The Box Ball System was introduced by Takahashi and Satsuma in 1990 as a cellular automaton that exhibits solitons (travelling waves). We study a continuous version, where blocks of consecutive occupied boxes are replaced by occupied intervals of the real numbers, separated by empty intervals. The walk representation of a configuration is given by a zig-zag function, and the dynamics by Pitman's transformation. We describe how to identify solitons, and show that they are conserved under the dynamics. We also show that the soliton decomposition of some random zig-zag walks can be mapped to a bidimensional Poisson process, a representation that linearizes the dynamics. This extends discrete space results by Ferrari, Gabrielli, Nguyen, Rolla and Wang. This is joint work with Pablo Blanc, Pablo Ferrari and Davide Gabrielli.

After recalling some background on (positive) self-similar Markov processes, the Lamperti transformation, and the connexion with Lévy processes, I will present very succinctly an almost completed project jointly with Nicolas Curien (Orsay) et Armand Riera (Paris Sorbonne) on self-similar Markov trees. Roughly speaking the latter are the analogs of (positive) self-similar Markov processes in a branching setting. They can be constructed by applying a version of the Lamperti change of time on real trees to branching Lévy processes.

The Brownian continuum random tree emerges as a fundamental limit shape for various discrete tree models. It arises as the scaling limit of, for instance, the uniform spanning tree on the complete graph with N vertices or on the torus Z^d_N of size-length N, for d≥5 (Peres and Revelle (2005)). The Aldous-Broder Markov chain is a simple algorithm for generating random spanning trees. This Markov chain, defined on a graph G= (V, E), evolves through a sequence of rooted trees, each with a subset of V as its vertex set. In Evans, Pitman and Winter(2006), it was shown that the suitable rescaled Aldous-Broder Markov chain on a complete graph converges to the so-called root growth with regrafting process (RGRG process) weakly with respect to the Gromov-Hausdorff topology. In this talk, we study the convergence of the rescaled Aldous-Broder Markov chain on high-dimensional regular graphs, such as Z^d_N, towards the RGRG process. Joint work with Osvaldo Angtuncio-Hernández (Centro de Investigación en Matemáticas (CIMAT)) and Anita Winter (Universität Duisburg-Essen).

We are interested in studying the genealogical structure of alleles in a Bienaymé-Galton-Watson (BGW) process with neutral mutation and finitely-many alleles. Our focus is on the mother-dependent mutation model, where the distribution of mutations follows the multinomial distribution introduced in [2]. For this case, we construct the multi-typeversion of the tree of alleles defined by Bertoin in [1] and obtain the convergence (after re-normalization) of the allelic sub-populations towards a multi-type tree indexed CSBP. Based on joint work in progress with María Clara Fittipaldi and Saraí Hernández-Torres.

References
[1] Jean Bertoin. A limit theorem for trees of alleles in branching processes with rare neutral mutations. Stochastic Process. Appl. 120 (2010) 678–697.
[2] Airam Blancas, María Clara Fittipaldi and Saraí Hernández-Torres. Crossing bridges between percolation models and Bienaymé-Galton-Watson trees (2024). arXiv:2411.09621.

A Lévy processes resurrected in the positive half-line is a Markov process obtained by removing successively all jumps that make it negative. A natural question, given this construction, is whether the resulting process is absorbed at 0 or not. In this work, we give conditions for absorption and conditions for non absorption bearing on the characteristics of the initial Lévy process. First, we shall give a detailed definition of the resurrected process whose law is described in terms of that of the process killed when it reaches the negative half line. In particular, we will specify the explicit form of the resurrection kernel. Then we will see that when the initial Lévy process X creeps downward and satisfies certain additional condition, the resurrected process is absorbed at 0 with probability one, independently of its starting point. Some criteria for absorption and some criteria for non absorption will be given.The most delicate case is when X enters immediately in the negative half line and drifts to −∞. It is then possible to give a sufficient condition for absorption but up to now, even when X is the negative of a subordinator, we do not know whether this condition can be dropped or not. We shall take a closer look at the case of stable processes. 

References
[1] K. Bogdan, K. Burdzy and Z.-Q. Chen: Censored stable processes. Probab. Theory Related Fields, 127, no. 1, 89–152, (2003). 
[2] N. Ikeda, M. Nagasawa and S. Watanabe: A construction of Markov processes by piecing out. Proc. Japan Acad.42, 370–375, (1966).
[3] P. Kim, R. Song and Z. Vondraček: Positive self-similar Markov processes obtained by resurrection. Stochastic Process. Appl.156 (2023), 379–420.
[4] V. Wagner: Censored symmetric Lévy-type processes. Forum Math. 31, no. 6, 1351–13

Dynkin’s isomorphism Theorem [1] as well as its derivatives, are identities in law providing the law of the sum of two independent processes. Most of the time these two processes are respectively the local time process of a Markov process and a permanental process. In spite of their various applications, these identities still remain mysterious inthe sense that there is no natural reason for summing these two independent processes.We will present an explanation available in particular for the so-called "generalized second Ray-Knight Theorem" [2].

References
[1] Dynkin E.B.: Local times and quantum fields. Seminar on stochastic processes, 1983 (Gainesville,Fla.), 69-83, Progr. Probab. Statist., 7, Birkhauser Boston, MA(1984).
[2] Eisenbaum N., Kaspi H., Marcus M. B., Rosen J. and Shi Z.: A Ray-Knight theorem for symmetricMarkov processes. Ann. Probab. 28, no. 4, 1781-1796(2000).
[3] Sznitman A.-S.: An isomorphism theorem for random interlacements. Electron. Commun. Probab.17, 9(2012).

Drawing inspiration from proposals and successful open problem sessions [1, 2], we will share strategies for creating open problems and propose a few to spark ideas and collaboration.

References
[1] Louigi Addario-Berry, How to create an open problem, 2022. https://problab.ca/louigi/talks/op.pdf
[2] Vlada Limic, Vlada’s Point: A WORKshop. IMS Bulletin,43(8), 2014. https://imstat.org/2014/11/17/vladas-point-a-workshop/1

A classical problem in game theory is to establish conditions for non-cooperative game models that guarantee the existence of solutions known as Nash equilibria. In this talk, we will provide a general overview of some results obtained for this problem under separability-additivity conditions in the payoff functions and the transition law (ARAT). We will conclude by presenting a result where the payoff functions are not necessarily assumed to be bounded with respect to the state variable of the game.
(Joint work with David GONZÁLEZ–SÁNCHEZ and AND Fernando LUQUE–VÁSQUEZ)

References
[1] Fonseca–Morales Alejandra, González–Sánchez David and Luque–Vásquez Fernando. On stationary Nash equilibria in ARAT games with unbounded payoff functions. To appear.

A certain type of sawtooth Markov process, called extremal shot noise process (ESN), is investigated. This process generalizes extremal processes by incorporating a drift mechanism. After introducing these processes and stating some fundamental properties of their semigroup and infinitesimal generator, I will characterize their first passage times below a given point, as well as their local time at 0 when the boundary 0 is accessible. A primaryinterest in these processes lies in the fact that their zero set matches Mandelbrot’s random cutout sets— sets obtained after placing Poisson random covering intervals on the positive half-line. Based on this connection, a new proof of the Fitzsimmons-Fristedt-SheppTheorem, see [3] and e.g. [1, Chapter 7], which characterizes the random cutout sets, is presented. Lastly, I will explain how some ESNs arise in functional limit theorems for continuous-state branching processes with large immigration. A result of Iksanov and Kabluchko [2] on Bienaymé-Galton-Watson processes with immigration is re-established in the context of continuous-time and space processes by following different arguments. Other regimes with very large immigration are also found. The results of the talk are extirpated from two joint work with Linglong Yuan (University of Liverpool), see [4,5]. 

(Joint with Linlong YUAN)

References
[1] Bertoin, J. (1999). Subordinators: Examples and Applications. Bernard, P. (eds) Lectures on Probability Theory and Statistics. Lect. Notes Math. 1717. Springer, Berlin.
[2] A. Iksanov, and Z. Kabluchko, Functional limit theorems for Galton–Watson processes with very active immigration. Stochastic Process. Appl.128291–305. 2018
[3] Fitzsimmons, P. , Fristedt, B. and Sheep, L. (1985). The set of real numbers left uncovered by random covering intervals. Z. Wahrscheinlichkeitstheor. Verw. Geb.70175–189.
[4] C. Foucart and Linglong Yuan, Extremal shot noise processes and random cutout sets, 2024, ArXiv-2302.03082 (to appear in Bernoulli)
[5] C. Foucart and Linglong Yuan, Weak convergence of continuous-state branching processes with large immigration, 2024, ArXiv-2311.04045 (to appear in SPA).

While (exchangeable) random probability measures have a huge tradition in Probability and Statistics, little is known on their tails. The few available results are derived using subordinators, and hence the methodology only applies to measures that have a representation as a normalized subordinator—such as the Dirichlet process—but fails to apply to the entire class of exchangeable probability measures—such as some Pitman–Yor processes. Here we provide the first characterization of the tails of a random probability measure that cannot be fully described by a subordinator representation. Based on a family oftransport maps defined on the space of probability measures, we show that the right tailof a Pitman–Yor process is heavy-tailed if the its expected measure is itself heavy-tailed;the Dirichlet process is the only member of this class that fails to obey this convenientproperty. Consequences of the main results, including aspects related to the posterior inference are discussed.

We prove a large deviation principle for the normalized bridge and excursion of aspectrally positiveα-stable Lévy process [1]. This LDP demonstrates a “one big jump”phenomenon, which often appears in the context of LDPs for random walks with heavy-tailed step distributions. We use our LDP in order to derive precise tail asymptotics fornatural functionals of the excursion, specifically its area and maximum. We prove ourresults in (a small variant of) the Skorokhod M1 topology, which is better suited to thecontext than the more usual J1 topology. (Joint with Léo DORT and Grégory MIERMONT)

References
[1] Léo Dort, Christina Goldschmidt and Grégory Miermont. A large deviation principle for the normalized excursion of an α-stable Lévy process without negative jumps, ALEA Lat. Am. J. Probab.Math. Stat.,21, pp.1625–1653 (2024)

The convex hulls of Lévy processes have been a topic of extensive study over the last few decades. While many studies focus on intrinsic volumes and their limiting behaviors,‘sensitive’ properties such as differentiability or the lack of corner points have eluded study except in the case of Brownian motion (Lévy, 1948; Cranston & Hsu & March,1989; Alexander & Eldan, 2019) or the space-time convex hull of a one-dimensional Cauchy process (Bertoin, 2000). Recently, we obtained a nearly exhaustive description of when the space-time convex hull of a one-dimensional Lévy process is smooth, as well as a description of how smooth it is (GC & Mijatovic & Kramer-Bang, 2024). Recent advances have shown that one-dimensional techniques and certain multi-dimensional extensions ofclassical results can provide sufficient conditions for the spatial convex hull of a multi-dimensional Lévy process to be corner-free. (This is joint work with Loïc Chaumont and Aleksandar Mijatovic.)

The study of Forward-Backward Stochastic Differential Equations (FBSDEs) can be traced back to the second half of the past century. Motivated by this class of systems, we present some recent results to consider the case when a noise (exogenous or endogenous) has some influence in the evolution of the stochastic system. This is motivated by mean-field games, but also by optimization problems arising when external factor modifies the dynamics of a controlled system, which will be exemplified along the presentation. In particular, McKean-Vlasov differential equations where the law is conditioned to a discontinuous Poissonian-type of process are analyzed. This is a joint work with J.Ricalde-Guerrero.

Fractional Brownian motion (fBm) is a cornerstone in stochastic modeling, with its dynamics highly sensitive to the Hurst parameter H. As H approaches zero, the behavior of fBm transitions to regimes with pronounced singularities and connections to log-correlated fields, making it an ideal framework for modeling rough stochastic phenomena in finance and beyond. Building on recent developments, this work investigates the additive functionals of fBm in the small H regime, with particular focus on local times and their fluctuations. Using techniques from Gaussian processes and fractional calculus, we provide a comprehensive analysis of the scaling limits and chaotic decompositions of these functionals. Our results extend the understanding of rough stochastic systems, bridging mathematical theory with practical modeling challenges.

Proton beam therapy is a relatively modern way of treating cancer. In short, protons are accelerated to around 2/3 the speed of light and projected into the body in the direction of cancerous tissues. Subatomic interactions slow the protons down causing energy deposition into tissue. The less energy protons have, the greater the number of subatomic interactions and the greater the rate at which energy is deposed. This results in the proton beam having an approximate “end point” where the majority of the initial energy in the beam is deposited. Rather obviously, this needs to be positioned into cancerous tissues and accuracy is essential; in particular if the tumour is positioned next to or within organs/bones where irreparable damage could be caused by the proton beam. Modelling the deposition of energy into the human body is often undertaken using software not dissimilar to (if not actually the same as) the software used that simulates particle experiments in large accelerators such as CERN. Simulations essentially reconstruct every nuclear interaction, incorporating e.g. CT scan data, and can take hours to prepare and execute. Whilst reasonably accurate, such simulations do not allow for “on the fly adjustments” to be made as patients move through their course of treatment. The only known mathematical model for the way in which energy is deposited into the patient suffers from being one dimensional and does not connect directly to the underlying particle physics. In this talk we discuss what we believe to be the first mathematical model of a proton beam, grounded in the subatomic physics, taking the form of a (7+1)-dimensional SDE which has both diffusive and jump components. A fundamental modelling question of this class of SDEs pertains to constructing a well-defined meaning of “rate of energy deposition” in the three dimensions of space and how relates to Monte Carlo simulation. Ultimately this boils down to the existence of an occupation density for the SDE, which, in turn requires us to work with fundamental ideas from Malliavin calculus. This work is part of a larger body of research that is currently being undertaken in collaboration with the proton beam treatment centre at University College London Hospital.

The main object of this article is to study the long time behavior for the distribution of a branching (birth and death)-diffusion process, motivated by population dynamics in changing environment (cf. a recent paper by Calvez, Henry, Méléard, Tran). The birthrates are bounded but death rates are unbounded. Our analysis is based on the spectral properties of the associated Feynman Kac semigroup. We require a standard spectral gap property for this semigroup with a dominant eigenfunction vanishing at infinity. Some examples of diffusions, diffusions with jump, pure jump dynamics are given for which it is true. Note that we consider situations where the underlying diffusion process doesn’t come down rapidly from infinity but the compactness properties follow from the divergence of the death rate at infinity. Note also that we work in an unbounded domain with unbounded death rate, a situation very different of the case of bounded domains. In the three different cases (critical, subcritical, supercritical), we prove the convergence in law of the branching diffusion process suitably normalized and conditioned to non-extinction. We also prove the existence of theQ-process. The main tool is the convergence of suitably normalized moments of the process, which follows from recursive relations for these moments. This is a joint work with Pierre Collet and Jaime San Martin.

References
[1] V. Bansaye, B. Cloez & P. Gabriel, Ergodic behavior of non-conservative semigroups via generalized Doeblin’s conditions, Acta Appl. Math.166(2020), p. 29–72.
[2] V. Calvez, B. Henry, S. Méléard & V.C. Tran, Dynamics of lineages in adaptation to agradual environmental change, Ann. H. Lebesgue5(2022), p. 729–777.
[3] P. Cattiaux, P. Collet, A. Lambert, S. Martńez, S. Méléard & J. San Martín. Quasi-stationarity distributions and diffusion models in population dynamics, Ann. Probab.37(2009),no. 5, p. 1926–1969.
[4]B. Cloez B. & P. Gabriel P., On an irreducibility type condition for the ergodicity of noncon-servative semigroups, C. R. Math. Acad. Sci. Paris358(2020), no. 6, p. 733–742.
[5]S.C. Harris, E. Horton, A.E. Kyprianou, M. Wang, Yaglom limit for critical non local branching Markov processes. The Annals of Probability, vol. 50, No 6 (2022), p.2373–2408.
[6]H. Hering, Subcritical branching diffusions. Compositio Mathematica, Tome 34 No. 3 (1977)

In this talk, we provide a series representation for the q-scale function for spectrally negative Lévy processes whose jumps part has bounded variation paths. Such a series representation is in terms of completely known parameters of the associated Lévy process. We use our results to prove Doney’s conjecture in the case when the Lévy process does not have a Gaussian component. Joint work with Ehyter Martín-González and AntonioMurillo-Salas.

In this talk, we will introduce the phenomenon of abrupt decorrelation for random processes under a given statistical distance. More precisely, we are interested in determining if random perturbed dynamical systems, rapidly and sharply, loses correlation with respect to its initial state. In particular, we will show that such phenomenon is present for linear stochastic differential equations with a limiting distribution, under the total variation, Wasserstein (or Kantorovich-Rubinstein) and Kullback-Liebler divergence (or relative entropy) distances. A typical example of such class of SDEs is the so-called Ornstein-Uhlenbeck process. This is a joint work with Sergio López (Fac. de Ciencias,UNAM) and Leandro Pimentel (Federal University of Rio de Janeiro).

The central topic in population genetics is to know how a population evolves under certain given conditions. In biology, some questions that arise from a sample taken from a population are primarily retrospective; that is, we seek to know what happened in the past so that today the population has the current particularities or, in other words, what were the evolutionary mechanisms responsible for the characteristics observed in the sample taken at present? The theory of coalescences arises to infer the past of the present sample [2]. In this talk, I will introduce coalescence theory and present some results for more recent models studied in the last decades, which have active and inactive individuals, also known as the seed-bank effects [1, 3, 4, 5, 6].

References
[1] Blath, J., González Casanova, A., Kurt N., Wilke Berenguer M. A new coalescent for seed-bank models. The Annals of Applied Probability, 50(3):741-759, 2013.
[2] Fu, Y. X. and W.H. Li, Coalescing into the 21st century: An overview and prospects of coalescent theory Theoretical Population Biology, 56(1):1-10, 1999.
[3] González Casanova, A., L. Peñaloza and A. Siri-Jégousse. The shape of a seed bank tree. Journal ofApplied Probability, 59(3), 631-651, 2022.
[4] González-Casanova, A., L. Peñaloza and A. Siri-Jégousse. Seed bank Cannings graphs: How dormancy smoothes random genetic drift. Latin American Journal of Probability and Mathematical Statistics,1165-1186, 2023.
[5] H. Wences A., L. Peñaloza, M. Steinrücken and A. Siri-Jégousse. The TMRCA of general genealogies in populations of variable size. Submitted to Theoretical Population Biology. 2024
[6] Kaj, I., Krone S. M., and Lascoux M. Coalescent theory for seed bank models Journal of Applied Probability, 38(2): 285-300, 2001.

In this talk, we explore recent advances in linear backward parabolic SPDEs, focusing on novel a priori estimates for weak solutions. Building on the foundational work of Y. Hu, J. Ma, and J. Yong [1] in strong solutions, we introduce Lp-estimates that rely on minimal assumptions regarding coefficient regularity, terminal data, and external forces. A key innovation is a new Itô formula for the Lp-norm, which generalizes the classic L2 result and allows us to enhance the regularity of the solution’s first component up to L∞.

References
[1] Ying Hu, Jin Ma, and Jiongmin Yong. On semi-linear degenerate backward stochastic partial dif- ferential equations. Probab. Theory Related Fields, 123(3):381–411, 2002.

We consider a version of the continuous-time multi-armed bandit problem where decision opportunities arrive at Poisson arrival times, and study its Gittins index policy. When driven by a Lévy processes, the Gittins index can be written explicitly in terms of the Weiner-Hopf factorization of a Lévy process Poissonian observation times, and is shown to converge to that in the classical Lévy bandit of Kaspi and Mandelbaum [1].This is joint work with Kei Noba, Kazutoshi Yamazaki.

References
[1] Kaspi, H., Mandelbaum, A. Multi armed bandits driven by Lévy processes. Ann. Appl. Probab.52, 541–565, 1995

Abstract in pdf.

We begin by introducing the notion of Brownian motion indexed by the Brownian tree. We will then present the main aspects of a theory, developed in two recent works with Armand Riera, that describes the evolution of this tree-indexed process between visits to 0. The theory applies to fairly general continuous Markov processes indexed by Lévy trees. Despite the radically different setting, we will see that our results share strong similarities with the celebrated Itô excursion theory for linear Brownian motion.

I am going to make some elementary discussions on the presence of the Euclidean norm of four-dimensional Brownian motion in the continuum Derrida–Retaux system.

Joint work with E. Aïdékon (Shanghai), B. Derrida (Paris), T. Duquesne (Paris).

In this talk, we study the forward integral, in the Russo and Vallois sense, with respect to Hölder continuous stochastic processes Y with exponent bigger than 1/2. Here, the integrands have the form f(Y), where f is a bounded variation function. As a consequence of our results, we show that this integral agrees with the generalized Stieltjes integral given by Zähle and that, in the case that Y is fractional Brownian motion, this forward integral is equal to the divergence operator plus a trace term, which is related to the local time of Y. Moreover, the definition of the forward integral allows us to obtain a representationof the solutions to forward stochastic differential equations with a possibly discontinuous coefficient and, as a consequence of our analysis, to figure out some explicit solutions.

Joint work with Johanna GARZÓN AND Jorge LEÓN.

We study the behavior of the solution of an Allen-Cahn equation with a double well reactive term under the action of a small space time white noise perturbation. We consider the SPDE with Dirichlet boundary conditions on a suitably large space interval starting from the identically null function that corresponds to the local maximum of the potential. Our main goal is the description, in the small noise limit, of the onset ofthe phase separation, with the emergence of spatial regions in each of the two phases. The time scale and the spatial structure are determined by a suitable Gaussian process that appears as the random counterpart of the linearized A-C equation. This issue was initially examined by De Masi et al. [Ann. Probab.22, (1994), 334-371] in the relatedcontext of a class of reaction-diffusion models obtained as a superposition of a speededup stirring process and a spin flip dynamics.

Joint work with Stella Brassesco (IVIC-Caracas) and Glauco Valle (IM-UFRJ).

Consider a large population whose size is fixed over the generations and in which every once in a while some randomly chosen individual experiences a beneficial mutation, leading to a slightly increased reproduction rate of its offspring. In the so-called Gerrish-Lenski parameter regime, typically a finite number of such clonal offspring populations is present together with one resident type. These subpopulations perform a contest for becoming resident, a phenomenon addressed as clonal interference. We show that, in the limit of infinite population size, the rescaled logarithmic sizes of the contending mutant families constitute a system of interacting piecewise linear trajectories driven by a Poisson point process, and relate the speed of adaptation in this system to heuristic predictions from [1] and [2]. Our analysis is carried out for a continuous-time Moran model and for strong selection with random fitness increments. We conjecture that our results extendto a wide range of reproduction dynamics as well as to moderate selection. The talk isbased on the joint work [3].

References
[1] E. Baake, A. González Casanova„ S. Probst, and A. W., Modelling and simulating Lenski’s long-term evolution experiment, Theor. Popul. Biol.127(2019), 58–74.
[2] P. J. Gerrish and R. Lenski, The fate of competing beneficial mutations in an asexual population, Genetica 102(1998), 127–144.
[3] F. Hermann, A. González Casanova, R. Soares dos Santos, A. Tobias, and A. W., From clonal interference to Poissonian interacting trajectories, 2024, arXiv:2407.00793