Research interests

Rolando Rebolledo

The numbering of references follows the order of the list of publications provided as a separate document.

**1
Investigating weak topologies for probability measures**

My 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 I 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 first 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 my main papers on the method I built up is the memoir [9].

Furthermore,
several extensions and improvements of the method followed in
[10]–[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. My 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, we analysed discretisation 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].

**2
An incursion in Stochastic Mechanics**

I
came to Stochastic Mechanics in 1988, to work in collaboration with
physicists in Quantum Optics, (see references [29], [30], [31]). In
particular [31] introduced the concept of an *entropic diffusion*
which gives coherent grounds for building up Nelson’s approach to
Stochastic Mechanics. Several lecture notes in Spanish were published
in proceedings of various Chilean meetings. This
was a short passage through this theory. I arrived at the opinion
that Stochastic Mechanics is more of a simulation of Quantum
Mechanics based on classical stochastic processes. And the
probability space was depending 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
my opinion, researchers in Probability need to be aware of both
models.

**3
Non commutative stochastic analysis and open quantum systems**

Applications of Probability to Scattering Theory motivated a joint research with Claudio Fernandez. Simultaneously, I became interested in non commutative probability through the seminal work of Accardi, Parthasarathy and Meyer. References [36] to [38], [40], [42], and [45] to [47] are connected with the beginning of this direction of research.

A systematic study of Quantum Markov Semigroups started then. The focus was, firstly, set on their qualitative analysis. An important part of this research program was carried jointly with Franco Fagnola. References [57] to [66], [71], [75], [77], [81], [82], are related to this. Namely, those papers were aimed at answering some fundamental questions like: Under which conditions on its generator there exists an invariant state for a given quantum Markov semigroup? Is the system ergodic? Is it recurrent or transient?

In parallel, I started a research on the so called "quantum decoherence" phenomenon, connected with classical reductions of quantum Markov semigroups (see [67], [68], [69], [75], [80]).

Classical reductions have recently been used too in designing statistical inference on open quantum systems (see [84]).

On the other hand, classical dilations of quantum Markov semigroups have been investigated in a number of papers (see [80] as well as [72], [74] both written jointly with Carlos Mora).

Even though the paradigm of Markov approximation to open quantum systems still having a number of relevant questions to be solved, it is currently important for applications to consider non Markov approaches. This is part of a research started in 2007 with Andrzej Kossakowski (see [73], [76], [78], [79]).

**4
Applications to physics, engineering and finance**

Throughout my career as a research fellow in Mathematics I have been very often inspired by physical problems. Perhaps Quantum Optics has been the most influential field in my latest research as it follows from the papers [30], [31], [43], [47], [52]. However, engineering applications are not missing: in [50] we applied Stochastic Differential Equations to Electricity. Moreover, in a series of papers with Eckhard Platen (see e.g. [44], [46], [49]), applications of Stochastic Analysis to Finance have been developed.