About • Papers • CV

I am an Associate Professor (Maître de conférences) in Probability at the Institut de Mathématiques de Bourgogne (IMB), Université Bourgogne Europe, within the SPOC team at the UFR Sciences & Techniques.

I received my PhD (2012) from Université de Rennes 1 (IRMAR) under the supervision of Mihai Gradinaru, and held a 2012–2013 postdoc at Université de Neuchâtel with Michel Benaïm.

My research focuses on probability theory, with a particular emphasis on random walks in random and non-random environments, Markov chains, stochastic differential equations and continuous-time stochastic processes, especially their long-time behavior (recurrence vs transience, scaling limits, phase transitions, and related phenomena). More details below:

• Time-inhomogeneous diffusions: scale-invariant drifts in time and space, random environments, scaling limits, phase transition phenomena, recurrence and transience.

  We studied one-dimensional time-inhomogeneous diffusions of the form $$dX_t = dB_t - \tfrac{1}{2}V'(t,X_t)\,dt,$$ with scale-invariant potentials.

  In the deterministic case, $V(t,x)= c\,t^{-\beta}x^{\alpha}$, this yields distorted Brownian motions with regimes precisely characterized in terms of recurrence, transience, asymptotic distributions, and explosion rates. The scaling limits can take the form $$\frac{X_t}{t^\gamma}\;\xrightarrow[t\to\infty]{}\;\mu,$$ where the behavior may be subdiffusive, diffusive, or superdiffusive, and where the limit law $\mu$ is not necessarily Gaussian. Roughly speaking, the values of $\gamma$ and $\mu$ depend on the position of $(\alpha,\beta)$ with respect to the critical line $2\beta=\alpha+1$.

  In the random case, $V(t,x) = c\,t^{-\beta}W'(x)$ with $W$ a two-sided Brownian motion, these models become time-inhomogeneous Brox-type diffusions. We obtained quenched and annealed convergence results, identified invariant and quasi-invariant measures, and described distinct diffusive behaviors together with convergence speeds. In the critical regime $\beta=1/4$ (the scale invariance of $W^\prime$ corresponds in some sense to $\alpha=-1/2$), the scaling limit takes the form $$\mathcal{L}\!\left(\frac{X_t}{\sqrt{t}}\right)-\mu_{T_t W}\;\longrightarrow\;0,$$ in total variation, where $\mu_\omega$ is a random measure associated with a random dynamical system and $$T_tW(x)=\frac{W(e^t x)}{e^{t/2}}.$$ When $\beta$ is above the critical threshold, one recovers a classical diffusive scaling limit. The case $\beta$ below the threshold remains open.

• Persistent random walks: functional scaling limits, VLMC increments, anomalous diffusion, recurrence and transience, phase transition phenomena.

  Persistent random walks are long-memory processes of the form $$S_n=\sum_{k=1}^n X_k,$$ where $(X_k)$ is generated by a variable-length Markov chain (VLMC). We established recurrence/transience criteria in one dimension without stationarity assumptions, and functional scaling limits described by anomalous diffusions (arcsine Lamperti processes) with phase transitions driven by persistence times. In higher dimensions, Fourier and series criteria (à la Chung–Fuchs) characterize recurrence, with subtle differences between persistent walks and their Markovian skeletons, including counterexamples to classical conjectures.

  In terms of scaling, one obtains $$\frac{X_{Lt}-m(L)t}{\sigma(L)} \;\xrightarrow[L\to\infty]{}\; Z(t),$$ where the limit process $Z$ may be a Brownian motion, a stable Lévy process, or an anomalous diffusion retaining memory of the underlying dynamics.

• Maximum Entropy Random Walks (MERWs): infinite setting, random environments, scaling limits, combinatorics, growth processes, phase transitions and localization phenomena.

  More recently, we have focused on Maximum Entropy Random Walks (MERWs), a class of processes that maximize entropy production along trajectories. For a finite irreducible graph with adjacency matrix $A$, a MERW is defined by the transition kernel $$P(x,y)=\frac{\psi(y)}{\lambda\,\psi(x)},\qquad A\psi=\lambda\psi,$$ where $\psi$ is the Perron–Frobenius eigenvector and $\lambda$ its eigenvalue. This construction yields a path measure whose entropy rate is $\ln(\lambda)$, maximal among all positive recurrent chains on the same graph.

  The finite case is explicit, but extending MERWs to infinite graphs raises subtle challenges of existence, uniqueness, and approximation. In recent work, we analyzed MERWs on $\mathbb{Z}$ with loops, both random and non-random, using explicit combinatorial representations of Perron–Frobenius eigenvectors. This allowed us to determine precisely the asymptotic behavior of these walks and to show, in particular, that essentially all such MERWs have positive speed. In the random environment case, MERWs exhibit localization phenomena directly connected to the Anderson localization phenomenon, where disorder induces spatial concentration of eigenfunctions and of the path measure itself.

  Beyond one-dimensional settings, we studied MERWs on infinite directed acyclic graphs, notably weighted Bratteli diagrams, which naturally model growth processes such as the Young lattice. In this context, MERWs coincide with central Markov chains, a rich framework that yields new combinatorial identities. Applications include retrieving and extending hook-length formulas for trees, interpreting the binary search tree process as a MERW, and connecting preferential attachment models to entropy maximization. These results also highlight links with Poisson–Dirichlet limits via the Chinese restaurant process, and with limit shapes of Young diagrams under the Plancherel measure.

  Finally, we investigated MERWs on weighted spider networks as a tractable infinite model. This led to unified proofs of scaling limits, the discovery of phase transitions, and localization phenomena arising from the presence of central nodes (with potential applications to models of polymers or proteins). Altogether, MERWs offer a powerful setting where classical processes such as the Bessel(3) diffusion can be reinterpreted as entropy-maximizing dynamics, bridging probability, combinatorics, and statistical physics.

For my publications and preprints, see my research page, as well as arXiv, HAL, ORCID, Google Scholar, ResearchGate.