English Version VF Julian Tugaut

Associate Professor
Hors Classe
Université Jean Monnet

    In 2025, Boris Nectoux and me are offering a PhD fellowship on the reflected McKean-Vlasov diffusions. Feel free to contact us if you are interested.

    If you are a master student, that you want to learn more about the nonlinear diffusions (and about their wide applications) and that you would like to follow an internship (eventually to prepare a PhD) on this subject, do not hesitate to send me an email to the address: julian (dot) tugaut (at) univ-st-etienne (dot) fr

    My works, since my PhD, are dealing with probabilities. More precisely, I work on stochastic analysis. By making research on this domain, I also touch to the functional analysis and analysis of partial differential equations.


    Let us remind that the Brownian motion is a collection of Gaussian random variables with independent increments and with almost surely continuous trajectories. It is the probabilistic interpretation of the Laplacian. One can see it as the heat process with the microscopical viewpoint.

    Let \(b\) and \(\sigma\) be two functions from \(\mathbb{R}_+\times\mathbb{R}^d\) to \(\mathbb{R}^d\). A diffusion is a solution to a stochastic differential equation of the form \[ X_t=X_0+\int_0^t\sigma(s,X_s){\rm d}B_s+\int_0^tb(s,X_s){\rm d}s\,, \] where \(X_0\) is a random variable satisfying good properties and where the integration involving the Brownian motion in the previous line is in the Itō sense. A classical example of such diffusion is the following: \[ X_t=X_0+\sigma B_t-\int_0^t\nabla V(X_s){\rm d}s \] where \(\sigma\) is a positive real and \(V\) is a potential on \(\mathbb{R}^d\). There is a natural link between this diffusion \((X_t)_t\) and a partial differential equation. Indeed, a classical result is as follows: the law of \(X_t\) is absolutely continuous with respect to the Lebesgue measure, for any \(t>0\). Moreover, its density, that we denote as \(u\), that is to say defined as \(u(t,x){\rm d}x:=\mathbb{P}\left(X_t\in\left[x;x+{\rm d}x\right]\right)\), is a solution to the following partial differential equation: \[ \frac{\partial u}{\partial t}={\rm div}\left\{\frac{\sigma^2}{2}\nabla u+u\nabla V\right\}\,. \] The stochastic equation admits a unique invariant probability measure (and in the same vein, the partial differential equation admits a unique stationary solution with total mass equal to \(1\)). This invariant probability measure is \(\displaystyle Z_{\sigma}^{-1}{\rm e}^{-\frac{2}{\sigma^2}V(x)}{\rm d}x\) where the quantity \(Z_{\sigma}^{-1}\) allows a suitable renormalization. Under some assumptions, we also have the long-time convergence towards this unique steady state. Namely, \(\displaystyle\lim_{t\to+\infty}u(t,x)=Z_\sigma^{-1}{\rm e}^{-\frac{2}{\sigma^2}V(x)}\).


    In the latter example, the diffusion is homogeneous since the potential \(V\) does not depend on the time. In my work, I am interested in diffusions which are not homogeneous (and possibly associated to a nonlocal partial differential equation), namely of the form: \[ X_t=X_0+\sigma B_t-\int_0^t\nabla_x W(s,X_s){\rm d}s\,. \]     Here, the potential \(W(s,\cdot)\) does depend on the time \(s\). My work may be applied in a non-conservative force case. However, here, I choose to present it in a typical case of nonlinear diffusion.


    What is the link between the deep neural networks, the charge and the discharge of the cathod in a lithium-ion battery, the plasmas, the polymers, the nonlinear filtering, the finance, the molecular dynamics, the sampling, the population dynamics and the econophysic? There is an underlying nonlinear diffusion.

    The dynamics of lithium-ion batteries are complex and are often approximated by models consisting of nonlinear partial differential equations about the internal ionic concentrations in a grid of atoms. The microscopical viewpoint corresponds to a mean-field system of particles. A crucial question about this model consists in estimating the first time that the concentration goes from a given minimum to another one.

    Optimizing multi-modal functions is a key in the training of the deep neural networks. The metastable behaviour of the linear diffusions is well-known. Namely, the diffusion remains an exponentially long time in a local minimum before exiting from its basin of attraction. Despite very few is known about the metastable behaviour for nonlinear diffusions, these nonlinear diffusions are widely used.

    The main goal of my research is to deal theoretically and numerically with this kind of questions and the related ones: can we estimate the exit-time and the exit-location of a broad variety of nonlinear diffusions in nonconvex settings?



    An example of inhomogeneous diffusion is the so-called self-stabilizing one, the latter being a particular instance of McKean-Vlasov diffusions and it is the probabilistic interpretation of the granular media. This latter example may be applied to several phenomena. Let us cite four of them: the plasmas, the social interactions, the contraction of the muscular cells, the interconnectivity of the neurons.

    Let us consider a large number of particles (with the physical meaning) which enter in inelastic collision in a random medium. The velocity is conserved and the kinetic energy dissipates. Taking a suitable renormalization as the number of particles goes to infinity yields the McKean-Vlasov diffusion \[ X_t=X_0+\sigma B_t-\int_0^t\nabla V(X_s){\rm d}s-\int_0^t\Big[\nabla F\ast\mathcal{L}\left(X_s\right)\Big](X_s){\rm d}s \] to represent the move of one particle (among the whole system). The process is generated by three competing forces: a Brownian motion (the random part of the model), a so-called confining potential \(V\) (which corresponds to the friction that is the exterior force acting on the particles) and the so-called interacting potential \(F\) (which corresponds to the inelastic collisions between the particles).

    Under suitable regularity assumptions on \(V\) and \(F\), the equation admits a unique strong solution \(\left(X_t\right)_{t\in\mathbb{R}_+}\) provided that the initial random variable \(X_0\) admits a moment of a given order, the latter one directly depending on the growth of the two potentials). Moreover, the second moment is uniformly bounded with respect to the time so that the family \(\left\{\mathcal{L}\left(X_t\right)\right\}_{t\in\mathbb{R}_+}\) is tight.

    The diffusive term allows an automatic regularization of the law. Hence, \(\mathcal{L}\left(X_t\right)\) is absolutely continuous with respect to the Lebesgue measure for any \(t>0\). Furthermore, its density - that we denote as \(u_t(x)\) (that is to say we have \(\mathbb{P}\left(X_t\in[x;x+{\rm d}x]\right)=u_t(x){\rm d}x\)) - satisfies the so-called granular media equation that is folowing nonlinear partial differential equation: \[ \frac{\partial}{\partial t}u_t(x)={\rm div}\left\{\frac{\sigma^2}{2}\nabla u_t(x)+u_t(x)\nabla V(x)+u_t(x)\nabla F\ast u_t(x)\right\}\,. \] This partial differential equation macroscopically expresses the microscopical behaviour of the particles system.

    Indeed, a well-chosen coupling associates the McKean-Vlasov diffusion to the mean-field system of particles. Let us look at \(N\) such independent McKean-Vlasov diffusions: \[ X_t^i=X_0^i+\sigma B_t^i-\int_0^t\nabla V(X_s^i){\rm d}s-\int_0^t\nabla F\ast u_s(X_s^i){\rm d}s\,. \] The Brownian motions (and the initial random variables) are assumed to be independent. We also consider the mean-field interacting particles system: \[ Y_t^i=X_0^i+\sigma B_t^i-\int_0^t\nabla V(Y_s^i){\rm d}s-\frac{1}{N}\sum_{j=1}^N\int_0^t\nabla F\left(Y_s^i-Y_s^j\right){\rm d}s\,. \] Intuitively, the empirical measure \(\displaystyle\frac{1}{N}\sum_{j=1}^N\delta_{Y_t^j}\) converges to \(u_t\) as \(N\) goes to infinity. This phenomenon is denoted as propagation of chaos. By using the exchangeability of the particles, the coupling implies the convergence towards \(0\) of the quantity \(\displaystyle\sup_{t\in[0;T]}\mathbb{E}\left\{\left|\left|X_t^1-Y_t^1\right|\right|^2\right\}\) for any \(T>0\).

    The behaviour of \(u_t(x)\) as \(t\) goes to infinity is not immediate. In the same vein, the uniqueness of the invariant probability measure is not. Concerning the exit-problem, it is not too. Nonetheless, we now have partial results on these questions (steady states, convergence, rate of convergence and basins of attraction as well as metastability).

    Some collaborators: Samuel Herrmann (Full Professor at Burgundy University, France), Hong Duong (Assistant Professor at University of Birmingham, United Kingdom), Bartłomiej Dyda (Assistant Professor at Uniwersytet Wroclawski, Poland), Pierre Del Moral (Senior Researcher INRIA, France), Aline Kurtzmann (Associate Professor at Lorraine University, France), Olivier Alata (Full Professor at Jean Monnet University, France), Gonçalo Dos Reis (Lecturer at University of Edinburgh, United Kingdom), Jean-François Jabir (Assistant Professor at Higher School of Economics, Russia), Mario Maurelli (Assistant Professor at Pisa, Italy), David Ruiz Baños (Assistant Professor at Oslo, Norway), Paul-Éric Chaudru de Raynal (Associate Professor at Nantes, France), Pierre Monmarché (Associate Professor at Sorbonne University, France), Milica Tomašević (CNRS junior researcher, France)...

    Some visits outside France:.