• PhD Research Project (2020-Present)  Qualitative Behaviour of Conditioned Random Dynamics.

 Advisors:  Jeroen S W Lamb and Martin Rasmussen.

 Financial support: Imperial College President's PhD Scholarship and by the  EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1).

• Research Internship Abroad (2019)  Piecewise smooth and reversible dynamical systems in  R^{2n+1}: existence of homoclinic trajectories and applications.

Advisors: Ricardo Miranda Martins and Jeroen S W Lamb.

Financial support: Scholarship from FAPESP (São Paulo Research Foundation). Project here.

• Master Research Project (2018-2020) Structural stability of piecewise smooth dynamical systems on torus and spheres

Advisors: Ricardo Miranda Martins.

Financial support: Scholarship from FAPESP (São Paulo Research Foundation). Project here.

• Undergraduate Research Project (2016-2017) The Center-Focus problem, Abelian integrals and Hilbert's 16th problem.

Advisors: Ricardo Miranda Martins.

Financial support: Scholarship from FAPESP (São Paulo Research Foundation). Project here.

• Alfréd Rényi Institute of Mathematics - Workshop on statistical properties of chaotic dynamics in and out of equilibrium (19th - 23rd Aug 2024).

Poster Presentation Quenched exponential decay of correlations for predominantly expanding multimodal circle maps.  Workshop page.

• Penn State University -  Semi-annual Workshop in Dynamical Systems and Related Topics (2nd - 5th November 2023).

Poster Presentation Quenched exponential decay of correlations for predominantly expanding multimodal circle maps.  Conference page.

• ICIAM - 10th International Congress on Industrial and Applied Mathematics  (20th - 25th  August 2023).

Delivered a presentation with the title On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape.  Conference page.

•  Scuola Normale Superiore - Probabilistic methods in dynamics (29th March - 7th April 2023).

Poster Presentation Quenched exponential decay of correlations for predominantly expanding multimodal circle maps.  Conference page.

• CIRM - Probabilistic techniques for random and time-varying dynamical systems (3rd - 7th October 2022).

Poster Presentation On the quasi-ergodicity of absorbing Markov chains, including random logistic maps with escape.  Conference page.

• Krakow IOWTA - 33rd International Workshop on Operator Theory and its Applications (05th -10th Sept 2022)

Delivered a presentation with the title Quasi-ergodic measures for absorbing Markov processes. Workshop page.

• University of Bath - Pólya urns, stochastic approximation and quasi-stationary distributions: new developments (11th -14th April 2022)

Delivered a presentation with the title Existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains: a Banach lattice approach. Workshop page.

• Freie Universität Berlin - Workshop on Quasi-stationary and Quasi-ergodic measures workshop (31st March - 1st Apr 2022)

Local workshop among  Freie Universität Berlin, Max Planck Institute and Imperial College London, organised by Maximilian Engel (FUB) and Zachary Adams (MPG). Delivered a presentation on the logistic Markov process with escape.

• 7th International Conference on Random Dynamical Systems (21st - 25th April 2022).

Delivered a presentation with the title Existence and Uniqueness of Quasi-stationary and Quasi-ergodic Measures for Absorbed Markov Processes”.  Event Page.

• Oxford - October 2021 CDT in Maths of Random Systems Workshop. (22nd October 2021).

Delivered a presentation with the title Existence and Uniqueness of Quasi-stationary and Quasi-ergodic Measures for Absorbed Markov Processes}. Workshop Page.

• Institute of Mathematics, VAST - Graduate School - Mathematics of Random Systems: Analysis, Modelling and Algorithms. (06th -10th September 2021)

Delivered a presentation with the title Existence and Uniqueness of Quasi-stationary and Quasi-ergodic Measures for Absorbed Markov Processes. Event Page.

• XXIV UNICAMP's Undergraduate Research Congress (October 2016)  

Poster presentation Piecewise Linear Dynamic Systems: Local and Global Theory. Proceedings here. (only in Portuguese).

1. Conditioned stochastic stability of equilibrium states on uniformly expanding repellers (2nd May 2024).

-with Bernat Bassols-Cornudella, Jeroen Lamb.
Abstract: We propose a notion of conditioned stochastic stability of invariant measures on repellers: we consider whether quasi-ergodic measures of absorbing Markov processes, generated by random perturbations of the deterministic dynamics and conditioned upon survival in a neighbourhood of a repeller, converge to an invariant measure in the zero-noise limit. Under suitable choices of the random perturbation, we find that equilibrium states on uniformly expanding repellers are conditioned stochastically stable. In the process, we establish a rigorous foundation for the existence of “natural measures”, which were proposed by Kantz and Grassberger in 1984 to aid the understanding of chaotic transients.  DOI: arXiv:2405.01343v1.

2. Random Young towers and quenched decay of correlations for predominantly expanding multimodal circle maps (28th March 2023).

-with Giuseppe Tenaglia.  

Abstract: In this paper, we study the random dynamical system $f_\omega^n$  generated by a family of maps $\{f_{\omega_0}: \mathbb S^1 \to \mathbb S^1\}_{\omega_0 \in [-\varepsilon,\varepsilon]},$ $f_{\omega_0}(x) = \alpha \xi (x+\omega_0) +a\ (\mathrm{mod }\ 1),$ where $\xi: \mathbb S^1 \to \mathbb R$ is a non-degenerated map, $a\in [0,1)$, and $\alpha,\varepsilon>0$. Fixing a constant $c\in (0,1)$, we show that for $\alpha$ sufficiently large and $\varepsilon > \alpha^{-1+c},$ the random dynamical system $f_\omega^n$ presents a random Young tower structure and quenched decay of correlations. DOI: arXiv:2303.16345

3. On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape (2nd September 2022).

-with Vincent P. H. Goverse, Jeroen S. W. Lamb, Martin Rasmussen. 

Abstract: In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu$ on $M$ where the transition kernel $\mathcal P$ admits an eigenfunction $0\leq \eta\in L^1(M,\mu)$. We find conditions on the transition densities of $\mathcal P$ with respect to $\mu$, which ensure that $\eta(x) \mu(d x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $\mu$-almost surely.  We apply this result to the random logistic map $X_{n+1} = \omega_n X_n (1-X_n)$  absorbed at $\mathbb R \setminus [0,1],$ where $\omega_n$ is an i.i.d sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$ DOI: https://doi.org/10.1017/etds.2023.69.

4. The Lyapunov spectrum for conditioned random dynamical systems  (8th April 2022).

-with Dennis Chemnitz, Hugo Chu, Maximilian Engel, Jeroen S. W. Lamb, Martin Rasmussen.

Abstract: We establish the existence of a full spectrum of Lyapunov exponents for memoryless random dynamical systems with absorption. To this end, we crucially embed the process conditioned to never being absorbed, the Q-process, into the framework of random dynamical systems, allowing us to study multiplicative ergodic properties. We show that the finite-time Lyapunov exponents converge in conditioned probability and apply our results to iterated function systems and stochastic differential equations. DOI: https://doi.org/10.48550/arXiv.2204.04129. To appear in Annales Henri Poincaré.

5. Existence and uniqueness of quasi-stationary and quasi-ergodic measures for absorbing Markov chains: a Banach lattice approach (27th November 2021, Stochastic Processes and their Applications). 

-with Jeroen S. W. Lamb, Guilermo Olicón-Méndez, Martin Rasmussen.

Abstract: We establish existence and uniqueness of quasi-stationary and quasi-ergodic measures for almost surely absorbed discrete-time Markov chains under weak conditions. We obtain our results by exploiting Banach lattice properties of transition functions under natural regularity assumptions. DOI: https://doi.org/10.1016/j.spa.2024.104364. 

6. A note on Vishik's normal form (12th November 2019, Journal of Differential Equations). 

-with Ricardo M. Martins, Douglas D. Novaes.

Abstract:  Vishik's Normal Form provides a local smooth conjugation with a linear vector field for smooth vector fields near contacts with a manifold. In the present study, we focus on the analytic case. Our main result ensures that for the analytic vector field and manifold, the conjugation with Vishik's normal form is also analytic. As an application, we investigate the analyticity of Poincaré Half Maps defined locally near contacts between the analytic vector field and manifold. DOI: https://doi.org/10.1016/j.jde.2021.02.011.

• Mathematica - Advanced Knowledge 

• Matlab/Octave - Advanced Knowledge 

• Python - Advanced Knowledge

• C/C++ - Intermediate Knowledge