English Version VF Julian Tugaut

Associate Professor
Hors Classe
Université Jean Monnet

    If you are a master student, that you want to learn more about the nonlinear diffusions (and about their applications) and that you would like to make 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. By making research on this domain, I also touch to the functional analysis and analysis of partial differential equations.



    More precisely, I work on stochastic analysis. 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. 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 \(\sigma>0\) is a constant and \(V\) is a potential on \(\mathbb{R}^d\). There exists a natural link between this diffusion \((X_t)_t\) and a partial differential equation. Indeed, a classical result is that 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 \(u\), 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 way, the partial differential equation admits a unique stationary solution with total mass equal to \(1\)) that is to say \(Z{\rm e}^{-\frac{2}{\sigma^2}V(x)}{\rm d}x\). Under some assumptions, we also have the long-time convergence towards this unique steady state.

    In the latter example, the diffusion is homogeneous since the potential \(V\) does not depend on the time. Personaly, I am interested in diffusions which are not time-homogeneous, that is to say 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 \(W\) does depend on the time. My work may be done in a non-conservative force case. However, I choose here to present 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 batterie, the plasmas, the polymers, the filtering, the finance, the molecular dynamic, 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 non-linear 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 one minimum to another one.

    Optimizing multi-modal functions is a key in the training of the deep neural networks. The metastable behavior of the linear diffusions is well-known. In other words, the diffusion remains an exponential time in a local minimum before exiting. Despite very few is known about the metastable behaviour for non-linear diffusions, these non-linear 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 non-linear diffusions in non-convex settings?



    An example of inhomogeneous diffusion is the one of McKean-Vlasov, which is the probabilistic interpretation of the granular media. This 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 which enter in inelastic collision in a random medium. The velocity is conserved and the kinetic energy dissipates. A good renormalization when the number of particles tends to infinity implies that the McKean-Vlasov diffusion \[ X_t=X_0+\sigma B_t-\int_0^t\nabla V(X_s){\rm d}s-\int_0^t\left[\nabla F\ast\mathcal{L}\left(X_s\right)\right](X_s){\rm d}s \] does represent the move of one particle (among an infinite number of ones). The process is generated by three competing forces: a Brownian motion (the random part of the model), a potential \(V\) denoted as confining (which corresponds to the friction) and the potential \(F\) denotes as interacting (which corresponds to inelastic collisions).

    Under regularity assumptions on \(V\) and \(F\), the equation admits a unique strong solution \(\left(X_t\right)_{t\in\mathbb{R}_+}\) if the initial random variable \(X_0\) admits a moment of a given order. 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 folowing granular media partial differential equation: \[ \frac{\partial}{\partial t}u_t(x)={\rm div}\left\{\frac{\sigma^2}{2}\nabla u_t(x)+u_t(x)\nabla V(x)+u_t(x)\nabla F\ast u_t(x)\right\}\,. \] The latter equation macroscopically translates the microscopical behavior of the particles system.

    Indeed, a well-chosen coupling associates it the McKean-Vlasov diffusion. Let us look at \(N\) such independent 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. We talk about 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 \(T>0\).

    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), Yulong Lu (Assistant Professor at University of Massachusetts Amherst, USA), David Ruiz Baños (Assistant Professor at Oslo, Norway), Paul-Eric 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:.