#### Origins

Rolando Rebolledo was born in Rancagua, August 12, 1947 and studied first in the Oscar Castro Secondary School until 1960. After that, his family moved to Santiago, where he finished the secondary school in 1963 at the “Instituto Nacional”. And at the Bachelor examination that year he was graded 31,9 over 35, becoming a national record.

His university studies conducted in the School of Engineering of the University of Chile, majoring in Mathematical Engineering in 1969 at the age of 22, with Distinction Unanimous. In 1968 he served as Assistant Professor in the School itself and after graduating, he was hired as full-time academic. Among his students at that time several early mathematical engineers, Rafael Correa, Guido del Pino, Antun Domic, are counted; students of the Faculty of Sciences of the University of Chile, including Victor González, Jorge Billeke and Günther Uhlman among many others.

#### The French period

In late 1970, after winning a grant from the French government, he traveled to France to begin his graduate studies. Professor Laurent Schwartz, whom he had known in the first ELAM in Rio de Janeiro in 1968, having read his memory of Mathematical Engineer containing several original results, advised him to enroll in the Doctorate of State. So began his research at the Laboratory of Probability, University Pierre et Marie Curie (Paris VI). In 1974, he was offered a position of Assistant in the Faculty of Reims. To obtain the degree of Doctor of State in 1979 in Paris VI (with the congratulations of the jury and the qualification “Très Honorable”, equivalent to Suma Cum Laude), under the direction of Professor Jacques Neveu, he rose to the position of “Professeur 2ème classe” at the University of Nice and then, to a “Professeur 1ère classe” position at University Paris-Sud in Orsay (Paris XI).

During the decade 1970-1980, his research focused on so-called Limit Theorems in the Theory of Stochastic Processes. For those years was built in France the General Theory of Processes, originated in the school of Strasbourg, led by Paul-André Meyer. Rebolledo’s work came to inaugurate a new branch of the theory, adding the analysis of weak topologies of measures via structural properties of processes. Already in 1978, he began a tour of various French and European universities (Copenhagen, Lund, later Dresden, Jena, Weierstrass Institute of the Academy of Sciences of Germany) presenting the nascent theory, and was invited to conferences in Hungary, and Czech Republic. His PhD thesis of State ([12]) made the synthesis of seven publications ([3], [4], [5], [6], [7], [9], [16]). Of these, the three most extensive ([8], [9], 16]) became a mandatory reference to the new theory of weak topology for stochastic processes, until the appearance of the book written by Jacod and Shyriaev “Limit Theorems for Stochastic Processes” in 1985. It should be noted that, at that time, the preparation of a “Docteur d’État” degree in mathematics was a solo sailor sort of. On the contrary, that was compensated by a great freedom given the candidate to study what he wanted. Together with Marie Duflo, Professor at Paris-Villetaneuse, Rebolledo began organizing workshops to assist students preparing the “Doctorat de 3è cycle” (PhD). This activity became a melting pot, which eventually settled in Orsay. Several students completed their degrees through this workshop activity. Rebolledo particularly, directed PhD thesis of both, Odile Pons-Ledru and Raúl Gouet, the latter now Professor at the University of Chile, who studied at Orsay.

After the 1973 military putsch in Chile, a number of intellectuals were killed, imprisoned or forced to emigrate, leaving the country mired in a deep cultural darkness. Promoting solidarity among scientists, Rebolledo, together with an outstanding group of French scientists, founded the Association “Sciences-Chili.” Alfred Kastler (Nobel Prize in Physics), Roger Godement, Laurent Schwartz, Jacques Neveu, Didier Dacunha-Castelle, Marie Duflo, Jaen Bretagnolle, Jacques Azéma, Jean-Pierre Kahane, joined the association among many others. “Sciences-Chili” focused on human rights defense and promotion of science helping young students, in particular, to continue their studies. However, in the opinion of many Chilean scientists, a greater commitment to science within the country was necessary. As a result, a country return was considered. Rebolledo was part of those who, even having reached a respectable academic position at a foreign university, decided a return to his motherland, accepting the risks of the moment, to help the national scientific research to restart.

#### Return to Chile

Rebolledo returned to Chile in August 1981. Given the disadvantaged circumstances in which the University of Chile, his alma mater was, it was impossible for him to retake his position at the Faculty of Engineering of this University. Professor Rolando Chuaqui, who at the time headed the Institute of Mathematics of the Catholic University, invited him to settle there, and participate in the foundation of the forthcoming faculty of Mathematics. At the same time, he started advising his first doctoral students, and launched the Winter School on Probability and Statistics Santiago (EIPES) with his colleague Guido del Pino (see the published book [1]). This annual international event attracted a large number of scientists from around the world, jointly organized by the Catholic University and the CIENES. This school worked regularly for fifteen years, and was followed later by the Winter School of Stochastic Analysis and Applications organized by the Catholic University and the University of Valparaiso.

#### Promoting scientific organizations

In 1982, Rebolledo was elected President of the Chilean Mathematical Society (SOMACHI). At that time, this scientific society existed as an informal group, but did not have legal personality. The Board chaired by Rebolledo, managed to complete the process. The SOMACHI also achieved two important initiatives to promote scientific development. First, SOMACHI agreed with the Cultural Service of the French Embassy, a committee of experts to evaluate nominations for French postgraduate scholarships in Mathematics. Second, the Society established a comprehensive program of development of mathematics in Chile. This was done with the active support of a number of mathematicians freshly returned from abroad, including Ricardo Baeza, Jorge Soto Andrade, Rolando Chuaqui, Oscar Barriga, Servet Martínez, Rafael Correa, Ruby Rodriguez. This project was funded by the United Nations Program for Development (UNDP) and provided a budget to support students, visiting professors, scientific meetings, purchase of books and equipment.

Rebolledo was twice reelected at the head of the Chilean Mathematical Society and was kept as a member of its Council during other terms, always involved in its different activities. In parallel, he continued his participation in international scientific organizations. He was elected Fellow of the International Statistical Institute (ISI) in 1982, President of the Latin American Committee of the Bernoulli Society, later on elected member of the Council of that Society.

He was the first Latin-American mathematician nominated Chair of the Commission for Development and Exchanges (CDE) of the International Mathematical Union in 1994. He served this Chair two terms until 2002. Moreover, elected again President of the Chilean Mathematical Society in 1994, he promoted and took part in the foundation of the “Unión Matemática de Latinoamérica y del Caribe (UMALCA).”

To summarize, along his academic career Rebolledo has maintained his practice and arguments in favor of providing Chile with an adequate institutional framework for the development of science.

Scholar achievements, early work

Rolando Chuaqui and Rolando Rebolledo have been the first two Chilean professors to advise PhD theses in the country. Gladys Bobadilla, Raúl Fierro, Eugenio Saavedra, Henri Comman, Carlos Mora, Rely Pellicer, and more recently Mallén Arenas, Mauricio Tejo, Julián Agredo, obtained their degrees under the direction of Rebolledo.

He has advised several Magister theses as well, and collaborated with other national postgraduate programs. The first doctoral theses directed by him were enrolled in the line of the General Process Theory, addressing different aspects of convergence in law.

His early research was focused on weak topologies of probability measures associated to stochastic processes, via Martingale Theory and the General Theory of Processes. The first article published in this direction of research was [3] devoted to the convergence in distribution of continuous martingales. This was followed and improved by a series ([4], [5], [6]) of notes, which included a fundamental preliminary version of a Central Limit Theorem for (discontinuous) square integrable martingales, building up a new method to approach the convergence in distribution of stochastic processes. The first applications of this method appeared in [7] and [8]. One of the main papers on the efficient method developed by Rebolledo is the memoir [9]. Furthermore, several extensions and improvements of the method followed in [10] to [14]. The article [15] provided a complete description of tightness on the Skorokhod space D by means of stopping times. On the other hand, in [16] the more general result on the Central Limit Theorem for Local Martingales was established. His research turned then into the search of necessary and sufficient conditions for the validity of the Central Limit Theorem. This was achieved for the case of semimartingales in [17].

Semimartingale or martingale problems have been considered in [18] to [22] and [26]. In particular, various results on the approximation of diffusions were derived. Finally, in [23], a paper written jointly with Eckhard Platen, they analysed discretization procedures and the approximation of diffusions.

The study of metastable phenomena in stochastic particle models, motivated the search of a weaker topology on the space D replacing the customary Skorokhod’s topology. The articles [24], [25], [27], [28] were aimed at solving that problem.

Finally, the convergence of non-adapted processes was covered in [32], [33], [36], and random field convergence was studied in [35].

#### Scholar achievements, the Physics connection

Gradually, Rebolledo began a fruitful dialogue with researchers in physics. The Probability Seminar founded by him by the end of 1981 became soon, in 1983, Seminar on Stochastic Analysis and Mathematical Physics. It still exists under the same name. The main reason for this change of name can be found in his research of that time carried jointly with a group of talented students in Quantum Optics. Among them, one finds Carlos Saavedra, Juan Carlos Retamal, both currently well-known senior researchers. In addition, that year 1983, Rebolledo met Professor Miguel Orszag, who was the advisor of the aforementioned students, starting an uninterrupted and fruitful dialog, which led later to joint research projects, conferences, seminars, students co-advised.

These exchanges strongly influenced a philosophical view of Rebolledo on his own research. He started studying the extensive relationship between open dynamical systems and the foundations of the notion of chance itself. Since then, Rebolledo has consistently dedicated his efforts to developing this relationship in search of a synthesis of distinct branches of Mathematical Analysis and different expressions of probability theory that today serve to describe such systems. It is the sense that he wanted to give to Stochastic Analysis. And in this way, his contributions have been varied, ranging from classical theory to quantum or non-commutative stochastic processes, motivated by open quantum systems.

So, he came to Nelson’s Stochastic Mechanics in 1988, working in collaboration with physicists in Quantum Optics, (see references [29], [30], [31]). In particular in [31] Rebolledo introduced the concept of an entropic diffusion, which gives coherent grounds for building up Nelson’s Mechanics. Several lecture notes in Spanish were published in proceedings of various Chilean meetings. This was a short passage through this theory. He finally arrived at the opinion that Stochastic Mechanics is more of a simulation of Quantum Mechanics based on classical stochastic processes. The probability space depends on the observable chosen. Thus, to overcome this difficulty, a new approach to probability was necessary, and that was provided by non-commutative approaches (or Quantum Probability). That is, Quantum Mechanics intrinsically contains a model for probability, which extends the one proposed by Kolmogorov. In Rebolledo’s opinion, researchers in Probability need to be aware of both models.

In one of his trips invited by the Russian Academy of Sciences, he had for the first time the opportunity to meet K.R. Parthasarathy, who was part of a select group of mathematicians investigating in Quantum Probability, as well as Luigi Accardi and Robin Hudson, who had similar concerns. Thus, in the 90s, his systematic work on the Theory of Quantum Markov Semigroups starts. At the same time, he continued to apply Classical Stochastic Analysis in his interdisciplinary work with engineers in Power Systems ([44]), as well as in Finance (see [41], [43], [48]).

Faithful to his idea of integrating different aspects of Analysis in the nascent Stochastic Analysis, in the same 90s Rebolledo began his collaboration with specialists from Functional Analysis and Partial Differential Equations. Collaborating actively with Claudio Fernandez, Humberto Prado, Victor Cortés, Maria Angélica Astaburuaga, Marco Corgini, he founded the so-called Research Group on Stochastic Analysis and Mathematical Physics ([36], [37], [38], [42], [54], [56]).

A number of physicists joined this team, and gradually the research group became known especially through the organization of numerous national and international meetings bearing the same name of Stochastic Analysis and Mathematical Physics (see published books [3] to [7]).

#### Scholar achievements, the Presidential Chairs

In 1995, Rebolledo was awarded his first Presidential Chair, a prize promoted by the President of the Republic’s Assessment Committee on Science, after evaluation by an international panel of referees. He used his prize to consolidate the national research group on Stochastic Analysis, funding students, visitors, researchers, and scientific meetings.

In 1999, he obtained a second Presidential Chair award, similarly fully devoted to promote collective work on Open Systems Theory.

During that time period, the results obtained by Rebolledo and collaborators on the Theory of Markov Semigroups became widely known in the specialized scientific community ([45], [47], [49], [52], [57], [59], [60], [61], [62], [63], [64], [65], [67], [68], [69], [70]). As a result, Rebolledo was invited to lecture in a number of research meetings and conferences abroad.

#### The Research Centre for Stochastic Analysis (ANESTOC)

Later, in 2006, his research group was awarded a new important grant, a so-called Scientific Research Ring funded by the Chilean Bicentennial Program on Science, giving birth to the Centre for Stochastic Analysis (ANESTOC), a research network involving a number of investigators spread around different universities. During 2011, ANESTOC became in addition an Associated Research Team (“Équipe Associée”) of INRIA-France, extending its cooperation horizon to other applications of mathematical modeling like renewable energies and neurosciences.

The Centre increasingly extended its scientific collaboration leading a number of different grants, like MathAmSud, a French-Latin American agreement, a second Scientific Research Ring in 2012, and covering brand new research fields, like Biostochastics, a term coined to name stochastic modeling in Biology.

#### Scholar achievements, current interdisciplinary research

The atmosphere created at the Centre, which permitted broad scientific exchanges, has been particularly stimulating for Rebolledo’s research. Open system theory fully emerged as the main trend in his mature research work in both, classical and quantum versions. A part of his approach to open quantum systems without memory, appears as a collection of papers written about the theory of Quantum Markov Semigroups ([71], [72], [75], [77], [80], [81], [82], [84], [86], [87], [88], [90], [91], [92]), while others explored the effects of memory in both, classical and quantum cases ([73], [76], [78], [79], [85], [89]).

Professor Miguel Orszag together with Rolando Rebolledo joined their forces to promote the Centre ANESTOC as a wide research network, which started including a number of outstanding scientists from Mathematics, Physics and Biology. It is worth mentioning among them, Professor Pablo Marquet, chair of the Department of Ecology and Professor Sergio Navarrete, Director of the Centre for Marine Research (ECIM) of the Catholic University.

Different seminars and conferences allowed fruitful exchanges with biophysicists and biologists like Ramón Latorre, Alan Neely, Adrián Palacios from the Interdisciplinary Centre for Neuroscience Research in Valparaíso. As a result, Rebolledo co-directed various student’s theses with them. Similarly, he has systematically maintained his joint work with engineers, promoting different initiatives to solve energy problems in Chile.

Moreover, Rebolledo played recently an important role in the foundation of the Center for Innovation, Research and Information in Chile (CIRIC), a subsidiary of INRIA-France.

Those wishing to delve into the details of the Rebolledo’s mathematical production can consult the list of his works. However, if one had to highlight results that have already become part of the baggage of contemporary probability, one should mention first his Theorems on the Central Limit for Martingales and Semimartingales and tightness criteria based on stopping times. In more recent times, the results obtained with Franco Fagnola on qualitative analysis of Quantum Markov semigroups; his studies of Quantum Decoherence and criteria for classical reduction of Quantum Markov Semigroups. His work on non-Markovian open quantum systems, carried out jointly with Andrzej Kossakowski starts to be currently cited as well.

During his career, he has been a visiting professor in several universities abroad, which include those of Copenhagen, Lundt, Rome, Genoa, Messina, Lisbon, Dresden, Institute Weierstrass, Steklov Institute of the Russian Academy of Sciences, Australian National University, Louisiana State University, Simón Bolívar, IMPA (Rio de Janeiro), Sao-Paulo, Reims, Max Planck Institute. Apart from the two awards “Presidential Chair,”Rebolledo received twice the Catholic University medal for his results in scientific research. His work is now widely recognized internationally. He is often invited to various conferences in both, Probability and Mathematical-Physics. He is frequently consulted as an evaluator of articles and projects in his areas of expertise. He has participated in several doctoral committees in France, Italy and Australia. It can be said that it has formed a scientific school in Chile, which fits between the pioneers of the national mathematical development. This school has attracted many young scientists from different places who are interested in interdisciplinary research and the new mathematics that emerges from this vision of Stochastic Analysis and Mathematical Physics.

#### The philosophical key, concluding remarks

It is worth deepening in the method followed by Rebolledo in mathematical research. His method is based on a close relationship with philosophical reflection. He uses to express his deep conviction on the necessary interaction between different sciences as a consequence of the mutual transformation of human species and Nature. In his view, science is constantly evolving, and therefore, there cannot exist total and immanent theories. They are all subject to a creative evolution that, at each stage, generates new theories able to encompass the previous one as particular cases. It is in this general context that he has coined the concept of the open system as a relationship between a part of Nature, which contains our observable quantities, and the surrounding medium. Thus, the organization of life is determined by this interrelation, called by him interpoiesis.

And from these basic ideas, he derives the concept of chance and the generation of its different mathematical models (probability theories) as part of our evolutionary understanding of the natural phenomena (papers in Philosophy and History [1], [3], [4]).

In his concept, the training of researchers in mathematics needs philosophical reflection to be complete. His students have almost all participated in distinct versions of the “Philosophical Colloquia.” These colloquia started in 1983 to address issues of philosophy of science to graduate students with different specialties: mathematicians, physicists, and biologists. Furthermore, for several years he taught a course in general education within the university entitled “On chance and its mathematical modeling.” These lectures were addressed to non-mathematical students: psychology, art, medicine, economics, music, and theology. It is the philosophical reflection which has certainly permitted a good dialogue with researchers in physics, biology, engineering and different fields of mathematics.

In his view, Mathematics does not end in itself. It is a means by which circulates the production of scientific knowledge of humanity, in its infinite quest to transform this world into a better one.