Clément Moulin-Frier

Pierre Rouanet

Pierre-Yves Oudeyer

Explauto: an open-source Python library to study autonomous exploration in developmental robotics

I. INTRODUCTION

Last year at the ICDL-Epirob conference [START_REF] Moulin-Frier | Exploration strategies in developmental robotics: A unified probabilistic framework[END_REF], we presented a probabilistic framework unifying two important families of exploration mechanisms recently shown to be efficient to learn complex non-linear redundant sensorimotor mappings, i.e. the mappings between the motor actions made by a robot and the sensory effects they produce. This can be for example a robot learning how arm movements make physical objects move, or how movements of a virtual vocal tract modulates vocalization sounds. These two exploration mechanisms are: 1) goal babbling [START_REF] Rolf | Goal babbling permits direct learning of inverse kinematics[END_REF], [START_REF] Baranes | Active learning of inverse models with intrinsically motivated goal exploration in robots[END_REF] and 2) active learning driven by the maximization of empirically measured learning progress [START_REF] Schmidhuber | A possibility for implementing curiosity and boredom in model-building neural controllers[END_REF], [START_REF] Baranes | Active learning of inverse models with intrinsically motivated goal exploration in robots[END_REF]. Both principles can be combined in various ways to achieve efficient exploration strategies -as opposed to random motor babbling, where collected sensorimotor data are generally too sparse to efficiently feed machine learning algorithms. This is especially true when the motor and sensory spaces are highly dimensional, the mapping between them is non-linear and redundant, and there is limited time allowed for learning -the general case of robotic systems.

We now provide an open-source Python library, called Explauto , providing a unified API to design and compare various exploration strategies driving various sensorimotor learning algorithms in various simulated or robotics systems. Explauto aims at being collaborative and pedagogic, providing a platform to developmental roboticists where they can publish and compare their algorithmic contributions related to autonomous exploration and learning, as well as a platform for teaching and scientific diffusion. The library is available at this address: https://github.com/flowersteam/explauto.

II. THE EXPLAUTO LIBRARY

A. Architecture

The Explauto architecture is illustrated in Figure 1a. We describe it below from bottom to top. We call sensorimotor system the physical properties of the robot and the environment in which it evolves. Programmatically, if we call M the space of motor commands allowed for a robot -e.g. target positions of the joints of a robotic arm, and S the space of sensory effects it perceives -e.g. the 3D position of its end effector, the sensory motor system must implement a way to compute the sensory effect s ∈ S corresponding to a given motor action m ∈ M . In the current version of the library, we provide two simulated sensorimotor systems (a multi-joint arm acting on a plan and an under-actuated torque-controlled inverse pendulum) as well as a system allowing the control of real robots based on Dynamixel actuators by providing bindings to the Pypot library 1 . We also provide bindings to an existing articulatory synthesizer2 which can be installed as an optional package.

The cognitive architecture of an agent is then composed by a sensorimotor model, which iteratively learns internal forward and inverse models from experience, and an interest model which actively chooses where to explore in the sensorimotor space. A sensorimotor model implements both the iterative learning process from sensorimotor experience, i.e. from the iterative collection of (m, s) ∈ M × S pairs by interaction with the environment, and the use of the resulting internal model to perform forward and inverse predictions (or any kind of general prediction between sensorimotor subspaces). Explauto provides several sensorimotor models: simple nearest-neighbor look-up, non-parametric models combining classical regressions and optimization algorithms, online local mixtures of Gaussians, and discrete Lindstone distributions. An interest model implements the active exploration process, where sensorimotor experiments are chosen to improve the forward or inverse prediction of the sensorimotor model. It explores in a given interest space X ⊂ M × S, resulting e.g. in motor babbling strategies when X = M or in goal babbling strategies when X = S. Interest models also have to implement a sampling procedure in X. Explauto provides several sampling procedures: random sampling as well as sampling maximizing the learning progress in forward or inverse predictions.

Explauto comes with a high level programming interface to compare the built-in, optional or user-defined implementations of sensorimotor systems, sensorimotor models and interest models (Figure 1b). It automatically takes advantage of the multi-core processors commonly found in modern personal computers, allowing the parallel execution of multiple experiments (parallelism on multiple machines has not been tested yet). It also contains plotting facilities such as the visualisation of the motor commands and reached effects over time, as well as the learning curves (Figure 1c).

B. Extending the library

Explauto is distributed under an open-source license (GNU GPL V3) and can be easily extended by subclassing three abstract classes providing a common interface and the base processing procedures for sensorimotor systems (Environment class), sensorimotor models (SensorimotorModel class) and interest models (InterestModel class). We encourage anyone to contribute by sending us pull-requests from our Github public repository. For example, we are currently working on an interface with a Baxter robot, on the integration of a state-of-the-art sensorimotor learning algorithm (Infinite Mixture of Linear Experts, [START_REF] Damas | Online learning of single-and multivalued functions with an infinite mixture of linear experts[END_REF]), as well as new implementations of the learning progress measure [START_REF] Moulin-Frier | Exploration strategies in developmental robotics: A unified probabilistic framework[END_REF].

III. CONCLUSION

The Explauto library provides a common platform to the Developmental Robotics community allowing the integration and comparison of various exploration strategies driving various sensorimotor learning algorithms in various simulated or robotics systems. Many other exploration strategies could be easily integrated into the library, as for example Direction Sampling [START_REF] Rolf | Goal babbling with unknown ranges: A direction-sampling approach[END_REF], compression progress [START_REF] Schmidhuber | A possibility for implementing curiosity and boredom in model-building neural controllers[END_REF], empowerment [START_REF] Klyubin | All else being equal be empowered[END_REF] and thus be compared in a proper way on various sensorimotor systems and using various sensorimotor internal models.

Explauto also aims to be a tool for scientific diffusion, in particular teaching. The library comes with a series of short Python tutorials explaining by practice the concepts of sensorimotor learning by experience, motor and goal babblings and intrinsically-motivated exploration.

Fig. 1 .

 1 Fig. 1. Explauto is based on a unified architecture, provides a high level API allowing to compare various exploration strategies in a few lines of code and can be executed on physical robots such as the open-source humanoid Poppy [6] (see a demonstration video at https://vimeo.com/95405850)

https://github.com/poppy-project/pypot

http://www.bu.edu/speechlab/software/diva-source-code/

ACKNOWLEDGMENT

This work was partially financed by ERC Starting Grant EXPLORERS 240 007. The library embeds parts of the third-party models library from Fabien Benureau available at https://github.com/humm/models.


Jeremy Boy

Jean-Daniel Fekete

The CO2 Pollution Map: Lessons Learned from Designing a Visualization that Bridges the Gap between Visual Communication and Information Visualization

Keywords: design, Motion design, Design for interpretation, Pollution map

We present the design of the CO2 Pollution map, a web-based visualization that uses motion design techniques to assist users in interpreting semantic information about the data. We describe our motivations and inspirations, found in the fields of graphic and motion design, and present how free visual variables can be used to create an impression of what the data are about. We finish by reporting an early evaluation of the impact of the visualization, which we published in the form of a data journalism article.

INTRODUCTION

We present the design of the CO2 Pollution map, a web-based visualization that uses techniques borrowed from motion design to convey semantic information about the data. As visualizations are now often used as "new media" (e. g., for journalistic purposes), we need to bridge the gap between visual communication and information visualization (Infovis). Inspirations can be found in the practices of graphic and motion design. Graphic design can inform us on how to communicate using visual forms, and motion design can provide a wide range of design solutions for creating film-like visual effects. Here, we introduce a way in which these inspirations can impact visualization design choices, and illustrate it with our implementation of the CO2 Pollution map. This visualization presents a visual metaphor for pollution, built upon a particle system that both encodes the data and helps users interpret what they are about. We discuss some of the benefits and tradeoffs of this design, and conclude by presenting early insights into the impact of the design choices we made for the CO2 Pollution map.

BACKGROUND

Three Levels of Visual Communication

Ruedi Baur [START_REF] Baur | Les 101 Mots du Design Graphique a l'Usage de Tous[END_REF] states that the work of a graphic designer is to produce artifacts or documents (e. g., websites, printed documents, or signage systems) that communicate with a user on three levels: an interpretation level, an orientation level, and an information level. On the interpration level, the artifact or document should immediately reveal what it is, what its purpose is, and what it is about. On the orientation level, specific elements of the document should provide cues on how it should be read and used. On the information level, the document should reveal the content and meaning of the communication.

Pollution Maps?

A simple Google search of the terms "co2 pollution map" reveals that most maps that encode pollution related data use heatmap or choropleth encodings. While such encodings are quite efficient on the information and orientation levels-provided they include legends, they fail on the interpretation level, since they do not communicate directly (i. e., without the use of text) what the data are about. Such encodings could just as well represent population densities or temperature. Other ways of designing pollution maps include bubble chart overlays and cartograms,which also fail on the interpretation level.

To produce a visualization that is effective on the interpretation level, we need to create a visual metaphor that allows users to immediately connect the visual features of the display with the semantic nature of the data.

Designing for Interpretation

Michael Danziger [START_REF] Danziger | Information Visualization for the People[END_REF] declares that visualization designers "should try to identify design vectors that position and emphasize Infovis as part of our intellectual, social, and cultural experience, rather than a cryptic, computational artifact that requires an explicit set of skills to decode. [...] Without an immediate cue as to the nature of the data being visualized, a non-committed viewer is likely to skip the graphic." He encourages the development of a semantic design approach for visualization, based on the creation of a structured visual language. A noticeable, working example of this is Wattenberg and Viegas' much acclaimed Wind Map [START_REF] Viegas | Wind map[END_REF]. Interpreting this visualization is almost instantaneous, as the animated trails effectively convey the impression of wind blowing across the map. An amazingly rich source of inspiration for such semantic designs can be found in the film industry, and more precisely in animated films and special effects. Our design of the CO2 Pollution map builds upon some of these techniques.

DESIGNING CO2 EMISSIONS 3.1 Metaphor

To design for interpretation, we need to find the right visual metaphor. Unlike most metaphors used in Infovis, which serve to explicit structural components of the data, or to describe interaction techniques, this metaphor should convey semantic information about the data.

The first question to answer is: "What does the data's topic make me, and others, think of?" For the CO2 Pollution map, we claim that the dark-gray smoke that emanates from exhaust pipes and chimneys reminds us of the presence of CO2 in the air. Therefore, using the combined potentials of d3.js, javascript, and SVG, we extend the bubble chart overlay encoding by transforming bubbles into animated smoke particles, and introduce a visual metaphor for "CO2 emissions."

Particle system

In motion graphics and film post-production, a common way of designing smoke is to use a particle system [8]. Particle systems are a collection of graphic objects (particles) that are dynamic, chaotic, and generated from a source (emitter) [START_REF] Reeves | Particle systems-a technique for modeling a class of fuzzy objects[END_REF]. Important attributes of a particle system are: 1) emitter type, 2) particle type, 3) number of particles emitted (nParticles) within a given time period (tEmit), 4) evolution of individual particles over life, and 5) color of particles. Due to space limitations, we will not describe all the parameters of these attributes in detail. For more information, refer to [START_REF] Giant | Trapcode Particular User Guide[END_REF][START_REF] Reeves | Particle systems-a technique for modeling a class of fuzzy objects[END_REF]. To prevent our particle system from hindering the actual mapping of the data, which uses position (i. e., the particle emitter) and number of visual marks (i. e., nParticles) as encoding visual variables [START_REF] Jansen | An Interaction Model for Visualizations Beyond The Desktop[END_REF], we use five free visual variables for the visual metaphor, namely animation, shape, size, opacity, and color.

For 1), we use a box emitter. In two-dimensional space, boxes are rectangular areas in which particles are generated at random X, Y positions. We use the SVG bounding boxes of countries (previously rendered using a d3-geo-projection) for which data are available.

For 2), we use feathered spheres, as they are SVG compliant (i. e., circles with a blur effect), and well-suited for designing smoke.

For 3), we determine nParticles for each country by transforming the raw values into proportions of the maximum value (i. e., a proportion of the highest amount of CO2 emissions), which we then map to a [0, 20] scale. We set the maximum number of particles to 20 due to technical limitations. SVG elements populate the HTML Document Object Model (DOM) tree, and the more nodes there are, the more computational power is required for rendering the tree. Thus, the highest of all CO2 emissions (≃ 6000 million tones in China in 2007) is displayed using 20 particles, each of which roughly encodes 300 million tones of CO2. At this point, we also set tEmit to 4 seconds.

For 4), the life of each particle (tParticle) is set to match tEmit (i. e., tParticle = tEmit = 4s). This way, when a particle "dies," it is immediately replaced by a new one, insuring that at any point in time, nParticles are displayed for a given country. The evolution of particles over life is composed of three phases: birth (0s), full growth (tParticle/2 = 2s), and death (tParticle); and consists in the animation of two variables: size and opacity. At birth, particle size and opacity are set to 0. At full growth, particle size is set to 10 pixels, with a degree of randomness of [-5, 5] pixels, and opacity is set to 0.5. We use size randomness to ensure that particles never look exactly the same. This gives the smoke a more "realistic" look. In addition, the 0.5 opacity nicely blends overlapping particles, also contributing to the realistic look. At death, particle size and opacity are both set back to 0, and "dead" particles (tParticle> 4s) are removed.

Finally for 5), each particle is given a random color selected along a linear ["dark gray", "light gray"] color scale.

DISCUSSION

The CO2 Pollution map has the benefit of immediately conveying the semantic nature of the data. Informal feedback has confirmed that people interpret "pollution" from the smoke effect. In addition, the map helps uncover certain specificities of the data that other mappings cannot. While the number of visual marks encoding and the use of animation may be sub-par considering the desired pre-attentiveness of visual mappings, the overall density of smoke effectively reveals the concentration of CO2 in different regions. For example, while European countries individually emit relatively low amounts of CO2 (compared to China or the USA), the map shows that together they emit quite a lot (see Fig. 1). Choropleth maps cannot show this, as they do not reveal densities across country borders.

However, there are also some limitations to this design. As it is, extracting specific values is almost impossible. The 300 million tones per particle limitation forces a rough encoding of the data for several countries. A finer encoding could be achieved by re-implementing the visualization using HTML Canvas instead of SVG, because Canvas elements do not populate the DOM. Nevertheless, the use of animation and the randomness of particle positions within the emitter would still prevent precise readings. For example, Fig. 1 shows a high concentration of smoke over the center of the USA, while it should be expected to be centered around the North East and West coasts. This is due to the aggregation of the data on a national scale, and the box emitter we used. A solution would be to find non-aggregated data (i. e., on a local scale), which would allow emitters to be changed to points (i. e., invariant X, Y positions) and centered on individual cities. However, this would increase the density of the visual marks (i. e., nParticles), which would still make it almost impossible to extract precise values.

Overall, the purpose of this visualization is to give an impression of pollution concentration to the viewer. It is not designed to perform advanced visual analytic tasks. Tversky et al. [START_REF] Tversky | Animation: Can it facilitate[END_REF] have indeed emitted warnings about the use of animation, due to the difficulty of perceiving and conceptualizing it. However, Lockyer et al. [START_REF] Lockyer | Simple motion textures for ambient affect[END_REF] have shown that motion textures can successfully communicate affect. We believe more work is needed in the vein of Huber and Healey's analysis of the perceptual properties of motion [START_REF] Huber | Visualizing data with motion[END_REF] to explore how this impression, and the potential affect it communicates, might alter users' ability to perform complex analytical tasks.

Fig. 1 .

 1 Fig. 1. In the Smog: the CO2 Pollution map. Light gray countries indicate that data are unavailable.

EARLY EVALUATION AND CONCLUSION

We published the CO2 Pollution map in the beginning of 2014 in a data-journalism-like article [1] on the Mediapart Club website (french press) [2], and on visualizing.org. It was referenced by visualisingdata.com as one of the "Best of the visualisation web... January 2014" [3]. By April 2014, the visualization had received over 3000 individual browser connections, and the average connection time was ≃ 2min. These simple indicators suggest that the CO2 Pollution map successfully captures viewers' attention. At this early stage, we expect this success to be the result of the emphasis we put on designing for interpretation, i. e., on designing the visual metaphor, although further study is necessary to truly assert this.


Thomas Opitz

Jérome Aze

Sandra Bringay

Cyrille Joutard

Caroline Lavergne

Mollevi

Breast Cancer and Quality of Life: Medical Information Extraction from Health Forums

Keywords: breast cancer, quality of life study, query expansion, social media, text mining

Internet health forums are a rich textual resource with content generated through free exchanges among patients and, in certain cases, health professionals. We tackle the problem of retrieving clinically relevant information from such forums, with relevant topics being defined from clinical auto-questionnaires. Texts in forums are largely unstructured and noisy, calling for adapted preprocessing and query methods. We minimize the number of false negatives in queries by using a synonym tool to achieve query expansion of initial topic keywords. To avoid false positives, we propose a new measure based on a statistical comparison of frequent co-occurrences in a large reference corpus (Web) to keep only relevant expansions. Our work is motivated by a study of breast cancer patients' health-related quality of life (QoL). We consider topics defined from a breast-cancer specific QoLquestionnaire. We quantify and structure occurrences in posts of a specialized French forum and outline important future developments.

Introduction

Health-related quality of life (QoL) is a multidimensional, subjective and dynamic concept with three main domains: physical, psychological and social functioning. Referring to the patient's perception of his treatment and illness, it constitutes a relevant alternative clinical endpoint. In 2012, the number of new cases of breast cancer in France amounts to 48,763 according to the French National Cancer Institute (INCa ). The overall relative survival rate 5 years after diagnosis is estimated at around 2 89% on average. Modern treatments are still exhausting to undergo such that many clinical research projects are now addressing the patients' QoL. We focus on stories told by patients in online health forums, where numerous topics deal with how to cope with symptoms of the illness and secondary treatment effects. Hancock et al. [START_REF] Hancock | The truth about lying in online dating profiles[END_REF] showed that anonymous communication via computers facilitates the expression of affective states such as emotions, opinions and doubts compared to more traditional communication contexts. Recently, the impact of social media on health outcomes has been studied extensively. Merolli et al. [START_REF] Merolli | Health outcomes and related effects of using social media in chronic disease management: A literature review and analysis of affordances[END_REF] survey the positive effects of social media in chronic disease management. Subirats et al. [START_REF] Subirats | Circles of Health: Towards an advanced social network about disabilities of neurological origin[END_REF] affirm that social networks promote knowledge democratization and user-empowerment. In this study, we aim to capture topics of interest in patient stories related to breast cancer. Oncologists may have difficulty to obtain such useful patient-centred information directly from patients. It may 1) assist them to investigate how patients use forums in relation to their individual needs, 2) explain consistency or disparity between different patient health outcomes, 3) help collect patients' individual perceptions and preferences, 4) help improve the construction and validation of questionnaires for traditional clinical studies.

Methods

We seek for posts in specialized forums related to a list of topics of interest coming from questionnaires usually filled in by the patients. The studied French CancerDuSein.org forum with data from 2011 to 2013 holds 16,961 posts from 675 users in more than 1,050 threads, of which some contain more than 500 replies. Semiautomatic information extraction methods for these data remain a significant technological challenge. The text structure is not standardized (slang, spelling and grammatical errors, etc.) and simple text search for relating topics of interest and posts is not effective. Whereas a query such as bouche sèche (=dry mouth) returns many false negatives and misses related occurrences such as langue sèche (=dry tongue), too general queries like bouche return many false positives. Query expansion is therefore necessary, achieved in the following by using web resources to produce and validate topic expansions. Our method consists of 4 steps: 1) message collection, 2) manual topic identification and morphological expansion, 3) automatic synonym-based topic expansion, 4) global synthesis.

Step 1: Message collection. Classical pre-processing steps are applied to overcome difficulties owing to the specificity of the language. Named entities like user names or drugs are identified. A part-of-speech tagger is used to retrieve lemmas and 3 4 to detect unknown words. We replace unknown words if a nearby word is found in a dictionary designed for the project (French words , named entities).

Step 2: Topic identification and morphological expansion. We define a list of topics of interest based on the breast-cancer specific module QLQ-BR23 of the QLQ-C30 questionnaire [START_REF] Turney | Mining the Web for synonyms: PMI-IR versus LSA on TOEFL[END_REF] proposed by the European Organisation for Research and Treatment of Cancer (EORTC), which evaluates function scales, symptom scales and the global health status. An expert manually defines the topics using questionnaire items and scales. Initially, we represent a topic through a topic set TI={TTI} composed of nI initial topic terms TTI, each composed of one or several lemmas without stop words. Basic morphological variations (nouns, verbs, adjectives) with the same word stem are taken into account and yield nM additional topic terms TTM in the morphologically expanded topic set TIM={TTI}∪{TTM}. Topic sets will be used to formulate the query for detecting topic occurrences in forum posts, see Step 4.

Step 3: Automatic synonymic topic expansion. Morphological variations do not cover all lexical variations found in forums. We extend TIM by adding synonymic topic terms TTS such that TIMS=TIM∪{TTS}. Medical terms from the MeSH are eligible for 6 substitution by any related term that appears in the same MeSH hierarchy level. However, the MeSH is limited to technical vocabulary used primarily by health professionals. In addition, we use an online synonym tool to enrich our list of 7 proposed topic expansions. Only synonyms containing no stop words are considered for expansion. Since synonymy depends strongly on the context, we propose a webbased validation to rank synonymic expansions. The originality of our approach is to focus not on the number of occurrences as done by [START_REF] Roche | WeMiT: Web-Mining for Translation[END_REF][START_REF] Salton | Term-weighting approaches in automatic text retrieval[END_REF], but on the similarity of contexts built for the initial topic set TIM and for each potential expansion TTS. The contexts are constructed from the most frequent co-occurrences within a large corpus of documents. For TIM, we use the nI+nM topic terms of TIM queries submitted to an internet search engine . We define a topic context as a weighted feature space based on 8 the top Nsnip search engine snippets for each query. Features taken into account are unigrams and bigrams of lemmatized nouns, verbs and adjectives. The weight of a feature f is calculated according to its frequency in the (nI+nM) x Nsnip snippets, written N(f|TIM), and the frequency in the French language , written N(f|Fr). A formula similar to the common Tf-Idf approach (cf. [9]) is applied to downweight weakly discriminant features:

! (1)
In our experimentations, we double the weight of bigrams, which are strongly discriminant features. Denote by Ei, i=1,…,k the weight of the top k features fi ordered by decreasing weight after removal of features that are part of the proposed topic expansion TTS (expected weights). The context of each topic expansion is constructed analoguously: we submit the query <TTS> to the chosen internet search engine and replace TIM by TTS and (nI+nM) by 1 in Eq. ( 1). Denote by Oi, i=1,…,k, the weight of the features fi, i=1,...,k, in the expansion context (observed weights). To rank a topic expansion TTS with respect to its proximity to TIM, various measures could be applied (cf. [START_REF] Salton | Term-weighting approaches in automatic text retrieval[END_REF]). Here, we propose to use a similarity score S based on a variant of the chisquared statistics

! (2)
The score S takes values between 0 for disjoint contexts and 1 for identical contexts. The truncation at 0 avoids penalizing occurrences that are more frequent in the expansion context than in the topic context. Elevating to the power 2 penalizes the version of the Medical Subject Headings http://mesh.inserm.fr/mesh/ http://synonymo.fr situation where observed weights are considerably smaller than expected weights. We include those topic expansions TTS into the topic set TIMS whose score is sufficiently high, e.g. S > S0 for some fixed threshold 0<S0<1.

Step 4: Global synthesis. We detect topic occurrences in lemmatized forum posts based on queries built from the expanded topic set TIMS. Keywords constituting a topic term are combined by the AND operator "&".Topic terms are then joined by the OR operator "|". The query must be matched in a range of at most Nlem lemmas in a post. Retrieved occurrences can be represented in terms of the number of occurrences per topic, message IDs, phrases of occurrence, etc.

Results

The QLQ-BR23 questionnaire with 23 items is represented by 13 topics, which are aggregated to form 5 topic groups: body image, sexuality, secondary effects (exempt hair loss), hair loss, breast symptoms. For each topic, we calculate the score S for at most 2500 topic expansions. The number of snippets is fixed to Nsnip=40 for each search engine query. We compare contexts based on the k=50 most relevant features of the topic context. Our experimentations suggest adding synonymic expansions TTS with score superior to 0.2 to the set TIMS. The final query used to retrieve occurrences in forum posts is composed of 31 initial topic terms in TI, 70 additional morphological topic expansions in TIM and 596 automatically validated synonymic topic expansions.

When fixing the score threshold S0 below 0.2 (e.g., to 0.1), we found a considerable increase in the number of false positives such that precision degrades, whereas we also find a significant number of additional true positives leading to an improvement in recall. We decided for a manual selection of relevant topic expansions with score between 0.1 and 0.2 by an expert. Figure 1 presents the number of retrieved posts. We attribute these occurrences as follows: occurrences of initial topic terms given by TI always count for type "initial", occurrences of morphological variants count for "morphological" but not for "automated", and only the remaining occurrences are attributed to "automated". The manual validation of a stratified subsample of size 400 by two experts results in a precision of 0.89 with a validation agreement of 0.86 (Cohen's kappa). Automatic topic expansion has significantly increased the number of retrieved posts without loss in precision. However, only a small number of posts related to sexuality and body image were detected.

!

Discussion

A study of QoL through user content posted in health forums presents an interesting alternative to QoL analysis based on the collection of auto-questionnaires. The substantial number of occurrences retrieved for certain topics in our example is promising for a deeper analysis based on a classification of emotions (fear, joy…) associated to the occurrences. The small number of posts related to sexuality and body image is surprising. Lexical variations referring to such subjects may be more heterogeneous and hence more difficult to detect automatically. Moreover, the rare occurrence of these topics is an interesting result in itself since it hints at subjects that play a minor role in the actualities of patients' daily life. Moreover, self-censorship of forum users could be stronger on certain highly intimate subjects. For a proper statistical treatment of our results, we plan to develop classification methods for the retrieval of forum users' patient history from forum data. Finally, to reduce the number of false negatives returned by the topic query, it may be more efficient to construct topics not from existing questionnaires, but in a joint effort of clinical experts and text mining specialists to tailor topics for easier detection. Although forum content is usually publicly available, we stress that results of such approaches should be fully anonymized in order to respect the privacy of individual forum users.

Figure 1 .

 1 Figure 1. Number of retrieved forum posts by topic group and by type of topic term.

http://medicament.comprendrechoisir.com/ http://www.cis.uni-muenchen.de/~schmid/tools/TreeTagger/ http://www.aspell.net

www.yahoo.fr

Acknowledgement

The study was supported by a grant from the French Public Health Research Institute (www.iresp.net) within the 2009-2013 Cancer Plan.


Petra Isenberg
email: petra.isenberg@inria.fr

Tobias Isenberg
email: tobias.isenberg@inria.fr

Michael Sedlmair
email: michael.sedlmair@univie.ac.at

Jian Chen
email: jichen@umbc.edu

Torsten Möller
email: torsten.moeller@univie.ac.at

Visualization According To Research Paper Keywords

We analyzed visualization paper keywords supplied for 4366 papers accepted to three main visualization conferences. We describe main keywords, topic areas, and 10-year historic trends from authorchosen keywords for papers published in the IEEE Visualization conference series (now called IEEE VIS) since 2004. Furthermore, we present the KeyVis Web application that allows visualization researchers to easily browse the 2600+ keywords used for IEEE VIS papers of the past 10 years, aiming at more informed and, hence, more effective keyword selections for future visualization publications and efficient search for related work.

INTRODUCTION

The field of visualization is an extremely diverse and, thus, exciting field to be a part of. Yet, its diversity also creates enormous challenges. There are different levels of appreciation for all aspects of visualization research, communication challenges between visualization researchers, and the challenge of communicating visualization as a research science to the outside. These issues lead, in particular, to the frequently asked question "what is visualization?"-among funding agencies or even between colleagues. Yet, while "what is visualization?" is a fundamental question, it is little discussed within our community. What is particularly missing in this picture is a bottom-up analysis: What types of visualization research are actually happening as expressed by single research contributions in the visualization conferences and journals. Our analysis is one of the first steps in this direction. We analyze author-assigned keywords from the three IEEE VisWeek/VIS conferences of the past ten years and based on this analysis, we contribute:

• a conceptual map of all visualization work as indexed by individual authors

• KeyVis, a Web-based search tool that makes the keyword metadata available to a broad set of people.

RELATED WORK

Similar to our goal, in other disciplines specific techniques have been used to analyze the scientific literature more broadly: to get a better sense of global research trends, links and patterns within the scientific literature. Co-word analysis is one approach among others (e. g., co-citation analysis) that has tackled the problem by analyzing the scientific literature according to the co-occurrence of keywords, words in titles, abstracts, or even in the full texts of scientific articles [START_REF]Mapping the Dynamics of Science and Technology[END_REF]. Co-word analysis has been used in different research areas, e. g., polymer chemistry [START_REF] Callon | Co-word analysis as a tool for describing the network of interactions between basic and technological research: The case of polymer chemistry[END_REF], acid rain research [START_REF] Law | Policy and the mapping of scientific change: A co-word analysis of research into environmental acidification[END_REF], or education [START_REF] Ritzhaupt | An investigation of distance education in North American research literature using coword analysis[END_REF]. Liu et al.'s work [START_REF] Liu | CHI 1994-2013: Mapping two decades of intellectual progress through co-word analysis[END_REF] is most related to ours. The authors examined papers of the ACM CHI conference from 1994-2013, identified research themes and their evolution, and classified individual keywords as popular, core, or backbone. We employ similar approaches but extend the work by an expert coding process. We also naturally differ as our focus is on a different research community with different keywords, trends, and patterns and a different historical evolution.

CO-WORD ANALYSIS METHODOLOGY

For our analysis of the visualization research literature we collected keywords freely assigned by the authors to their research papesr accepted to IEEE VisWeek/VIS from 2004-2013. The dataset contained 1097 published papers (excluding posters). Out of these, 58 contained no author-assigned keywords, yielding a set of 1039 papers we considered in our analysis. These papers contained a total of 2823 unique keywords. Next, we engaged in an extensive manual cleaning pass in which we consolidated keywords that were the same but presented either as singulars/plurals, with spelling mistakes, or as acronyms. This yielded a cleaned dataset that contained 2629 unique keywords. Based on this data, we engaged in a manual expert coding in order to find higher-level clusters of keyword topics.

The resulting set of keywords contained 156 unique higher-level keywords that occurred a total of 4026 times across all papers (this number is lower than for the cleaned data as potentially generated duplicates per paper were removed).

To analyze the keyword datasets we first filtered the data and and removed keywords that occurred less than ten times and also excluded higher-level terms (visualization, information visualization, scientific visualization, visual analytics). Next, we generated document-keyword matrices and performed a correlation computation. On each resulting correlation matrix we performed a hierarchical clustering using Ward's method and a squared Euclidean distance metric. We also generated a keyword network in which two keywords were linked if their correlation was > 0 and each link was assigned its respective correlation value.

RESULTS

Individual Keywords

The top three most frequently occurring keywords across all three conferences were volume rendering (148×), hardware and computation (89×), and graphs (73×). Interestingly, there was little overlap between the most common keywords per individual conference. Interaction emerged for both IEEE InfoVis and VAST as the only shared keyword amongst the top three per conference:

VAST:

analysis process (53), interaction (31), applications (24) InfoVis: graphs (73), interaction (69), evaluation (53) Vis/SciVis: volume rendering (148), hardware and computation (89), flow (70)

When expanding the analysis to the top 10 keywords, application emerges as a keyword common to all three conferences. InfoVis and VAST also share the keyword evaluation (ranked 14 in Vis/SciVis) while VAST and Vis/SciVis share the keyword time (ranked 12 in InfoVis). We can conclude that all three sub-communities share some joint concerns (e. g., applications, evaluation, time) but that they do have their respective foci.

Next, we were interested in historic trends for individual keywords. We used Tableau to calculate linear trend lines for the 15 most frequently used keywords. According to this calculation, the terms interaction, evaluation, multidimensional/multivariate, and machine learning & statistics are significantly on the rise. In contrast, volume visualization, flow visualization, and hardware and computation have been significantly decreasing in frequency of occurrence over the past 10 years (see Fig. 1). It is interesting to note that two very core and frequent keywords for the IEEE Vis/SciVis conference are in significant decline. This could perhaps be due to the fact the many fundamental research questions have been tackled and that researchers are now using more specific or other keywords.

Keyword Clusters

Next we were interested to see which keywords were often used together to index papers and would, thus, describe the relatedness between visualization topics. We used the methodology outlined in Sect. 3 to derive topic clusters and network diagrams. Fig. 2 shows the keyword map in which keywords are connected if their correlation was ≥ 0.11, meaning that keywords were frequently used together to index individual papers. Node colors further indicate cluster memberships. In the following, we describe all clusters by their two most common keywords and the color used in Fig. 2.

Based on analyzing clusters with high density (keywords closely connected to one another) and centrality (many connections to other clusters) within the network [START_REF]Mapping the Dynamics of Science and Technology[END_REF], we found that three motor themes emerged for the community based on high density and centrality: 1-dark pink: surfaces, numerical methods/mathematics, 2-dark blue: flow, topology, 3-gray: volume rendering, hardware and computation. All three themes are traditional topics of IEEE Vis/SciVis which also has the longest history of the three IEEE VIS conferences.

Undeveloped but central themes in the network were characterized by high centrality and low density: 1-purple: displays, toolkits/systems/environments, 2-light green: data and data management, dimensionality reduction, 3-light pink: large scale data, cameras & views, 4-light purple: time, focus+context.

KEYVIS.ORG: A KEYWORD SEARCH TOOL

To make our data accessible for others, we created a webpage that makes author and expert keywords and related papers search-and browsable: http://www.keyvis.org/. Visitors can search all 2629 unique author-assigned keywords, find out which keywords cooccurred how frequently, which manual expert clusters they belong to, and the actual research papers they appear on. Our main goal was to generate an easy-to-use, lightweight interface to our keyword data in order to: (a) support visualization researchers in making more informed decisions when picking keywords for their papers, and (b) give a new lens on identifying relevant related work. We have used the site ourselves for choosing keywords and finding related work that we were not aware of before. We hope that others will find it similarly useful. In the long run, our goal is to maintain the website as a platform for visualization keyword access and analysis.

NEXT STEPS

Our analysis is a first step towards two larger research goals: Creating a common vocabulary: One can think about the problem of creating a common vocabulary for visualization more broadly. By identifying key terms and providing clear definitions, sub-communities in visualization may be able to communicate more clearly about similar approaches and, this, in turn can help to also collaborate more effectively with people outside the community. Finally, a common vocabulary can also facilitate to more easily understand emerging and declining research trends within the field.

Establish a comprehensive taxonomy of visualization research: Keyword analysis could be a first step to develop a taxonomy of visualization research. This could serve two purposes. One the one hand, a taxonomy will help to better communicate "what is visualization" to other disciplines, i. e., researchers and practitioners not part of the VIS community. On the other hand, we are hoping to be able to facilitate the crucial step of matching reviewers with papers and grants such that the peer review process improves and new contributions are seen in the right context.

Figure 1 :

 1 Figure 1: Keywords significantly on the rise or in decline.

Figure 2 :

 2 Figure 2: Keyword map from clustering of the coded keywords; showing only connected nodes with correlation strength ≥ 0.11. Circle areas correspond to number of occurrences of the keyword in the dataset, link width corresponds to correlation strength, and color distinguished node clusters. We chose a link threshold based on visual inspection of the resulting graph to generate a manageable and readable layout. Isolated nodes were removed from the image.


Viorel Barbu

Michael R Öckner

Francesco Russo

Stochastic Porous Media Equations in R d Equations de milieux poreux dans R d

Keywords: Stochastic, porous media, Wiener process, maximal monotone graph, distributions. 2010 MSC: 76S05, 60H15, 35K55, 35K67

Existence and uniqueness of solutions to the stochastic porous media equation dX -∆ψ(X)dt = XdW in R d are studied. Here, W is a Wiener process, ψ is a maximal monotone graph in R × R such that ψ(r) ≤ C|r| m , ∀r ∈ R. In this general case, the dimension is restricted to d ≥ 3, the main reason being the absence of a convenient multiplier result in the space

When ψ is Lipschitz, the well-posedness, however, holds for all dimensions on the classical Sobolev space

d+2 , we prove the finite time extinction with strictly positive probability.

Résumé

Nous étudions existence et unicité pour les solutions d'une équation de milieux poreux dX -∆ψ(X)dt = XdW dans R d . Ici W est un processus de Wiener, ψ est un graphe maximal monotone dans R × R tel que ψ(r) ≤ C|r| m , ∀r ∈ R. Dans ce contexte général, la dimension est restreinte à d ≥ 3, essentiellement compte tenu de l'absence d'un résultat adéquat de multiplication dans l'espace H = {ϕ ∈ S ′ (R d ); |ξ|(F ϕ)(ξ) ∈ L 2 (R d )}, pour d ≤ 2. Lorsque ψ est Lipschitz, le problème est néanmoins bien posé pour toute dimension dans l'espace de Sobolev classique H -1 (R d ). Si ψ(r)r ≥ ρ|r| m+1 et m = d-2 d+2 , nous prouvons une propriété d' extinction en temps fini avec probabilité strictement positive.

Introduction

Consider the stochastic porous media equation

dX -∆ψ(X)dt = XdW in (0, T ) × R d , X(0) = x on R d , (1.1)
where ψ is a monotonically nondecreasing function on R (eventually multivalued) and W(t) is a Wiener process of the form

Email addresses: vbarbu41@gmail.com (Viorel BARBU), roeckner@math.uni-bielefeld.de (Michael R ÖCKNER), francesco.russo@ensta-paristech.fr (Francesco RUSSO) Preprint submitted to Journal de mathématiques pures et appliquées September 9, 2014

W(t) = ∞ k=1
µ k e k β k (t), t ≥ 0.

(1.2)

Here {β k } ∞ k=1 are independent Brownian motions on a stochastic basis {Ω, F , F t , P}, µ k ∈ R and {e k } ∞ k=1 is an orthonormal basis in H -1 (R d ) or H -1 (see (2.2) below) to be made precise later on.

On bounded domains O ⊂ R d with Dirichlet homogeneous boundary conditions, equation (1.1) was studied in [START_REF] Barbu | Existence and uniqueness of nonnegative solutions to the stochastic porous media equation[END_REF], [START_REF] Barbu | Existence of strong solutions for stochastic porous media equation under general monotonicity conditions[END_REF], [START_REF] Barbu | Stochastic porous media equations and self-organized criticality[END_REF], under general assumptions on ψ : R → R (namely, maximal monotone multivalued graph with polynomial growth, or even more general growth conditions in [START_REF] Barbu | Existence of strong solutions for stochastic porous media equation under general monotonicity conditions[END_REF]). It should be said, however, that there is a principial difference between bounded and unbounded domains, mainly due to the multiplier problem in Sobolev spaces on R d . If d ≥ 3 and O = R d , existence and uniqueness of solutions to (1.1) was proved in [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF] (see, also, [START_REF] Röckner | Non-monotone stochastic generalized porous media equations[END_REF]) in a general setting which covers the case O = R d (see Theorems 3.9, Proposition 3.1 and Example 3.4 in [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF]). However, it should be said that in [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF] ψ is assumed continuous, such that rψ(r) → ∞ as r → ∞, which we do not need in this paper.

We study the existence and uniqueness of (1.1) under two different sets of conditions requiring a different functional approach. The first one, which will be presented in Section 3, assumes that ψ is monotonically nondecreasing and Lipschitz. The state space for (1.1) is, in this case, H -1 (R d ), that is, the dual of the classical Sobolev space H 1 (R d ). In spite of the apparent lack of generality (ψ Lipschitz), it should be mentioned that there are physical models described by such an equation as, for instance, the two phase Stefan transition problem perturbed by a stochastic Gaussian noise [START_REF] Barbu | The two phase stochastic Stefan problem[END_REF]; moreover, in this latter case there is no restriction on the dimension d.

The second case, which will be studied in Section 4, is that where ψ is a maximal monotone multivalued function with at most polynomial growth. An important physical problem covered by this case is the self-organized criticality model

dX -∆H(X -X c )dt = (X -X c )dW, (1.3)
where H is the Heaviside function and X c is the critical state (see [START_REF] Barbu | Stochastic porous media equations and self-organized criticality[END_REF], [START_REF] Barbu | Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise[END_REF], [START_REF] Barbu | Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions[END_REF]). More generally, this equation with discontinuous ψ covers the stochastic nonlinear diffusion equation with singular diffusivity D(u) = ψ ′ (u).

It should be mentioned that, in this second case, the solution X(t) to (1.1) is defined in a certain distribution space H -1 (see (2.2) below) on R d and the existence is obtained for d ≥ 3 only, as in the case of continuous ψ in [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF]. The case 1 ≤ d ≤ 2 remains open due to the absence of a multiplier rule in the norm • H -1 (see Lemma 4.1 below).

In Section 5, we prove the finite time extinction of the solution X to (1.1) with strictly positive probability under the assumption that ψ(r)r ≥ ρ|r| m+1 and m = d-2 d+2 • Finally, we would like to comment on one type of noise. Existence and uniqueness can be proved with g(t, X(t)) by replacing X(t) under (more or less the usual) abstract conditions on σ (see, e.g., [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF], [START_REF] Röckner | Non-monotone stochastic generalized porous media equations[END_REF]). The main reason why in this paper we restrict ourselves to linear multiplicative noise is that first we want to be concrete, second the latter case is somehow generic (just think of taking the Taylor expansion of σ(t, •) up to first order), and third for this type of noise we prove finite time extinction in Section 5.

Preliminaries

To begin with, let us briefly recall a few definitions pertaining distribution spaces on R d , whose classical Euclidean norm will be denoted by | • |.

Denote by S ′ (R d ) the space of all temperate distributions on R d (see, e.g., [START_REF] Hörmander | Linear partial differential operators[END_REF]) and by H the space

H = {ϕ ∈ S ′ (R d ); ξ → |ξ|F (ϕ)(ξ) ∈ L 2 (R d )}, (2.1)
where F (ϕ) is the Fourier transform of ϕ. We denote by L 2 (R d ) the space of square integrable functions on

R d with norm | • | 2 and scalar product •, • 2 . In general | • | p will denote the norm of L p (R d ) or L p (R d ; R d ), 1 ≤ p ≤ ∞.
The dual space H -1 of H is given by

H -1 = {η ∈ S ′ (R d ); ξ → F (η)(ξ)|ξ| -1 ∈ L 2 (R d )}. (2.
2)

The duality between H and H -1 is denoted by •, • and is given by

ϕ, η = R d F (ϕ)(ξ)F (η)(ξ)dξ (2.3)
and the norm of H denoted by • 1 is given by

ϕ 1 = R d |F (ϕ)(ξ)| 2 |ξ| 2 dξ 1 2 = R d |∇ϕ| 2 dξ 1 2
.

(2.4)

The norm of H -1 , denoted by • -1 is given by

η -1 = R d |ξ| -2 |F (η)(ξ)| 2 dξ 1 2 = (-∆) -1 η, η 1 2 .
(2.5) (We note that the operator -∆ is an isomorphism from H onto H -1 .) The scalar product of H -1 is given by

η 1 , η 2 -1 = (-∆) -1 η 1 , η 2 . (2.6)
As regards the relationship of H with the space L p (R d ) of p-summable functions on R d , we have the following.

Lemma 2.1. Let d ≥ 3. Then we have

H ⊂ L 2d d-2 (R d ) (2.7)
algebraically and topologically.

Indeed, by the Sobolev embedding theorem (see, e.g., [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF], p. 278), we have

|ϕ| 2d d-2 ≤ C|∇ϕ| 2 , ∀ϕ ∈ C ∞ 0 (R d ),
and, by density, this implies (2.7), as claimed.

It should be mentioned that (2.7) is no longer true for 1 ≤ d ≤ 2. However, by duality, we have

L 2d d+2 (R d ) ⊂ H -1 , ∀d ≥ 3. (2.8)
Denote by H 1 (R d ) the Sobolev space

H 1 (R d ) = {u ∈ L 2 (R d ); ∇u ∈ L 2 (R d )} = {u ∈ L 2 (R d ); ξ → F (u)(ξ)(1 + |ξ| 2 ) 1 2 ∈ L 2 (R d )} with norm |u| H 1 (R d ) = R d (u 2 + |∇u| 2 )dξ 1 2 = R d |F u(ξ)| 2 (1 + |ξ| 2 )dξ 1 2
and by H -1 (R d ) its dual, that is,

H -1 (R d ) = {u ∈ S ′ (R d ); F (u)(ξ)(1 + |ξ| 2 ) -1 2 ∈ L 2 (R d )}.
The norm of H -1 (R d ) is denoted by | • | -1 and its scalar product by •, • -1 . We have the continuous and dense embeddings

H 1 (R d ) ⊂ H, H -1 ⊂ H -1 (R d ).
It should be emphasized, however, that H is not a subspace of L 2 (R d ) and so L 2 (R d ) is not the pivot space in the duality •, • given by (2.3).

Given a Banach space Y, we denote by L p (0, T ; Y) the space of all Y-valued p-integrable functions on (0, T ) and by C([0, T ]; Y) the space of continuous Y-valued functions on [0, T ]. For two Hilbert spaces H 1 , H 2 let L(H 1 , H 2 ) and L 2 (H 1 , H 2 ) denote the set of all bounded linear and Hilbert-Schmidt operators, respectively. We refer to [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF], [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF] for definitions and basic results pertaining infinite dimensional stochastic processes.

Equation (1.1) with the Lipschitzian ψ

Consider here equation (1.1) under the following conditions.

(i) ψ : R → R is monotonically nondecreasing, Lipschitz such that ψ(0) = 0.

(ii) W is a Wiener process as in (1.2), where

e k ∈ H 1 (R d ), such that C 2 ∞ := 36 ∞ k=1 µ 2 k (|∇e k | 2 ∞ + |e k | 2 ∞ + 1) < ∞, (3.1)
and

{e k } is an orthonormal basis in H -1 (R d ).
We insert the factor 36 for convenience here to avoid additional large numerical constants in subsequent estimates.

X ∈ L 2 (Ω; C([0, T ]; H -1 (R d ))) ∩ L 2 ([0, T ] × Ω; L 2 (R d )) (3.2) • 0 ψ(X(s))ds ∈ C([0, T ]; H 1 (R d )), P-a.s. (3.3) X(t) -∆ t 0 ψ(X(s))ds = x + t 0 X(s)dW(s), ∀t ∈ [0, T ], P-a.s. (3.4) Remark 3.3. The stochastic (Itô-) integral in (3.4
) is the standard one from [START_REF] Da Prato | Stochastic equations in infinite dimensions[END_REF] or [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]. In fact, in the terminology of these references,

W is a Q-Wiener process W Q on H -1 , where Q : H -1 → H -1
is the symmetric trace class operator defined by

Qh := ∞ k=1 µ 2 k e k , h -1 e k , h ∈ H -1 . For x ∈ H -1 , define σ(x) : Q 1/2 H -1 → H -1 by σ(x)(Q 1/2 h) = ∞ k=1 (µ k e k , h -1 e k • x), h ∈ H. (3.5)
By (3.1), each e k is an H -1 -multiplier such that

|e k • x| -1 ≤ 2 (|e k | ∞ + |∇e k | ∞ ) |x| -1 , x ∈ H -1 . (3.6) Hence, for all x ∈ H -1 , h ∈ H -1 , ∞ k=1 |µ k e k , h -1 e k x| -1 ≤        ∞ k=1 µ 2 k |e k x| 2 -1        1/2 |h| -1 ≤ 2C ∞ |x| -1 |h| -1 = 2C ∞ |x| -1 |Q 1/2 h| Q 1/2 H -1 ,
and thus σ(x) is well-defined and an element in L(Q 1/2 H -1 , H -1 ). Moreover, for x ∈ H -1 , by (3.5), (3.6),

σ(x) 2 L 2 (Q 1/2 H -1 ,H -1 ) = ∞ k=1 |σ(x)(Q 1/2 e k )| 2 -1 = ∞ k=1 |µ k e k x| 2 -1 = ∞ k=1 µ 2 k |e k x| 2 -1 ≤ C 2 ∞ |x| 2 -1 . (3.7) Since {Q 1/2 e k | k ∈ N} is an orthonormal basis of Q 1/2 H -1 , it follows that σ(x) ∈ L 2 (Q 1/2 H -1 , H -1
) and the map x → σ(x) is linear and continuous (hence Lipschitz) from

H -1 to L 2 (Q 1/2 H -1 , H -1
:= t 0 σ(X(s))dW Q (s), t ∈ [0, T ],
is well-defined as a continuous H -1 -valued martingale and by Itô's isometry and (3.7)

E t 0 X(s)dW(s) 2 -1 = ∞ k=1 µ 2 k E t 0 |X(s)e k | 2 -1 ds ≤ C 2 ∞ E t 0 |X(s)| 2 -1 ds, t ∈ [0, T ]. (3.8)
Furthermore, it follows that

t 0 X(s)dW(s) = ∞ k=1 t 0 σ(X(s))(Q 1/2 e k )dβ k (s) = ∞ k=1 t 0 µ k e k X(s)dβ k (s), t ∈ [0, T ], (3.9)
where the series converges in L 2 (Ω; C([0, T ]; H -1 )). In fact,

• 0 X(s)dW(s) is a continuous L 2 -valued martingale, because X ∈ L 2 ([0, T ] × Ω; L 2 (R d ))
and, analogously to (3.7), we get

σ(x) 2 L 2 (Q 1/2 H -1 ,L 2 ) ≤ C 2 ∞ |x| 2 2 , x ∈ L 2 (R d ). 5
In particular, by Itô's isometry,

E t 0 X(s)dW(s) 2 2 ≤ C 2 ∞ E t 0 |X(s)| 2 2 ds, t ∈ [0, T ].
Furthermore, the series in (3.8) even converges in L 2 (Ω; C([0, T ]; L 2 (R d ))). We shall use the facts presented in this remark throughout this paper without further notice.

Theorem 3.4. Let d ≥ 1 and x ∈ L 2 (R d ). Then, under assumptions (i), (ii), there is a unique strong solution to equation (1.1). This solution satisfies

E sup t∈[0,T ] |X(t)| 2 2 ≤ 2|x| 2 2 e 3C 2 ∞ t .
In particular,

X ∈ L 2 (Ω; L ∞ ([0, T ]; L 2 (R d ))). Assume further that ψ(r)r ≥ αr 2 , ∀r ∈ R, (3.10)
where α > 0. Then, there is a unique strong solution X to (1.1) for all x ∈ H -1 (R d ).

Proof of Theorem 3.4. We approximate (1.1) by

dX + (ν -∆)ψ(X)dt = XdW(t), t ∈ (0, T ), X(0) = x on R d , (3.11)
where ν ∈ (0, 1). We have the following.

Lemma 3.5. Assume that ψ is as in assumption (i). Let x ∈ L 2 (R d ). Then, there is a unique (F t ) t≥0 -adapted solution X = X ν to (3.11) in the following strong sense:

X ν ∈ L 2 (Ω, C([0, T ]; H -1 (R d ))) ∩ L 2 ([0, T ] × Ω; L 2 (R d )), (3.12)
and P-a.s.

X ν (t) = x + (∆ -ν) t 0 ψ(X ν (s))ds + t 0 X ν (s)dW(s), t ∈ [0, T ]. (3.13)
In addition, for all ν ∈ (0, 1),

E sup t∈[0,T ] |X ν (t)| 2 2 ≤ 2|x| 2 2 e 3C 2 ∞ T . (3.14)
If, moreover, ψ satisfies (3.10), then for each x ∈ H -1 (R d ) there is a unique solution X ν satisfying (3.12), (3.13).

Proof of Lemma 3.5. Let us start with the second part of the assertion, i.e., we assume that ψ satisfies (3.10) and that x ∈ H -1 (R d ). Then the standard theory (see, e.g., [20, Sections 4.1 and 4.2]) applies to ensure that there exists a unique solution X ν taking value in H -1 (R d ) satisfying (3.12), (3.13) above. Indeed, it is easy to check that (H1)-(H4) from [20, Section 4.1] are satisfied with

V := L 2 (R d ), H := H -1 (R d ), Au := (∆ -ν)(ψ(u)), u ∈ V, and H -1 (R d ) is equipped with the equivalent norm |η| -1,ν := η, (ν -∆) -1 η 1/2 , η ∈ H -1 (R d ),
(in which case, we also write H -1 ν ). Here, as before, we use •, • also to denote the dualization between H 1 (R d ) and H -1 (R d ). For details, we refer to the calculations in [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Example 4.1.11], which because p = 2 go through when the bounded domain Ω there is replaced by R d . Hence [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Theorem 4.2.4] applies to give the above solution X ν .

In the case when ψ does not satisfy (3.10), the above conditions (H1), (H2), (H4) from [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF] still hold, but (H3) not in general. Therefore, we replace ψ by ψ+λI, λ ∈ (0, 1), and thus consider A λ (u) := (∆-ν)(ψ(u)+λu), u ∈ V := L 2 (R d ) and, as above, by [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Theorem 4.2.4], obtain a solution X ν λ , satisfying (3.12), (3.13), to

dX ν λ (t) + (ν -∆)(ψ(X ν λ (t)) + λX ν λ (t))dt = X ν λ (t)dW(t), t ∈ [0, T ], X ν λ (0) = x ∈ H -1 (R d ).
(3.15)

In particular, by (3.12),

E sup t∈[0,T ] |X ν λ (t)| 2 -1 < ∞. (3.16)
We want to let λ → 0 to obtain a solution to (3.11). To this end, in this case (i.e., without assuming (3.10)), we assume from now on that x ∈ L 2 (R d ). The reason is that we need the following.

Claim 1. We have X ν λ ∈ L 2 ([0, T ] × Ω; H 1 (R d )) and E sup t∈[0,T ] |X ν λ (t)| 2 2 + 4λE T 0 |∇X ν λ (s)| 2 2 ds ≤ 2|x| 2 2 e 3C 2 ∞ T ,
for all ν, λ ∈ (0, 1).

Furthermore, X ν λ has continuous sample paths in L 2 (R d ), P-a.s.

Proof of Claim 1. We know that

X ν λ (t) = x + (∆ -ν) t 0 (ψ(X ν λ (s)) + λX ν λ (s))ds + t 0 X ν λ (s)dW(s), t ∈ [0, T ]. (3.17) Let α ∈ (ν, ∞). Recalling that (α -∆) -1 2 : H -1 (R d ) → L 2 (R d
) and applying this operator to the above equation, we find

(α -∆) -1 2 X ν λ (t) = (α -∆) -1 2 x + t 0 (∆ -ν)(α -∆) -1 2 (ψ(X ν λ (s)) + λX ν λ (s))ds + t 0 (α -∆) -1 2 σ(X ν λ (s))Q 1/2 dW(s), t ∈ [0, T ]. (3.18)
Applying Itô's formula (see, e.g., [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Theorem 4.2.5] with

H = L 2 (R d )) to |(α -∆) -1 2 X ν λ (t)| 2 2 , we obtain, for t ∈ [0, T ], |(α -∆) -1 2 X ν λ (t)| 2 2 = |(α -∆) -1 2 x| 2 2 +2 t 0 (∆ -ν)(α -∆) -1 2 ψ(X ν λ (s)), (α -∆) -1 2 X ν λ (s) ds -2λ t 0 (|∇((α -∆) -1 2 X ν λ (s))| 2 2 + ν|(α -∆) -1 2 X ν λ (s)| 2 2 )ds + t 0 (α -∆) -1 2 σ(X ν λ (s))Q 1/2 2 L 2 (H -1 ,L 2 ) ds +2 t 0 (α -∆) -1 2 X ν λ (s), (α -∆) -1 2 σ(X ν λ (s))Q 1/2 dW(s) 2 . (3.19) But, for f ∈ L 2 (R d ), we have (α -∆) -1 2 (∆ -ν)(α -∆) -1 2 f = (P -I) f,
where

P := (α -ν)(α -∆) -1 .
For the Green function g α of (α -∆), we then have, for f ∈ L 2 (R d ),

P f = (α -ν) R d f (ξ)g α (•, ξ)dξ.
Hence, by [START_REF] Röckner | Non-monotone stochastic generalized porous media equations[END_REF]Lemma 5.1], the integrand of the second term on the right-hand side of (3.19) with

f := X ν λ (s) (∈ L 2 (R d ) for ds-a.e. s ∈ [0, T ]) can be rewritten as ψ( f ), (P -I) f 2 = - 1 2 R d R d [ψ( f ( ξ))-ψ( f (ξ))][ f ( ξ)-f (ξ)]g α (ξ, ξ)d ξ dξ - R d (1 -P1(ξ)) • ψ( f (ξ)) f (ξ)dξ.
Since ψ is monotone, ψ(0) = 0 and P1 ≤ 1, we deduce that

ψ( f ), (P -I) f ≤ 0.
Hence, after a multiplication by α, (3.19) implies that, for all t ∈ [0, T ] (see Remark 3.3),

α|(α -∆) -1 2 X ν λ (t)| 2 2 + 2λ t 0 |∇( √ α(α -∆) -1 2 X ν λ (s))| 2 2 ds ≤ α|(α -∆) -1 2 x| 2 2 + t 0 ∞ k=1 µ 2 k α(α -∆) -1 (e k X ν λ (s)), e k X ν λ (s) 2 ds +2 t 0 α(α -∆) -1 X ν λ (s), σ(X ν λ (s))Q 1/2 dW(s) 2 .
Hence, by the Burkholder-Davis-Gundy (BDG) inequality (with p = 1) and since α

(α -∆) -1 is a contraction on L 2 (R d ), E sup s∈[0,t] | √ α(α -∆) -1 2 X ν λ (s)| 2 2 +2λE t 0 |∇( √ α(α -∆) -1 2 X ν λ (s))| 2 2 ds ≤ | √ α(α -∆) -1 2 x| 2 2 + C 2 ∞ E t 0 |X ν λ (s)| 2 2 ds +6E        t 0 ∞ k=1 µ 2 k α(α -∆) -1 X ν λ (s), e k X ν λ (s) 2 2 ds        1/2 . (3.20)
The latter term can be estimated by

C ∞ E        sup s∈[0,t] |α(α -∆) -1 X ν λ (s)| 2 t 0 |X ν λ (s)| 2 2 ds 1/2        ≤ 1 2 E sup s∈[0,t] | √ α(α -∆) -1 2 X ν λ (s)| 2 2 + 1 2 C 2 ∞ E t 0 |X ν λ (s)| 2 2 ds, (3.21)
where we used that

√ α(α-∆) -1 2 is a contraction on L 2 (R d ).
Note that the first summand on the right-hand side is finite by (3.16)

, since the norm | √ α(α -∆) -1 2 • | 2 is equivalent to | • | -1 .
Hence, we can subtract this term after substituting (3.21) into (3.20) to obtain

E sup s∈[0,t] | √ α(α -∆) -1 2 X ν λ (s)| 2 2 +4λE t 0 |∇( √ α(α -∆) -1 2 X ν λ (s)| 2 2 ds ≤ 2| √ α(α -∆) -1 2 x| 2 2 + 3C 2 ∞ E t 0 |X ν λ (s)| 2 2 ds, t ∈ [0, T ]. (3.22)
Obviously, the quantity under the sup

s∈[0,t]
on the left-hand side of (3.22) is increasing in α. So, by the monotone convergence theorem, we may let α → ∞ in (3.22) and then, except for its last part, Claim 1 immediately follows by Gronwall's lemma, since [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Theorem 4.2.5]), it follows from (3.17) that, for λ, λ ′ ∈ (0, 1) and t ∈ [0, T ],

√ α(α -∆) -1 2 is a contraction in L 2 (R d ) and x ∈ L 2 (R d ). The last part of Claim 1 then immediately follows from [19, Theorem 2.1]. Applying Itô's formula to |X ν λ (t) -X ν λ ′ (t)| 2 -1,ν (see
|X ν λ (t) -X ν λ ′ (t)| 2 -1,ν +2 t 0 ψ(X ν λ ) -ψ(X ν λ ′ ) + (λX ν λ -λ ′ X ν λ ′ ), X ν λ -X ν λ ′ 2 ds = t 0 σ(X ν λ (s) -X ν λ ′ (s) 2 L 2 (Q 1/2 H -1 ,H -1 ν ) ds +2 t 0 X ν λ (s) -X ν λ ′ (s), σ(X ν λ (s) -X ν λ ′ (s))dW Q (s) -1,ν . (3.23)
Our assumption (i) on ψ implies that

(ψ(r) -ψ(r ′ ))(r -r ′ ) ≥ (Lip ψ + 1) -1 |ψ(r) -ψ(r ′ )| 2 , for r, r ′ ∈ R,
where Lip ψ is the Lipschitz constant of ψ. Hence (3.23), (3.7) and the BDG inequality (for p = 1 imply that, for all t ∈ [0, T ])

E sup s∈[0,t] |X ν λ (s) -X ν λ ′ (s)| 2 -1,ν +2(Lip ψ + 1) -1 E t 0 |ψ(X ν λ (s)) -ψ(X ν λ ′ (s))| 2 2 ds ≤ 2(λ + λ ′ )E t 0 (|X ν λ (s)| 2 2 + |X ν λ ′ (s)| 2 2 )ds + C 2 ∞ t 0 |X ν λ (s) -X ν λ ′ (s)| 2 -1,ν ds +2E        t 0 ∞ k=1 µ 2 k X ν λ (s) -X ν λ ′ (s), (X ν λ (s) -X ν λ ′ (s))e k 2 -1,ν ds        1/2
. By (3.7) and Young's inequality, the latter term is dominated by

1 2 E sup s∈[0,t] |X ν λ (s) -X ν λ ′ (s)| 2 -1,ν + 1 2 C 2 ∞ E t 0 |X ν λ (s) -X ν λ ′ (s)| 2 -1,ν ds.
Hence, because of x ∈ L 2 (R d ) and Claim 1, we may now apply Gronwall's lemma to obtain that, for some constant C independent of λ ′ , λ (and ν),

E sup t∈[0,T ] |X ν λ (t) -X ν λ ′ (t)| 2 -1,ν + E T 0 |ψ(X ν λ (s)) -ψ(X ν λ ′ (s))| 2 2 ds ≤ C(λ + λ ′ ). (3.24)
Hence there exists an (F t )-adapted continuous

H -1 -valued process X ν = (X ν (t)) t∈[0,T ] such that X ν ∈ L 2 (Ω; C([0, T ]; H -1 )). Now, by Claim 1, it follows that X ν ∈ L 2 ([0, T ] × Ω; L 2 (R d )).
Claim 2. X ν satisfies equation (3.13) (i.e., we can pass to the limit in (3.17) as λ → 0).

Proof of Claim 2. We already know that

X ν λ -→ X ν and • 0 X ν λ (s)dW(s) -→ • 0 X ν (s)dW(s) in L 2 (Ω; C([0, T ]; H -1
)) as λ → 0 (for the second convergence see the above argument using (3.7) and the BDG inequality). So, by (3.17) it follows that

• 0 (ψ(X ν λ (s)) + λX ν λ (s)))ds, λ > 0,
converges as λ → 0 to an element in L 2 (Ω; C([0, T ]; H 1 ). But, by (3.24) and Claim 1, it follows that

• 0 (ψ(X ν λ (s)) + λX ν λ (s))ds -→ • 0 ψ(X ν (s))ds (3.25) as λ → 0 in L 2 (Ω; L 2 ([0, T ]; L 2 (R d )))
. Hence Claim 2 is proved. Now, (3.14) follows from Claim 1 by lower semicontinuity. This completes the proof of Lemma 3.5.

Proof of Theorem 3.4 (continued). We are going to use Lemma 3.5 and let ν → 0. The arguments are similar to those in the proof of Lemma 3.5. So, we shall not repeat all the details. Now, we rewrite (3.11) as

dX ν + (I -∆)ψ(X ν )dt = (1 -ν)ψ(X ν )dt + X ν dW(t) (3.26)
and apply Itô's formula to ϕ(x) = 1 2 |x| 2 -1 (see, e.g., [START_REF] Prévôt | A concise course on stochastic partial differential equations[END_REF]Theorem 4.2.5]). We get, for x ∈ H -1 , by (3.8) and after taking expectation,

1 2 E|X ν (t)| 2 -1 + E t 0 R d ψ(X ν (s))X ν (s)dξ ds = 1 2 |x| 2 -1 + (1 -ν)E t 0 ψ(X ν (s)), X ν (s) -1 ds + 1 2 E t 0 ∞ k=1 µ 2 k |X ν e k | 2 -1 ds ≤ 1 2 |x| 2 -1 + E t 0 |ψ(X ν )| -1 |X ν | -1 ds + 1 2 C 2 ∞ E t 0 |X ν (s)| 2 -1 ds, ∀t ∈ [0, T ]. Recalling that | • | -1 ≤ | • | 2 ,
we get, via Young's and Gronwall's inequalities, for some C ∈ (0, ∞) that

E|X ν (t)| 2 -1 + α 2 E T 0 |X ν (s)| 2 2 ds ≤ C|x| 2 -1 , t ∈ [0, T ], ν ∈ (0, 1), (3.27)
because, by assumption (i), ψ(r)r ≥ α|ψ(r)| 2 , ∀r ∈ R, with α := (Lip ψ + 1) -1 . Here we set α = 0 if (3.10) does not hold. Now, by a similar calculus, for X ν -X ν ′ we get

|X ν (t) -X ν ′ (t)| 2 -1 + 2 t 0 R d (ψ(X ν ) -ψ(X ν ′ ))(X ν -X ν ′ )dξ ds ≤ C t 0 ψ(X ν ) -ψ(X ν ′ ), X ν -X ν ′ -1 ds +C t 0 (ν|ψ(X ν )| 2 + ν ′ |ψ(X ν ′ )| 2 )|X ν -X ν ′ | -1 ds +C t 0 |X ν -X ν ′ | 2 -1 ds + ∞ k=1 t 0 µ k (X ν -X ν ′ ), e k (X ν -X ν ′ ) -1 dβ k , t ∈ [0, T ].
Taking into account that, by assumption (i),

(ψ(x) -ψ(y))(x -y) ≥ α|ψ(x) -ψ(y)| 2 , ∀x, y ∈ R d ,
we get, for all ν, ν ′ > 0,

|X ν (t) -X ν ′ (t)| 2 -1 + α t 0 |ψ(X ν (s)) -ψ(X ν ′ (s))| 2 2 ds ≤ C 1 t 0 |X ν (s) -X ν ′ (s)| 2 -1 ds + α 2 t 0 |ψ(X ν (s)) -ψ(X ν ′ (s))| 2 2 ds +C 2 (ν + ν ′ ) t 0 (|ψ(X ν (s))| 2 2 + |ψ(X ν ′ (s))| 2 2 )ds + ∞ k=1 t 0 µ k (X ν (s) -X ν ′ (s)), e k (X ν (s) -X ν ′ (s)) -1 dβ k (s), t ∈ [0, T ].
So, similarly to showing (3.24) in the proof of Lemma 3.5, by (3.14), if x ∈ L 2 (R d ), and by (3.27), if x ∈ H -1 (R d ) and ψ satisfies (3.10), by the Burkholder-Davis-Gundy inequality, for p = 1, we get, for all ν, ν ′ ∈ (0, 1),

E sup t∈[0,T ] |X ν (t) -X ν ′ (t)| 2 -1 + E T 0 |ψ(X ν (s)) -ψ(X ν ′ (s))| 2 2 ds ≤ C(ν + ν ′ ).
The remaining part of the proof is now exactly the same as the last part of the proof of Lemma 3.5.

Remark 3.6. Theorem 3.4 is a basic tool for the probabilistic (double) representation of equation (1.1), which holds when ψ is Lipschitz, as it is proved in [START_REF] Barbu | A stochastic Fokker-Planck equation and double probabilistic representation for the stochastic porous media type equation[END_REF]. If (1.1) is not perturbed by noise, and ψ is possibly discontinuous, its probabilistic representation was performed in [START_REF] Blanchard | Probabilistic representation for solutions of an irregular porous media type equation[END_REF], [START_REF] Barbu | Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case[END_REF], [START_REF] Barbu | A stochastic Fokker-Planck equation and double probabilistic representation for the stochastic porous media type equation[END_REF] with extensions and numerical simulations located in [START_REF] Belaribi | A probabilistic algorithm approximating solutions of a singular pde of porous media type[END_REF], [START_REF] Belaribi | Probabilistic and deterministic algorithms for space multidimensional irregular porous media equation[END_REF].

Equation (1.1) for maximal monotone functions ψ with polynomial growth

In this section, we assume d ≥ 3 and we shall study the existence for equation (1.1) under the following assumptions:

(j) ψ : R → 2 R is a maximal monotone graph such that 0 ∈ ψ(0) and

sup{|η|; η ∈ ψ(r)} ≤ C(1 + |r| m ), ∀r ∈ R, (4.1)
where 1 ≤ m < ∞.

(jj) W(t) = ∞ k=1
µ k e k β k (t), t ≥ 0, where {β k } ∞ k=1 are independent Brownian motions on a stochastic basis {Ω, F , F t , P},

µ k ∈ R, and e k ∈ C 1 (R d ) ∩ H -1 are such that {e k } is an orthonormal basis in H -1 and ∞ k=1 µ 2 k (|e k | 2 ∞ + |∇e k | 2 d + 1) < ∞. (4.2)
The existence of {e k } as in (jj) is ensured by the following lemma.

Lemma 4.1. Let d ≥ 3 and let e ∈ L ∞ (R d ; R d ) be such that ∇e ∈ L d (R d ; R d ). Then xe -1 ≤ x -1 (|e| ∞ + C|∇e| d ), ∀x ∈ H -1 , (4.3)
where C is independent of x and e.

Proof. We have

xe -1 = sup{ x, eϕ ; ϕ 1 ≤ 1} ≤ x -1 sup{ eϕ 1 ; ϕ 1 ≤ 1}. (4.4)
On the other hand, by Lemma 2.1 we have, for all ϕ ∈ C ∞ 0 (R d ),

eϕ 1 ≤ |e∇ϕ + ϕ∇e| 2 ≤ |e∇ϕ| 2 + |ϕ∇e| 2 ≤ |e| ∞ |∇ϕ| 2 + |ϕ| p |∇e| d ≤ |e| ∞ ϕ 1 + C ϕ 1 |∇e| d ,
where p = 2d d-2 • Then, by (4.4), (4.3) follows, as claimed. Remark 4.2.

(i) It should be mentioned that, for d = 2, Lemma 4.1 fails and this is the main reason our treatment of equation (1.1) under assumptions (j), (jj) is constrained to d ≥ 3.

(ii) We note that Remark 3.3 with the rôle of H -1 (R d ) replaced by H -1 remains true in all its parts under condition (jj) above. We shall use this below without further notice.

We denote by j : R → R the potential associated with ψ, that is, a continuous convex function on R such that ). An H -1 -valued adapted process X = X(t) is called strong solution to (1.1) if the following conditions hold:

∂ j = ψ, i.e., j(r) ≤ ζ(r -r) + j(r), ∀ζ ∈ ψ(r), r, r ∈ R.
X is H -1 -valued continuous on [0, T ], P-a.s., (4.5)
X ∈ L p (Ω × (0, T ) × R d ). (4.6)
There is η ∈ L p m (Ω × (0, T ) × R d ) such that η ∈ ψ(X), dt ⊗ P ⊗ dξ -a.e. on (0, T ) × Ω × R d (4.7)

and P-a.s.

X(t) = x + ∆ t 0 η(s)ds + ∞ k=1 µ k t 0 X(s)e k dβ k (s) (4.8) in D ′ (R d ), t ∈ [0, T ].
Here D ′ (R d ) is the standard space of distributions on R d . Theorem 4.4 below is the main existence result for equation (1.1).

Theorem 4.4. Assume that d ≥ 3 and that

x ∈ L p (R d ) ∩ L 2 (R d ) ∩ H -1 , p := max(2, 2m).
Then, under assumptions (j), (jj), there is a unique solution X to (1.1) such that

X ∈ L 2 (Ω; C([0, T ]; H -1 )). (4.9)
Moreover, if x ≥ 0, a.e. in R d , then X ≥ 0, a.e. on (0, T ) × R d × Ω.

Theorem 4.4 is applicable to a large class of nonlinearities ψ : R → 2 R and, in particular, to

ψ(r) = ρH(r) + αr, ∀r ∈ R, ψ(r) = ρH(r -r c )r,
where ρ > 0, α, r c ≥ 0, which models the dynamics of self-organized criticality (see [START_REF] Barbu | Stochastic porous media equations and self-organized criticality[END_REF], [START_REF] Barbu | Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise[END_REF], [START_REF] Barbu | Stochastic porous media equations and self-organized criticality: convergence to the critical state in all dimensions[END_REF]). Here H is the Heaviside function.

As mentioned earlier, Theorem 4.4 can be compared most closely to the main existence result of [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF]. But there are, however, a few notable differences as we explain below. The function ψ arising in [START_REF] Ren | Stochastic generalized porous media and fast diffusion equations[END_REF] is monotonically increasing, continuous and are assumed to satisfy a growth condition of the form N(r) ≤ rψ(r) ≤ C(N(r) + 1)r, ∀r ∈ R, where N is a smooth and ∆ 2 -regular Young function defining the Orlicz class L N . In contrast to this, here ψ is any maximal monotone graph (multivalued) with arbitrary polynomial growth.

Proof of Theorem 4.4. Consider the approximating equation

dX λ -∆(ψ λ (X λ ) + λX λ )dt = X λ dW, t ∈ (0, T ), X λ (0) = x, ( 4.10)
where

ψ λ = 1 λ (1 -(1 + λψ) -1
), λ > 0. We note that ψ λ = ∂ j λ , where (see, e.g., [START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF])

j λ (r) = inf |r -r| 2 2λ + j(r); r ∈ R , ∀r ∈ R.
We have the following result.

Lemma 4.5. Let x ∈ H -1 ∩ L p (R d ) ∩ L 2 (R d ), p := 2m, d ≥ 3.
Then (4.10) has a unique solution

X λ ∈ L 2 (Ω; C([0, T ]; H -1 )) ∩ L ∞ ([0, T ]; L p (Ω × R d )). (4.11)
Moreover, for all λ, µ > 0, we have

E sup 0≤t≤T (X λ (t) -X µ (t)) 2 -1 ≤ C(λ + µ) (4.12) E|X λ (t)| p p ≤ C|x| p p , ∀t ∈ [0, T ], (4.13)
E T 0 R d |ψ λ (X λ )| p m dt dξ ≤ C|x| p p , ∀λ > 0, (4.14) E sup 0≤t≤T X λ (t) 2 -1 ≤ C x 2 -1 , ∀λ > 0, (4.15)
where C is independent of λ, µ.

Proof. We consider for each fixed λ the equation (see (3.11))

dX ν λ + (ν -∆)(ψ λ (X ν λ ) + λX ν λ )dt = X ν λ dW X ν λ (0) = x, (4.16)
where

ν > 0. Let x ∈ L 2 (R d ) ∩ L p (R d ) ∩ H -1
. By Claim 1 in the proof of Lemma 3.5, (4.16) has a unique solution

X ν λ ∈ L 2 (Ω; L ∞ ([0, T ]; L 2 (R d ))) ∩ L 2 (Ω × [0, T ]; H 1 (R d )) with continuous sample paths in L 2 (R d ).
As seen in the proof of Theorem 3.4, we have, for ν → 0,

X ν λ → X λ strongly in L 2 (Ω; C([0, T ]; H -1 (R d ))) weak-star in L 2 (Ω; L ∞ ([0, T ]; L 2 (R d ))),
and, by (3.14), along a subsequence also, where X λ is the solution to (4.10). It remains to be shown that X λ satisfies (4.11)-(4.15). In order to explain the ideas, we apply first (formally) Itô's formula to (4.16) for the function ϕ(x) = 1 p |x| p p . We obtain

1 p E|X ν λ (t)| p p + E t 0 R d (ν -∆)(ψ λ (X ν λ ) + λX ν λ )|X ν λ | p-2 X ν λ ds dξ = 1 p |x| p p + p -1 2 E t 0 R d ∞ k=1 µ 2 k |X ν λ e k | 2 |X ν λ | p-2 dt dξ.
(4.17)

Taking into account that X ν λ , ψ λ (X ν λ ) ∈ L 2 (0, T ; H 1 (R d )), P-a.s., by Claim 1 in the proof of Lemma 3.5, we have

t 0 R d (ν -∆)(ψ λ (X ν λ ) + λX ν λ )|X ν λ | p-2 X ν λ ds dξ ≥ λ(p -1) t 0 R d |∇X ν λ | 2 |X ν λ | p-2 dξ ds,
and by (4.2) we have

E t 0 R d ∞ k=1 µ 2 k |X ν λ e k | 2 |X ν λ | p-2 ds dξ ≤ C ∞ E t 0 R d |X ν λ | p dξ ds < ∞.
Then, we obtain by (4.17) via Gronwall's lemma

E|X ν λ (t)| p p ≤ C|x| p p , t ∈ (0, T ), (4.18)
and, by (4.1),

E t 0 R d |ψ λ (X ν λ )| p m dt dξ ≤ C|x| p p , t ∈ [0, T ]. (4.19)
It should be said, however, that the above argument is formal, because the function ϕ is not of class C 2 on L 2 (R d ) and we do not know a priori if the integral in the left side of (4.17) makes sense, that is, whether

|X ν λ | p-2 X ν λ ∈ L 2 (0, T ; L 2 (Ω; H 1 (R d ))
). To make it rigorous, we approximate X ν λ by a sequence {X ν,ε λ } of solutions to the equation

dX ν,ε λ + A ν,ε λ (X ν,ε λ )dt = X ν,ε λ dW, X ν,ε λ (0) = x. (4.20)
Here, A ν,ε λ = 1 ε (I -(I + εA ν λ ) -1 ), ε ∈ (0, 1), is the Yosida approximation of the operator

A ν λ x = (ν -∆)(ψ λ (x) + λx), ∀x ∈ D(A ν λ ) = H 1 (R d ). We set J ε = (I + εA ν λ ) -1 and note that J ε is Lipschitz in H = H -1 (R d ) as well as in all L q (R d ) for 1 < q < ∞. Moreover, we have |J ε (x)| q ≤ |x| q , ∀x ∈ L q (R d ), (4.21) see [3], Lemma 3.1. Since A ν,ε λ is Lipschitz in H, equation (3.1
) has a unique adapted solution X ν,ε λ ∈ L 2 (Ω; C([0, T ]; H) and by Itô's formula we have

1 2 E|X ν,ε λ (t)| 2 -1 ≤ 1 2 |x| 2 -1 + C 1 ∞ k=1 µ 2 k E t 0 |X ν,ε λ (s)e k | 2 -1 ds,
which, by virtue of (jj), yields

E|X ν,ε λ (t)| 2 -1 ≤ C 2 |x| 2 -1 , ∀ε > 0, x ∈ H. (4.22) Similarly, since A ν,ε λ is Lipschitz in L 2 (R d ) (see Lemma 4.6 below), we have also that X ν,ε λ ∈ L 2 (Ω; C([0, T ]; L 2 (R 2 )
)) and, again by Itô's formula applied to the function |X ν,ε λ (t)| 2 2 , we obtain

E|X ν,ε λ (t)| 2 2 ≤ 1 2 |x| 2 2 + C 3 ∞ k=1 µ 2 k E ∞ 0 |X ν,ε λ (s)e k | 2 2 ds,
which yields, by virtue of (jj),

E|X ν,ε λ (t)| 2 2 ≤ C 4 |x| 2 2 , ∀t ∈ [0, T ]. (4.23) Claim 1. For p ∈ [2, ∞) and x ∈ L p (R d ), we have that X ν,ε λ ∈ L ∞ W ([0, T ]; L p (Ω; L p (R d )) ∩ L 2 (Ω; L 2 (R d )))
, where here and below the subscript W refers to (F t )-adapted processes.

Proof. For R > 0, consider the set

K R = {X ∈ L ∞ W ([0, T ]; L p (Ω; L p (R d )) ∩ L 2 (Ω; L 2 (R d ))), e -pαt E|X(t)| p p ≤ R p , e -2αt E|X(t)| 2 2 ≤ R 2 , t ∈ [0, T ]}. Since, by (4.20), X ν,ε λ is a fixed point of the map X F -→ e -t ε X + 1 ε t 0 e -t-s ε J ε (X(s))ds + t 0 e -(t-s) ε X(s)dW(s), obtained by iteration in C W ([0, T ]; L 2 (Ω; H ∩ L 2 (R d ))
), it suffices to show that F leaves the set K R invariant for R > 0 large enough. By (4.21), we have

       e -pαt E e -t ε x + 1 ε t 0 e -t-s ε J ε (X(s))ds p p        1 p ≤ e -( 1 ε +α)t |x| p + e -αt t 0 1 ε e -(t-s) ε (E|X(s)| p p ) 1 p ds ≤ e -( 1 ε +α)t |x| p + R 1 + αε , (4.24)
and, similarly, that

       e -2αt E e -t ε x + 1 ε t 0 e -(t-s) ε J ε (X(s))ds 2 2        1 2 ≤ e -( 1 ε +α)t |x| 2 + R 1 + αε . (4.25)
Now, we set

Y(t) = t 0 e -(t-s) ε X(s)dW(s), t ≥ 0. We have dY + 1 ε Y dt = X dW, t ≥ 0, Y(0) = 0.
Equivalently,

d(e t ε Y(t)) = e t ε X(t)dW(t), t > 0; Y(0) = 0.
By Lemma 5.1 in [START_REF] Krylov | Itô's formula for the L p -norm of stochastic W 1 p -valued processes[END_REF], it follows that e t ε Y is an L p (R d )-valued (F t )-adapted continuous process on [0, ∞) and

E|e t ε Y(t)| p p = 1 2 p(p -1) ∞ k=1 µ 2 k E t 0 R d |e s ε Y(s)| p-2 |e s ε X(s)e k | 2 ds.
This yields via Hypothesis (jj)

E|e t ε Y(t)| p p ≤ 1 2 (p -1)E t 0 |e s ε Y(s)| p p ds + CE t 0 |e s ε X(s)| p p ds, ∀t ∈ [0, T ],
and, therefore,

E|Y(t)| p p ≤ C 1 e -(α+ 1 ε )pt E t 0 |e s ε X(s)| p p ds ≤ R p e -pαt εC 1 p(1 + εα) , ∀t ∈ [0, T ].
Similarly, we get

e -2αt E|Y(t)| 2 2 ≤ R 2 εC 1 2(1 + εα) , ∀t ∈ [0, T ].
Then, by formulae (4.24), (4.25), we infer that, for α large enough and R > 2(|x| p + |x| 2 ), F leaves K R invariant, which proves Claim 1.

Claim 2. We have, for all p ∈ [2, ∞) and x ∈ L p (R d ), that there exists C p ∈ (0, ∞) such that

ess sup t∈[0,T ] E|X ν,ε λ (t)| p p ≤ C p for all ε, λ, ν ∈ (0 , 1). (4.26)
Proof. Again invoking Lemma 5.1 in [START_REF] Krylov | Itô's formula for the L p -norm of stochastic W 1 p -valued processes[END_REF], we have by (4.20) that X ν,ε λ satisfies

E|X ν,ε λ (t)| p p = |x| p p -p E t 0 R d A ν,ε λ (X ν,ε λ )X ν,ε λ |X ν,ε λ | p-2 dξ ds +p(p -1) ∞ k=1 µ 2 k E t 0 R d |X ν,ε λ | p-2 |X ν,ε λ e k | 2 dξ ds. (4.27) On the other hand, A ν,ε λ (X ν,ε λ ) = 1 ε (X ν,ε λ -J ε (X ν,ε λ )
) and so we have

R d A ν,ε λ (X ν,ε λ )X ν,ε λ |X ν,ε λ | p-2 dξ = 1 ε R d |X ν,ε λ | p dξ - 1 ε R r J ε (X ν,ε λ )|X ν,ε λ | p-2 X ν,ε λ dξ.
Recalling (4.21), we get, via the Hölder inequality,

R d A ν,ε λ (X ν,ε λ )X ν,ε λ |X ν,ε λ | p-2 dξ ≥ 0,
and so, by (4.27) and Hypothesis (jj), we obtain, via Gronwall's lemma, estimate (4.26), as claimed.

Claim 3. We have, for ε → 0,

X ν,ε λ -→ X ν λ strongly in L ∞ W ([0, T ]; L 2 (Ω; H)) and weakly * in L ∞ ([0, T ]; L p (Ω; L p (R d )) ∩ L 2 (Ω; L 2 (R d ))).
Proof. For simplicity, we write X ε instead of X ν,ε λ and X instead of X ν λ . Also, we set γ(r) ≡ ψ λ (r) + λr. Subtracting equations (4.20) and (4.16), we get via Itô's formula and because A ν,ε λ is monotone on H

1 2 E|X ε (t) -X(t)| 2 -1,ν + E t 0 R d (γ(J ε (X)) -γ(X))(X ε -X)dξ ds ≤ CE t 0 |X ε (s) -X(s)| 2 -1,ν ds,
and hence, by Gronwall's lemma, we obtain

E|X ε (t) -X(t)| 2 -1,ν ≤ CE T 0 R d |γ(J ε (X)) -γ(X)||X ε -X|dξ ds. (4.28)
On the other hand, it follows by (4.21) that

Ω×[0,T ]×R d |J ε (X)| 2 P(dω)dt dξ ≤ Ω×[0,T ]×R d |X| 2 P(dω)dt dξ, while, for ε → 0, J ε (y) -→ y in H -1 , ∀y ∈ H -1 , (because A ν,ε λ is maximal monotone in H -1 (R d )) and so, J ε (X(t, ω)) -→ X(t, ω) in H -1 (R d ) for all (t, ω) ∈ (0, T ) × Ω. Hence, as ε → 0, J ε (X) -→ X weakly in L 2 (Ω × [0, T ] × R d ), (4.29)
and, according to the inequality above, this implies that, for ε → 0,

|J ε (X)| L 2 ((0,T )×Ω×R d ) -→ |X| L 2 ((0,T )×Ω×R d ) .
Hence, J ε (X) -→ X strongly in L 2 (Ω×[0, T ]×R d ) as ε → 0. Now, taking into account that γ is Lipschitz, we conclude by (4.28), (4.29) and by estimates (4.23), (4.26) that Claim 3 is true. Now, we can complete the proof of Lemma 4.5. Namely, letting first ε → 0 and then ν → ∞ in (4.26), we get (4.13) and hence (4.14) as desired. Now, let us prove (4.12) and (4.15). Arguing as in the proof of Theorem 3.4, we obtain

1 2 |X ν λ (t)| 2 -1,ν + t 0 R d (ψ λ (X ν λ ) + λX ν λ )X ν λ dξ ds = 1 2 |x| 2 -1,ν + 1 2 t 0 R d ∞ k=1 µ 2 k |X ν λ e k | 2 -1,ν dξ ds + t 0 X ν λ , X ν λ dW -1,ν ds. (4.30) Keeping in mind that, by (4.3), |X ν λ e k | -1,ν ≤ C|X ν λ | -1,ν (|e k | ∞ + |∇e k | d ),
where C is independent of ν, we obtain by the Burkholder-Davis-Gundy inequality for p = 1 (cf. the proof of Theorem 3.4)

E sup t∈[0,T ] |X ν λ (t)| 2 -1,ν + λE T 0 |X ν λ | 2 2 ds ≤ C|x| 2 -1,ν .
Taking into account that lim

ν→0 |y| -1,ν = y -1 , ∀y ∈ H -1 ,
we obtain, as in Theorem 3.4 (see the part following (3.26)), that

E sup t∈[0,T ] X λ (t) 2 -1 + λE T 0 |X λ (t)| 2 2 dt ≤ C x 2 -1 , ∀λ > 0, (4.31)
where C is independent of λ. In particular, (4.15) holds.

Completely similarly, one proves (4.12). Namely, we have

d(X ν λ -X ν µ ) + (ν -∆)(ψ λ (X ν λ ) + λX ν λ -ψ µ (X ν µ ) -µX ν µ )dt = (X ν λ -X ν µ )dW
and again proceeding as in the proof of Theorem 3.4, we obtain as above that

1 2 |X ν λ (t) -X ν µ (t)| 2 -1,ν + t 0 R d (ψ λ (X ν λ ) + λX ν λ -ψ µ (X ν µ ) -µX ν µ )(X ν λ -X ν µ )dξ ds = 1 2 t 0 R d ∞ k=1 µ 2 k |(X ν λ -X ν µ )e k | 2 -1,ν ds + t 0 X ν λ -X ν µ , (X ν λ -X ν µ )dW -1,ν , t ∈ [0, T ].
Then, applying once again the Burkholder-Davis-Gundy inequality for p = 1, and the fact that, by Hypothesis (j), |ψ λ (r)| ≤ C|r| m , ∀ ∈ R with C independent of λ, we get, proceeding as in the proof of Theorem 3.4, that

E sup t∈[0,T ] |X ν λ (t) -X ν µ (t)| 2 -1 ≤ C(λ + µ),
where C is independent of ν, λ, µ. (For details, we refer to the proof of (3.10), (3.14) in [START_REF] Barbu | Stochastic porous media equations and self-organized criticality[END_REF]). Letting ν → 0 as in the previous case, we obtain (4.12), as claimed. This completes the proof of Lemma 4.5.

Above we have used the lemma below.

Lemma 4.6. A ν,ε λ is Lipschitz in L 2 (R d ). Proof. It suffices to check that J ε is Lipschitz in L 2 (R d ). We set γ(r) = ψ λ (r) + λr. We have, for x, x ∈ L 2 (R d ), J ε (x) -J ε ( x) -ε∆(γ(J ε (x)) -γ(J ε ( x))) = x -x.
Multiplying by γ(J ε (x))γ(J ε ( x)) in L 2 (R d ), we get

J ε (x) -J ε ( x), γ(J ε (x)) -γ(J ε ( x)) 2 ≤ |γ(J ε (x)) -γ(J ε ( x))| 2 |x -x| 2 .
Taking into account that (γ(r)γ(r))(rr) ≥ L|r -r|, ∀r, r ∈ R, and that γ is Lipschitz, we get

|J ε (x) -J ε ( x)| 2 ≤ C|x -x| 2 ,
as claimed.

Proof of Theorem 4.4 (continued). By (4.12)-(4.15), it follows that there is a process X ∈ L ∞ ([0, T ]; L p (Ω × R d )) such that, for λ → 0, To prove that η ∈ ψ(X), a.e. in Ω × (0, T ) × R d , it suffices to show that, for each ϕ ∈ C ∞ 0 (R d ), we have

X λ → X weak-star in L ∞ ([0, T ]; L p (Ω × R d )) λX λ → 0 strongly in L 2 ([0, T ]; L 2 (Ω × R d )) ψ λ (X λ ) → η weakly in L p m ([0, T ] × Ω × R d ) X λ → X strongly in L 2 (Ω; C([0, T ]; H -1 )).
lim sup λ→0 E T 0 R d ϕ 2 ψ λ (X λ )X λ dt dξ ≤ E T 0 R d ϕ 2 ηX dξ dt. ( 4.34)
Indeed, we have by convexity of j λ

E T 0 R d ϕ 2 ψ λ (X λ )(X λ -Z)dξ dt ≥ E T 0 R d ϕ 2 ( j λ (X λ ) -j λ (Z))dξ dt, ∀Z ∈ L p ((0, T ) × Ω × R d ),
and so, by (4.32) and (4.34), we see that

E T 0 R d ϕ 2 (η(X -Z))dtdξ ≥ E T 0 R d ϕ 2 ( j(X) -j(Z))dξ dt, ∀Z ∈ L p ((0, T ) × Ω × R d ),
because, for λ → 0, j λ (Z) → j(Z), and j λ (X λ ) → j(X), a.e. and thus, by Fatou's lemma

lim inf λ→0 E T 0 R d ϕ 2 j λ (X λ )dξ dt ≥ E T 0 R d ϕ 2 j(X)dξ dt.
Now, we take ϕ ∈ C ∞ 0 (R d ) to be non-negative, such that ϕ = 1 on B N and ϕ = 0, outside B N+1 where for a given N ∈ N, B N is the closed ball of R d with radius N. We get

E T 0 B N+1 ϕ 2 (η(X -Z))dξ dt ≥ E T 0 R d ϕ 2 ( j(X) -j(Z))dξ dt, ∀Z ∈ L p ((0, T ) × Ω × R d ). (4.35)
This yields

E T 0 B N+1 ϕ 2 η(X -Z)dξ dt ≥ E T 0 B N+1 ϕ 2 ζ(X -Z)dξ dt, ( 4
Ω × B N+1 ), that is, ψ(Z) = ζ ∈ L p ′ ((0, T ) × Ω × B N+1 ), ζ ∈ ψ(Z), a.e. . Since m p ≤ p ′ with 1 p ′ = 1 -1 p , by virtue of assumption (j), ψ is maximal monotone in L p ((0, T ) × Ω × B N+1 ) × L p ′ ((0, T ) × Ω × B N+1
), and so, the equation

J(Z) + ψ(Z) ∋ J(X) + η, (4.37)
where J(Z) = |Z| p-2 Z, has a unique solution (Z, η) (see, e.g., [START_REF] Barbu | Nonlinear differential equations of monotone types in Banach spaces[END_REF], p. 31). If, in (4.36), we take Z the solution to (4.37), we obtain that

E T 0 B N+1 ϕ 2 (J(X) -J(Z))(X -Z)dtdξ ≤ 0.
Then, choosing α = 2 p , yields

E T 0 B N+1 |ϕ α X| p-2 ϕ α X -|ϕ α Z| p-2 ϕ α Z (ϕ α X -ϕ α Z)dtdξ ≤ 0.
Consequently, this gives

E T 0 R d (J(ϕ α X) -J(ϕ α Z))(ϕ α X -ϕ α Z)dtdξ ≤ 0. (4.38)
On the other hand, we have

J(ϕ α X) -J(ϕ α Z) = (p -1)|λϕ α X + (1 -λ)ϕ α Z| p-2 (X -Z),
for some λ = λ(X, Z) ∈ [0, 1]. Substituting into (4.38) yields

|ϕ α (X -Z)| 2 = 0 a.e. in (0, T ) × Ω × B N+1 , Hence, X = Z on (0, T ) × Ω × B N .
Coming back to (4.37), this gives η ∈ ψ(X), dtdPdξ, a.e., because N is arbitrary.

To prove (4.34), we use the Itô formula in (4.16) to x → 1 2 ϕx 2 -1 to get, as in (4.30),

1 2 E ϕX ν λ (t) 2 -1 + E t 0 (-∆) -1 (ν -∆)(ψ λ (X ν λ ) + λX ν λ , ϕ 2 X ν λ ) ds ≤ 1 2 ϕx 2 -1 + 1 2 E t 0 ∞ k=1 µ 2 k ϕX ν λ e k 2 -1 ds.
Then, letting ν → 0, we obtain

1 2 E ϕX λ (t) 2 -1 + E t 0 ψ λ (X λ ) + λX λ , ϕ 2 X λ 2 ds ≤ 1 2 ϕx 2 -1 + 1 2 E t 0 ∞ k=1 µ 2 k ϕX λ e k 2 -1 ds. (4.39)
On the other hand, by (4.33) we get similarly

1 2 E ϕX(t) 2 -1 + E t 0 η(s), ϕ 2 X 2 ds = 1 2 ϕx 2 -1 + 1 2 E 1 0 ∞ k=1 µ 2 k ϕXe k 2 -1 , t ∈ [0, T ].
Comparing with (4.39), we obtain (4.34), as claimed. If x ≥ 0, a.e. in R d , it follows that X ≥ 0, a.e. in in Ω × (0, T ) × R d . To prove this, one applies Itô's formula in (4.16) to the function x → |x -| 2 2 and get (X ν λ ) -= 0, a.e. in Ω × (0, T ) × R d . Then, for ν → 0, we obtain the desired result. This completes the existence proof for

x ∈ L 2 (R d ) ∩ L p (R d ) ∩ H -1 . Uniqueness. If X 1 , X 2 are two solutions, we have d(X 1 -X 2 ) -∆(η 1 -η 2 )dt = (X 1 -X 2 )dW, t ∈ (0, T ), (X 1 -X 2 )(0) = 0, where η i ∈ ψ(X i ), i = 1, 2, a.e. in Ω × (0, T ) × R d .
Applying again, as above (that is, via the approximating device) Itô's formula in

H -1 to 1 2 ϕ(X 1 -X 2 ) 2 -1 , where ϕ ∈ C ∞ 0 (R d ), we get that 1 2 d ϕ(X 1 -X 2 ) 2 -1 -∆(η 1 -η 2 ), ϕ(X 1 -X 2 ) -1 = 1 2 ∞ k=1 µ 2 k ϕ(X 1 -X 2 )e k 2 -1 dt + (X 1 -X 2 ), ϕ(X 1 -X 2 )dW -1 = 0. Note that, since η 1 -η 2 ∈ L p m (Ω × (0, T ) × R d ), we have -E T 0 ∆(η 1 -η 2 ), ϕ(X 1 -X 2 ) -1 dt = E T 0 R d (η 1 -η 2 ), ϕ(X 1 -X 2 )dt dξ ≥ 0,
and, therefore,

E ϕ(X 1 (t) -X 2 (t)) 2 -1 ≤ C t 0 E ϕ(X 1 -X 2 ) 2 -1 ds, ∀t ∈ [0, T ],
and, since ϕ was arbitrary in C ∞ 0 (R d ), we get X 1 ≡ X 2 , as claimed. Remark 4.7. The self-organized criticality model (1.3), that is, ψ(r) ≡ H(r) = Heaviside function, which is not covered by Theorem 4.4 for 1 ≤ d ≤ 2, can, however, be treated in the special case By approximating W by a smooth W ε ∈ C 1 ([0, T ]; R) and letting ε → 0, after some calculation one concludes that the latter equation has a unique strong solution Y. We omit the details, but refer to [START_REF] Barbu | On a random scaled porous media equation[END_REF] for a related treatment.

W(t) = N j=1 µ j β j (t), µ j ∈ R,

The finite time extinction

Assume here that ψ satisfies condition (j) of the beginning of Section 4 and that W is of the form (jj). Moreover, one assumes that

ψ(r)r ≥ ρ|r| m+1 , ∀r ∈ R, (5.1)
where m is as in Hypothesis (j).

Theorem 5.1.

Let d ≥ 3 and m = d-2 d+2 . Let x ∈ L m+1 (R d ) ∩ L 2 (R d ) ∩ H -1
and let X = X(t); t ∈ [0, T ], be the solution to (1.1) given by Theorem 4.4. We set

τ = inf{t ≥ 0; X(t, •) -1 = 0}.
(5.2)

Then, for every t > 0,

X(t) = 0, ∀t ≥ τ, (5.3)
and

P[τ ≤ t] ≥ 1 -x 1-m -1 C * ργ m+1 (1 -e -C * (1-m)t ) . (5.4)
where γ -1 = sup{ u -1 |u| -1 m+1 ; u ∈ L m+1 } and C * > 0 is independent of the initial condition x. Proof. We follow the arguments of [START_REF] Barbu | Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise[END_REF]. The basic inequality is

X(t) 1-m -1 + ρ(1 -m)γ m+1 t r 1 [ X(s) -1 >0] ds ≤ X(r) 1-m -1 + C * (1 -m) t r X(s) 1-m -1 ds +(1 -m) t r X(s) (m+1) -1 X(s), X(s)dW(s) -1 , P-a.s., 0 < r < t < ∞, (5.5)
where C * is a suitable constant. (We note that, by virtue of (2.8), γ -1 < ∞.) To get (5.5), we apply the Itô formula in (4.10) to the semimartingale X λ (t) 2 -1 and to the function

ϕ ε (r) = (r + ε 2 ) 1-m 2 , r > -ε 2
, where X λ is the solution to (4.10).

We have .

dϕ ε ( X λ (t) 2 -1 ) +(1 -m)( X λ (t) 2 -1 + ε 2 ) -m+1 2 X λ (t), ψ λ (X λ (t)) + λX λ (t) 2 dt = 1 2 ∞ k=1 µ 2 k        (1-m) X λ (t)e k 2 -1 ( X λ (t) 2 -1 +ε 2 ) m+1 2 -(1-m 2 ) X λ (t)e k 2 -1 X λ (t) 2 -1 ( X λ (t) 2 -1 +ε 2 ) m+1 2        dt +2 ϕ ′ ε ( X λ (t) 2 -
(5.6)

Here, we have used the fact that, by Lemma 4.5, for λ → 0, , 0 ≤ r ≤ t < ∞, because, by (2.7), x -1 ≤ γ -1 |x| m+1 , ∀x ∈ L m+1 (R d ). Letting ε → 0, we get (5.5), as claimed. Now, we conclude the proof as in [START_REF] Barbu | Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise[END_REF]. Namely, by (5.5), it follows that

X λ → X in H -

X(s), X(s)dW(s)

-1

and, therefore, t → e -C * (1-m)t X(t) 1-m -1 is an {F t } supermartingale. Hence, X(t) -1 = 0 for t ≥ τ, because of Proposition 3.4, Chap. 2 of [START_REF] Revuz | Grundlehren der Mathematischen Wissenschaften[END_REF]. Moreover, taking expectation for r = 0, we get e -C * (1-m)t E X(t) 1-m -1 + ρ(1m)γ m+1 t 0 e -C * (1-m)s P(τ > s)ds ≤ x 1-m -1 .

This implies that P(τ > t)

1e -C * (1-m)t C * (1m) ≤ t 0 e -C * (1-m)s P(τ > s)ds ≤ x 1-m -1

ρ(1m)γ m+1 , and so (5.4) follows. This completes the proof.

Corollary 5.2. Let x ∈ H -1 ∩ L m+1 (R d ) ∩ L 2 (R d ) be such that x -1 < ργ m+1 C * . Let τ be the stopping time defined in (5.2). Then P(τ < ∞) > 0. In other words, there is extinction in finite time with positive probability.

Remark 5.3. In the case of bounded domain, Theorem 5.1 remains true for m ∈ d-2 d+2 , 1 (see [START_REF] Barbu | Finite time extinction of solutions to fast diffusion equations driven by linear multiplicative noise[END_REF]). One might suspect that also in this case the extinction property (5.4) holds for a larger class of exponents m. However, the analysis carried out in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF] for deterministic fast diffusion equations in R d shows that the extinction property is dependent not only on the exponent m, but also on the space L p (R d ), where the solution exists (the so called extinction space).

Remark 5.4.

• The analysis in this section holds, in particular, if all the coefficients µ k do vanish, i.e., in the deterministic framework. In that case, Theorem 5.1 implies the existence of a deterministic time τ > 0 so that t ≥ τ ⇒ X(t) -1 = 0, and so X(t) = 0, for all t ≥ τ.

• Let us set, for instance, ψ(u) = u m , d ≥ 3, m = d-2 d+2 •. Observe that (L 1 ∩ L ∞ )(R d ) ⊂ (L m+1 ∩ L 2 )(R d ) ∩ H -1 . Consider, for instance, as initial condition x ∈ (L 1 ∩ L ∞ )(R d ).

• By the Benilan-Crandall approach, see, e.g., Theorem 1 of [START_REF] Benilan | The continuous dependence on ϕ of solutions of u t -∆ϕ(u) = 0[END_REF], there is a solution u : [0, T ] × R d → R, of dX -∆ψ(X)dt = 0 in (0, T ) × R d ,

X(0) = x on R d , (5.7)
in the sense of distributions. u belongs to (L 1 ∩ L ∞ )((0, T ) × R d ) and also the η u = ψ(u). In this case, u fulfills mass conservation.

• By use of Theorem 4.4, there is another solution v : [0, T ] × R d → R d in the sense of distributions, such that v ∈ L p ((0, T ) → R d ), with p = max(1, 2m). Also, η v = ψ(v) ∈ L p m ((0, T ) × R d ). By Theorem 5.1, if x is small enough, there will be extinction, and so, v does not fulfill any mass conservation.

• In particular, there is no uniqueness for (5.7) in the sense of distributions. Remark that according to [START_REF] Brezis | Uniqueness of solutions of the initial-value problem for u t -∆ϕ(u) = 0[END_REF],

uniqueness is guaranteed in the class (L 1 ∩ L ∞ )((0, T ) × R d ).

Definition 4 . 3 .

 43 Let x ∈ H -1 and p := max(2, 2m

( 4 .

 4 32)It remains to be shown that X is a solution to(1.1) in the sense of Definition 4.3. By (4.10) and (4.32), we see that dX -∆ηdt = XdW, t ∈ (0, T ) X(0) = x. (4.33)

  .36) for all Z ∈ L p ((0, T ) × Ω × B N+1 ) and ζ ∈ L p ′ ((0, T ) × Ω × B N+1 ) such that ζ ∈ ψ(Z), a.e. in (0, T ) × Ω × B N+1 .We denote by ψ :L p ((0, T ) × Ω × B N+1 ) → L p ′ ((0, T ) × Ω × B N+1) the realization of the mapping ψ in L p ((0, T ) ×

  (i.e., spatially independent noise) via the rescaling transformation X = e W Y, which reduces it to the random parabolic equation ∂ ∂t Ye -W ∆ψ(

e

  -C * (1-m)t X(t) 1-m -1 + ρ(1m)γ m+1 t r e -C * (1-m)s 1 [ X s -1 >0] ds ≤ e -C * (1-m)r X(r)

  1 )X λ (t), X λ (t)dW(t) .

	This yields	ϕ ε ( X λ (t) 2 -1 ) + ρ(1 -m)	r	t	( X λ (s) 2 -1 + ε 2 ) -m+1 2	R d	|X λ | m+1 ds dξ
				≤ ϕ ε ( X λ (r) 2 -1 ) + C *		r	t	X λ (s) 2 -1 ( X λ (s) 2 -1 + ε 2 ) -1+m 2 ds
						t
				+2 r	r -1 )X t ϕ ′ ε ( X λ (s) 2 X(s) 2 -1 + ε 2 ) -m+1 2	R d	|X(s, ξ)| m+1 dξ ds
	≤ ϕ ε ( X(t) 2 -1 ) + C *	r	t	X(s) 2 -1 ( X(s) 2 -1 + ε 2 ) -m+1 2 ds
	t
	+2	ϕ ′
	r

λ (s), X λ (s)dW(s) -1

. Now, letting λ → 0, we obtain that X satisfies the estimate

ϕ ε ( X(t) 2 -1 ) + ρ(1m) ε ( X(s) 2 -1 )X(s), X(s)dW(s) -1

Acknowledgements. Financial support through the SFB 701 at Bielefeld University and NSF-Grant 0606615 is gratefully acknowledged. The third named author was partially supported by the ANR Project MASTERIE 2010 BLAN 0121 01.