τ 0 |ρ u (x) -ρ u (y)| ǫ 2 du,
which, by Grönwall's inequality, implies that:

|ρ t (x) -ρ t (y)| ≤ |x -y| exp |δ -1| 2ǫ 2 t . (31)
Moreover, we have:

ρ s (y) -ρ t (y) = δ -1 2 s t du ρ u (y) + B s -B t
which, by (30), entails the inequality:

|ρ s (y) -ρ t (y)| ≤ |δ -1| 2ǫ |s -t| + |B s -B t |. (32)
Putting inequalities (31) and (32) together yields the claimed inequality (29). Hence, we have, a.s., for all rationals x, y > γ and all t ≤ s < T ǫ (γ):

|ρ t (x) -ρ s (y)| ≤ |x -y| exp |δ -1| 2ǫ 2 t + δ -1 2 |s -t| + |B s -B t |
and, by density of Q ∩ (γ, +∞) in (γ, +∞), this inequality remains true for all x, y > γ.

Since, a.s., t → B t is continuous on R + , the continuity of ρ on U ǫ γ is proved.

Corollary 5. Almost-surely, we have:

∀x ≥ 0, ∀t ∈ [0, T 0 (x)), ρ t (x) = x + δ -1 2 t 0 du ρ u (x) + B t . (33)
Remark 15. We have already remarked in Section 3 that, for all fixed x ≥ 0, the process (ρ t (x)) t≥0 satisfies the SDE [START_REF] Tsatsoulis | Spectral Gap for the Stochastic Quantization Equation on the 2-dimensional Torus[END_REF]. By contrast, the above Corollary shows the stronger fact that, considering the modification ρ of the Bessel flow constructed in Proposition 2 above, a.s., for each x ≥ 0, the path (ρ t (x)) t≥0 still satisfies relation [START_REF] Tsatsoulis | Spectral Gap for the Stochastic Quantization Equation on the 2-dimensional Torus[END_REF].

Proof. Consider an almost-sure event A ∈ F as in Remark 7. On the event A, for all r ∈ Q + , we have:

∀t ∈ [0, T 0 (r)), ρ t (r) = r + δ -1 2 t 0 du ρ u (r) + B t .
Denote by B ∈ F any almost-sure event on which ρ satisfies the monotonicity property (15). We show that, on the event A ∩ B, the property (33) is satisfied. Suppose A ∩ B is fulfilled, and let x ≥ 0. Then for all r ∈ Q such that r ≥ x, we have: ∀t ≥ 0, ρ t (x) ≤ ρ t (r) so that T 0 (r) ≥ T 0 (x). Hence, for all t ∈ [0, T 0 (x)), we have in particular t ∈ [0, T 0 (r)), so that:

ρ t (r) = r + δ -1 2 t 0 du ρ u (r) + B t .
Since, for all u ∈ [0, t], ρ u (r) ↓ ρ u (x) as r ↓ x with r ∈ Q, by the monotone convergence theorem, we deduce that:

t 0 du ρ u (r) -→ t 0 du ρ u (x)
as r ↓ x with r ∈ Q. Hence, letting r ↓ x with r ∈ Q in the above equation, we obtain:

ρ t (x) = x + δ -1 2 t 0 du ρ u (x) + B t .
This yields the claim.

One of the main difficulties for proving Proposition 3 arises from the behavior of ρ t (x) at t = T 0 (x). However we will circumvent this problem by working away from the event t = T 0 (x). To do so, we will make use of the following property.

Lemma 5. Let δ < 2 and x ≥ 0. Then the function y → T 0 (y) is a.s. continuous at x.

Proof. The function y → T 0 (y) is nondecreasing over R + . Hence, if x > 0, it has left-and right-sided limits at x, T 0 (x -) and T 0 (x + ), satisfying:

T 0 (x -) ≤ T 0 (x) ≤ T 0 (x + ).

(34)

Similarly, if x = 0, there exists a right-sided limit T 0 (0 + ) satisfying T 0 (0) ≤ T 0 (0 + ). Suppose, e.g., that x > 0. Then we have:

E e -T 0 (x + ) ≤ E e -T 0 (x) ≤ E e -T 0 (x -) .

Now, by the scaling property of the Bessel processes (see, e.g., Remark 3.7 in [START_REF] Zambotti | Random Obstacle Problems, École d' Été de Probabilités de Saint-Flour XLV -2015[END_REF]), for all y ≥ 0, the following holds:

(yρ t (1)) t≥0 (d)
= (ρ y 2 t (y)) t≥0 , so that T 0 (y)

(d)
= y 2 T 0 (1). Therefore, using the dominated convergence theorem, we have: = E e -x 2 T 0 (1) = E e -T 0 (x) .

Similarly, we have E e -T 0 (x -) = E e -T 0 (x) . Hence the inequalities (35) are actually equalities; recalling the original inequality (34), we deduce that T 0 (x -) = T 0 (x) = T 0 (x + ) a.s.. Similarly, if x = 0, we have T 0 (0) = T 0 (0 + ) a.s.

Before proving Proposition 3, we need a coalescence lemma, which will help us prove that the derivative of ρ t at x is 0 if t > T 0 (x): Lemma 6. Let x, y ≥ 0, and let τ be a nonnegative (F t ) t≥0 -stopping time. Then, almostsurely:

ρ τ (x) = ρ τ (y) ⇒ ∀s ≥ τ, ρ s (x) = ρ s (y).

Proof. On the event {ρ τ (x) = ρ τ (y)}, the processes (X δ t (x)) t≥0 := (ρ t (x) 2 ) t≥0 and (X δ t (y)) t≥0 := (ρ t (y) 2 ) t≥0 both satisfy, on [τ, +∞), the SDE:

X t = ρ τ (x) 2 + 2 t τ X s dB s + δ(t -τ ).
By pathwise uniqueness of this SDE (see [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Theorem (3.5), Chapter IX), we deduce that, a.s. on the event {ρ τ (x) = ρ τ (y)}, X t (x) = X t (y), hence ρ t (x) = ρ t (y) for all t ≥ τ . Now we are able to prove Prop. 3.

Proof of Proposition 3. Let t > 0 and x > 0 be fixed. First remark that:

P(T 0 (x) = t) = 0.
Indeed, if δ > 0, then : P(T 0 (x) = t) ≤ P(ρ t (x) = 0) and the RHS is zero since the law of ρ t (x) has no atom on R + (it has density p δ t (x, •) w.r.t. Lebesgue measure on R + , where p δ t was defined in equation ( 12) above). On the other hand, if δ = 0, then 0 is an absorbing state for the process ρ, so that, for all s ≥ 0: P(T 0 (x) ≤ s) = P(ρ s (x) = 0) and the RHS is continuous in s on R + , since it is given by exp(-x 2 2s ) (see [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Chapter XI, Corrolary 1.4). Hence, also in the case δ = 0 the law of T 0 (x) has no atom on R + . Hence, a.s., either t < T 0 (x) or t > T 0 (x).

First suppose that t < T 0 (x). A.s., the function y → T 0 (y) is continuous at x, so there exists a rational number y ∈ [0, x) such that t < T 0 (y); since, by Remark [START_REF] Elworthy | Formulae for the derivatives of heat semi-groups[END_REF], t → ρ t (y) is continuous, there exists ǫ ∈ Q * + such that t < T ǫ (y). By monotonicity of z → ρ(z), for all s ∈ [0, t] and z ≥ y, we have: ρ s (z) ≥ ρ s (y) ≥ ǫ.

Hence, recalling Corollary 5, for all s ∈ [0, t] and h ∈ R such that |h| < |x -y|:

ρ s (x + h) = x + h + s 0 δ -1 2 du ρ u (x + h) + B s .
Hence, setting η h s (x) := ρs(x+h)-ρs(x) h ,we have:

∀s ∈ [0, t], η h s (x) = 1 - δ -1 2 t 0 η h u (x) ρ u (x)ρ u (x + h) du

  s in L p . Hence, the left-hand side of equality (25) converges in L p to the stochastic integral t 0 η s dB s . Letting ǫ → 0 in that equality, we thus obtain :t 0 η s dB s = D t -x as claimed.

E

  e -T 0 (x + ) = lim y↓x E e -T 0 (y) = lim y↓x E e -y 2 T 0(1)

  x)ρ s (x + h) .

Acknowledgements. I would like to thank Lorenzo Zambotti, my Ph.D. advisor, for all the time he patiently devotes in helping me with my research. I would also like to thank Thomas Duquesne and Nicolas Fournier, who helped me solve a technical problem, as well as Yves Le Jan for a helpful discussion on the Bessel flows of low dimension, and Lioudmila Vostrikova for answering a question on this topic.

Note that, for all s ∈ [0, t] and h ∈ R such that |h| < |x -y|, we have (s, x + h) ∈ [0, T ǫ (y)) × (y, +∞) ⊂ U . Hence, by Lemma 4, we have, for all s ∈ [0, t]

with the domination property:

valid for all |h| < |x -y|. Hence, by the dominated convergence theorem, we deduce that:

which yields the claimed differentiability of ρ t at x.

We now suppose that t > T 0 (x). Since the function y → T 0 (y) is a.s. continuous at x, a.s. there exists y > x, y ∈ Q, such that t > T 0 (y). By Remark [START_REF] Elworthy | Formulae for the derivatives of heat semi-groups[END_REF], the function t → ρ t (y) is continuous, so that ρ T 0 (y) (y) = 0. By monotonicity of z → ρ(z), we deduce that, for all z ∈ [0, y], we have :

By Lemma 6, we deduce that, leaving aside some event of proability zero, all the trajectories (ρ t (z)) t≥0 for z ∈ [0, y] ∩ Q coincide from time T 0 (y) onwards. In particular, we have:

Since, moreover, the function z → ρ t (z) is nondecreasing, we deduce that it is constant on the whole interval [0, y]:

∀z ∈ [0, y], ρ t (z) = ρ t (x).

In particular, the function z → ρ t (z) has derivative 0 at x. This concludes the proof.


M Bisquert

A Bégué

P Poncelet

M Teisseire

EVALUATION OF FUSION METHODS FOR CROP MONITORING PURPOSES

Keywords: MODIS-Landsat fusion, remote sensing, validation

In this contribution we present a local evaluation procedure of Landsat-MODIS fusion methods for crop monitoring purposes. Two fusion methods are applied to obtain a twoyear time series of Landsat-resolution images. The validation is applied at pixel level in order to analyze if the simulated images are capable of unmixing coarse-resolution pixels and obtaining an accurate temporal profile of the high-resolution pixels near the boundaries between two fields. The experiment has been conducted in two neighbor fields and results have shown that the temporal profile of these high-resolution pixels agrees with the temporal profile of a pure coarse-resolution pixel in the corresponding field. They highlight that the simulated images allow identifying the membership of the high-resolution simulated pixels to one field or the neighbor one.

INTRODUCTION

The knowledge of the vegetation phenology is important for many purposes including land cover classification and vegetation monitoring. In [START_REF] Bradley | Comparison of phenology trends by land cover class: a case study in the Great Basin, USA[END_REF] the vegetation phenology is used for obtaining an accurate regional land cover classification and afterwards for studying the effect of extreme meteorological conditions on the different land covers previously identified. The agricultural monitoring systems need to have dense temporal information, as it is known that agricultural production follows strong seasonal patterns and depends highly on climate and management practices [START_REF] Atzberger | Advances in Remote Sensing of Agriculture: Context Description, Existing Operational Monitoring Systems and Major Information Needs[END_REF]. For vegetation monitoring purposes, remote sensing has proved to be a very valuable tool.

In natural vegetation patches, such as forests or rangelands, a low/medium spatial resolution (hecto-to kilometric) is appropriate, and earth-observing systems providing this spatial information on a daily step exist (MODIS, MERIS, VGT…). For crop monitoring, the problem is different, as most of the fields need high/very high resolution (decametric to metric) images that usually have the constraint of having low temporal frequency (e.g. 16 days for Landsat), and in some cases, high economic cost of the images (high frequency image tasking is generally not affordable for agriculture applications). Furthermore, low temporal frequency has an additional problem in regions frequently cloudy. Actually the presence of clouds may lead to long periods were none image can be acquired. In this context fusion methods are highly valuable because they allow to simulate images at high spatial resolution in dates where only information at coarse spatial resolution is available [START_REF] Amorós-López | Multitemporal fusion of Landsat/TM and ENVISAT/MERIS for crop monitoring[END_REF]. Fusion methods usually combine the information of at least one pair of high and coarse spatial resolution images as in the STARFM (Spatio Temporal Adaptive Reflectance Fusion Method) [START_REF] Gao | On the blending of the Landsat and MODIS surface reflectance: predicting daily Landsat surface reflectance[END_REF] and ESTARFM (Enhanced Spatio Temporal Adaptive Reflectance Fusion Method) [START_REF] Zhu | An enhanced spatial and temporal adaptive reflectance fusion model for complex heterogeneous regions[END_REF]. In [START_REF] Bisquert | Soft Fusion of Heterogeneous Image Time Series[END_REF] a simple fusion method WA (Weighted Average) was developed which only needs two input images: a high spatial resolution image at no matter which date (obviously closer dates lead to better results) and a coarse spatial resolution image at the date to be simulated or at a close date. This method uses soft operators and is based on the temporal distance between the input images and the timestamp of the image to be simulated. Evaluation techniques of the fusion methods are usually based in global statistics, such as correlation coefficient, root mean square error (RMSE) and mean absolute deviation (MAD). However, the evaluation method should be adapted to the specific objective of the simulated images.

The objective of the present work is to go further the statistical evaluation of the fusion methods by proposing a thematic evaluation focused on time series crop monitoring. Simulated images are obtained by using the fusion methods ESTARFM and WA presented in [START_REF] Zhu | An enhanced spatial and temporal adaptive reflectance fusion model for complex heterogeneous regions[END_REF] and [START_REF] Bisquert | Soft Fusion of Heterogeneous Image Time Series[END_REF]. These methods were applied to fuse Landsat and MODIS (MODerate Imaging Spectroradiometer) images in an agricultural region of Brazil. In this sense we examine the quality of the temporal profile of a specific crop extracted from the temporal series of high resolution images (Landsat) including the simulated images, compared to the temporal profiles obtained from coarse-resolution images (MODIS). The quality of the simulated images is therefore analyzed at pixel level.

The remainder of this paper is organized as follows: Section 2 presents the data used, Section 3 describes the fusion methods and the evaluation methodology applied, Section 4 shows the evaluation results and Section 5 summarizes the main conclusions.

DATA

The study area is a well-known region of Brazil with important plantations of Eucalyptus [7]. Therefore the crop selected in this study corresponds to Eucalyptus. MODIS images are used as coarse-spatial resolution images. Specifically the product MOD13Q1 with 250 m spatial resolution and 16-day composites has been selected. 21 images of the year 2009 and 21 of the year 2010 have been downloaded.

As high resolution, the product Landsat Surface Reflectance Climate Data Record has been selected. This product provides surface reflectances from Landsat 5 satellite with 30 m spatial resolution and 16 days revisit cycle. Six Landsat images were available for 2009 (corresponding to the dates 02/01, 04/22, 05/24, 08/12, 08/25 and 09/13; unfortunately the image of 09/13 presented an important cloud coverage) and three for 2010 (03/02, 08/28 and 11/03). Figure 1 shows an example of the study area from the Landsat and the MODIS images. We can notice that parcels can be easily identified in the Landsat image, while not in the MODIS one.

METHODOLOGY

Considering that we have a dense temporal series of coarsespatial resolution images and a scarce temporal series of high resolution images, the first step consists in applying the fusion methods ESTARFM and WA [START_REF] Zhu | An enhanced spatial and temporal adaptive reflectance fusion model for complex heterogeneous regions[END_REF][START_REF] Bisquert | Soft Fusion of Heterogeneous Image Time Series[END_REF] for simulating NDVI (Normalized Difference Vegetation Index) images at the Landsat spatial resolution in those dates where no Landsat images are available. This leads to obtain an enriched temporal series of high resolution (real + simulated) images.

The fusion method developed in [START_REF] Bisquert | Soft Fusion of Heterogeneous Image Time Series[END_REF] combines a coarse-resolution image near the date to simulate (target) and a high-resolution image in a different date. This method is based on the temporal distance between the input images and the target image. The first step is radiometrically normalizing the Landsat images to the MODIS ones. Afterwards the MODIS images are up-scaled to the Landsat resolution. Finally the weighted average (WA) per pixel is obtained, where the weight is related to the temporal distance between each input image and the target image by using the following equation:

𝑊𝐴(𝐻, 𝐿) = ( ) * ( ) * ( ) ( ) (1)
where H and L reference respectively high and lowresolution pixels (Landsat and MODIS), and the µ's stand for the weights assigned to each image and are calculated as follows:

If t 0 t i < t then 𝜇 𝑡 (𝑡 𝑖 ) = (𝑡 𝑖 -𝑡 0 ) (𝑡 -𝑡 0 ) ⁄ If t t i < t E then 𝜇 (𝑡 ) = (𝑡 -𝑡 ) (𝑡 -𝑡 ) ⁄

where t i corresponds to the timestamp of the high or lowresolution image, and t 0 and t E are timestamps identified outside the temporal range of the input and output images being used.

The ESTARFM (Enhanced Spatio Temporal Fusion Method) [START_REF] Zhu | An enhanced spatial and temporal adaptive reflectance fusion model for complex heterogeneous regions[END_REF] needs two high resolution images and three coarse-resolution images as inputs and is much more complex.

For the thematic validation we intend to analyze if the simulated high-resolution pixels near a boundary between two parcels can be identified as belonging to one parcel or the neighbor one and reproduce the temporal profile of the corresponding field. The following experiment has been performed (Cf. Figure 2): pure coarse-resolution pixels have been identified in two neighboring eucalyptus fields, and four mixed coarse-resolution pixels were identified along the boundary of both fields. Several high-resolution pixels corresponding to the mixed coarse-resolution ones have been identified at each side of the boundary, belonging to one or the other field. The NDVI temporal profile for the high-resolution pixels have been extracted from the enriched Landsat time series (including the available and the simulated images) and averaged for each field. Moreover the temporal profiles of the pure coarse-resolution pixels inside each field have been extracted from the MODIS time series. This experiment allows verifying if the simulated images are capable to unmix pixels at high resolution belonging to the same mixed coarse-resolution pixel.

RESULTS AND DISCUSSION

In figure 3 we present the temporal profiles corresponding to the pure coarse-resolution pixels (extracted from MODIS temporal series) and the high resolution pixels (extracted from the enriched time series) belonging to the two fields: field A (Figure 3a) and field B (Figure 3b). Both fields correspond to eucalyptus plantations that have been cut at different dates (Figure 3). A good agreement between the temporal profile obtained from the Landsat enriched time series and the MODIS time series is observed for both fields. Moreover we observe how the profiles obtained from the high-resolution pixels of each field are different, which shows the ability of identifying the membership of the simulated pixels to one or the other field.

ESTARFM and WA are generally performing similar except for some dates where ESTARFM leads to some anomalies in the NDVI values. Good overall results are obtained for the temporal profile with both methods except when sharp changes are present; in this case none of the fusion methods analyzed is able to identify this sharp change but identifies a monotone one. Figure 3 shows a sharp change for each field (arrows) representing the clearcut harvest of the eucalyptus field. MODIS temporal series identifies this harvest date showing a sharp drop of the NDVI, while with the simulated images a monotone drop is observed.

Figure 4 illustrates two examples of the original and simulated images. One can observe how both simulated images are able to correctly identify the boundary between the two parcels analyzed which is not identifiable in the MODIS image. In the case of 05/24 both simulated images show similar values to the original one, however in the second one (08/28) the ESTARFM presents important errors in the left field. This is not the case in the WA method. MODIS pur (red) and mixed (white) pixels used for obtaining the temporal profiles are represented.

CONCLUSIONS

This study shows a local evaluation procedure of simulated images for monitoring purposes. A two-year time series of Landsat-resolution images is obtained including 9 Landsat images and 35 simulated images. The evaluation procedure has focused on a boundary between two fields which is not identifiable with the coarse-resolution images. We have shown that the simulated images allow identifying the highresolution pixels belonging to each field and obtaining a temporal profile which agrees with the temporal profiles obtained from pure coarse-resolution pixels inside each field in the case of monotone changes. Therefore the simulated images are highly valuable for monitoring purposes as well as for any application needing temporal series, such as for classification purposes. However the presence of abrupt changes (harvest, fire…) is not well achieved with the simulated images, an adaptation of the method is being considered for identifying these sharp changes.

Fig. 1 .Fig. 2 .

 12 Fig. 2. Eucalyptus parcels (A and B) selected for the validation at local scale. Two pure coarse-resolution pixels (1, 2) are identified in the parcels, while four mixed coarsepixels (3-6) are identified along the boundary between the two parcels. In each mixed coarse-pixel several high resolution pixels are identified belonging either to parcel A or to parcel B.

Fig. 3 .Fig. 4 .

 34 Fig.4. Example of original (Landsat and MODIS) and simulated (ESTARFM and WA) images for two dates of 2009.MODIS pur (red) and mixed (white) pixels used for obtaining the temporal profiles are represented.

Acknowledgements

This work was supported by public funds received in the framework of GEOSUD, a project (ANR-10-EQPX-20) of the program 'Investissements d'Avenir' managed by the French National Research Agency.