1. Nonlinear diffusions in population dynamics


2. Metastability of general McKean-Vlasov diffusions


3. Competition between common noise and idiosyncratic noise


4. Constrained nonlinear diffusions


5. Other perspectives

Nonlinear diffusions in population dynamics

    One of my research perspective is to use the nonlinear diffusions in population dynamics. This twofold perspective is made of two axes which are intrinsically related. First axis does correspond to the competition (or the cooperation) between two different populations, each of these being modeled by an interacting particles system. The second axis does correspond to the breakup of alignment inside a population of animals, each of these individuals being modeled by a particle in a system of interacting particles following the stochastic Cucker-Smale model with small additive noise.


    Firstly, I will introduce the first axis, which corresponds to the study of multi-agents multi-species system. To do so, I will here limit myself to the case of coupled self-stabilizing diffusions: \[ \left\{\begin{array}{l} X_t=X_0+\sigma B_t-\int_0^t\nabla V_1\left(X_s\right){\rm d}s-\int_0^t\Big[a\nabla F_{1,1}\ast\mu_s\left(X_s\right)+(1-a)\nabla F_{1,2}\ast\nu_s\left(X_s\right)\Big]{\rm d}s\,,\\ Y_t=Y_0+\sigma \widetilde{B_t}-\int_0^t\nabla V_2\left(Y_s\right){\rm d}s-\int_0^t\Big[a\nabla F_{2,1}\ast\mu_s\left(Y_s\right)+(1-a)\nabla F_{2,2}\ast\nu_s\left(Y_s\right)\Big]{\rm d}s\,,\\ \mathbb{P}\left(X_t\in {\rm d}x\right)=\mu_t({\rm d}x)\,,\\ \mathbb{P}\left(Y_t\in {\rm d}x\right)=\nu_t({\rm d}x)\,. \end{array}\right. \] Here, \(F_{1,1}\) and \(F_{2,2}\) correspond to the intra-species interaction potentials while \(F_{1,2}\) and \(F_{2,1}\) are the inter-species interaction potentials. And, \(V_1\) (respectively \(V_2\)) is the external potential acting on the first (respectively the second) population. Concerning \(B\) and \(\widetilde{B}\), they are independent Brownian motions. Moreover, \(X_0\) and \(Y_0\) are independent from the two Wiener processes.

    Some first results have been obtained in collaboration with Hong Duong. More precisely, in the case of coupled self-stabilizing diffusions, we have established the existence of a unique strong solution to the system of equations, a propagation of chaos through a coupling with the associated particles systems and we have exhibited suitable assumptions ensuring the non-uniqueness of the invariant probability measures. This paper has been published in Stochastics. Hong Duong, Greg Pavliotis and me are currently writting about the generalization to the case of \(M\) species and in particular we will completely characterize the invariant probability measures and we will establish the long-time convergence with entropy techniques. Finally, we will generalize the laws of Arrhenius and of Kramers'type that I have obtained for McKean-Vlasov diffusions (one-specy case) to the multi-species processes. Another question concerns the first time that the two groups of individuals begin to interact when the inter-species interaction has low-range. To study the asymptotics of such time will be allowed by the techniques that Jean-François Jabir and me have developed for the collision-problem.


    I now introduce the second axis of this perspective: the study of the Cucker-Smale model when putting an additive gaussian noise. The Cucker-Smale model is the first mathematical model to have been able to understand, in its deterministic version, the agglomeration of population of animals when the number of individuals is large. It consists in a kinetic system where the speed depends on the "mean" speed of the ensemble of individuals. The equation admits the following form: \[ \left\{\begin{array}{l} \frac{{\rm d}x_i}{{\rm d}t}=v_i\\ \frac{{\rm d}v_i}{{\rm d}t}=-\frac{1}{N}\sum_{j=1}^NH\left(\left|x_i-x_j\right|\right)\left(v_i-v_j\right)\,, \end{array}\right. \] where \(x_i\) denotes the position of the individual \(i\) and \(v_i\) its speed. Here, \(H\) is called the communication rate. In the original model, it is like so: \[ H(x)=\frac{K}{\left(\zeta^2+|x|^2\right)^\gamma}\,, \] \(K\), \(\zeta\) and \(\gamma\) being positive parameters. Of course, the further is the individual \(j\) from the individual \(i\), the less they interact to align their speed.

    Let us mention that this model does not contain any source of randomness. The one that we are interested in is a weakly noisy version of the latter model: \[ \left\{\begin{array}{l} {\rm d}x_i=v_i\,{\rm d}t\\ {\rm d}v_i=\sigma {\rm d}B_t^i-\frac{1}{N}\sum_{j=1}^NH\left(\left|x_i-x_j\right|\right)\left(v_i-v_j\right)\,{\rm d}t\,, \end{array}\right. \] where \(\sigma>0\) is constant and intended to be small and where the Brownian motions \(B^i\) are independent.

    The main result that we aim to establish is the breakup of alignment in the small-noise limit. Namely, in the noiseless model, we know that we have \[ \lim_{t\to+\infty}\sup_{i\neq j}|v_i-v_j|=0\,. \] Hence, we aim to capture the first time that two individuals of the noisy model would not be aligned anymore. More precisely, we wish to prove a Kramers'type law for this breakup time, as the diffusion coefficient tends to \(0\). This work is supported by the Jean Monnet University through a PhD fellowship, that we have given to a very good candidate: Hetranso AHNI. Mainly, his work consists in extending our results on the collision-problem to a Langevin framework and without assuming any independency between the processes, which will be a prominent result.

Metastability of general McKean-Vlasov diffusions

    The second perspective that I see in my research concerns the study of the metastability itself for nonlinear diffusions. Up to now, I succeeded in establishing results concerning the exit-times, the invariant probability measures and the long-time convergence. However, despite these problems are intrinsically related, very few is known about their joint use. For example, when the diffusion coefficient is constant and small, what are the basins of attraction (long-time problem) and the rate of convergence towards the steady states for McKean-Vlasov diffusions?

    Even with "simple" instances of such diffusions, as for example the self-stabilizing diffusion, the problem is open provided that the confining potential is not uniformly convex. Let us introduce here the simplest possible case: \[ {\rm d}X_t=\sigma{\rm d}B_t-\left(X_t^3-X_t\right){\rm d}t+\alpha\left(X_t-\mathbb{E}\left[X_t\right]\right){\rm d}t\,. \] Here, the external potential is \(V(x):=\frac{x^4}{4}-\frac{x^2}{2}\) and the interacting one is \(F(x):=\frac{\alpha}{2}x^2\). If \(\alpha\) is sufficiently large (in this explicit example, it is sufficient to ensure \(\alpha\geq\frac{1}{4}\)), the result that I published in Kinetic and Related Models specifies that if \(\mathcal{L}(X_0)\) is compactly supported in \((0;+\infty)\), then for \(\sigma\) small enough, \(\mathcal{L}(X_t)\) weakly converges towards the invariant probability measure with positive first moment.

    If the law of \(X_0\) does not satisfy this property, very few is known. In particular, we can observe that the stationary measures of the noiseless dynamics are reduced to \(\delta_{-1}\), \(\delta_0\) and \(\delta_1\) when \(\alpha\geq1\), namely when synchronization occurs (which ensures that the convexity of \(F\) compensates the lack of convexity of \(V\)) which pushes the different trajectories to synchronize themselves and to take a common value). However, if \(\alpha\) is less than \(1\), some other steady states emerge. These states are of the form \(p\delta_{a_1(p)}+(1-p)\delta_{a_2(p)}\) with \(p\) belonging to an interval with non-empty interior that is to say there is a continuum of such steady states. Here, \(a_1(p)\in(-1;0)\) and \(a_2(p)\in(0;1)\).

    Yet, for almost any \(p\) in this continuum (with respect to the Lebesgue measure), putting \(W_p(x):=V(x)+F\ast\left(p\delta_{a_1(p)}+(1-p)\delta_{a_2(p)}\right)(x)\), the potential \(W_p\) does not satisfy the hypothesis \(W_p(a_1(p))=W_p(a_2(p))\). However, any limit, as the noise vanishes, of any invariant probability measure is of the form \(p\delta_{a_1(p)}+(1-p)\delta_{a_2(p)}\) with \(W_p'(a_1(p))=W_p'(a_2(p))=0\) and \(W_p(a_1(p))=W_p(a_2(p))\). Thus, in the small-noise (albeit positive noise) case, these states are just metastable.

    I guess that if the initial law is one of this metastable state with \(p\notin\left\{0;1;\frac{1}{2}\right\}\) and if \(\Delta(p):=W_p(a_1(p))-W_p(a_2(p))\) is positive, the long-time limit for a sufficiently small diffusion coefficient \(\sigma\) is the invariant probability measure close to \(\delta_1\). More generally, if we consider the granular media equation (that is a nonlinear partial differential equation): \[ \frac{\partial \mu_t^\sigma}{\partial t}=\frac{\sigma^2}{2}\Delta\mu_t^\sigma+{\rm div}\left\{\mu_t^\sigma\left(\nabla V+\nabla F\ast\mu_t^\sigma\right)\right\}\,, \] as well as its noiseless version \[ \frac{\partial \mu_t^0}{\partial t}={\rm div}\left\{\mu_t^0\left(\nabla V+\nabla F\ast\mu_t^0\right)\right\}\,, \] we may establish that if \(\displaystyle\lim_{t\to+\infty}\mu_t^0=p\delta_{a_1(p)}+(1-p)\delta_{a_2(p)}\), the limiting probability as the time \(t\) goes to infinity of \(\mu_t^\sigma\) is the invariant probability near \(\delta_1\) provided that \(\Delta(p)>0\). In particular: \[ \delta_1=\lim_{\sigma\to0}\lim_{t\to+\infty}\mu_t^\sigma\neq\lim_{t\to+\infty}\lim_{\sigma\to0}\mu_t^\sigma=p\delta_{a_1(p)}+(1-p)\delta_{a_2(p)}\,. \] To prove this conjecture, I imagine the following simple approach: the control of the law through the exit-times, that is by using metastability. Namely, I will look for \(\displaystyle\mu_{t(\sigma)}^\sigma=\mathcal{L}(X_{t(\sigma)}^\sigma)\) where the time \(t(\sigma)\) follows a typical characteristic scale of metastability of the process. Hence, I will obtain the basins of attraction and the rate of convergence of the self-stabilizing diffusion in a simple case. More general cases where \(V\) is not as simple will follow. More general cases where the interaction term is not linear will follow. Finally, I will be interested in non-reversible cases.

    More generally, I will show similar results on the long-time (in the small-noise regime) of general McKean-Vlasov diffusions of the form \[ {\rm d}X_t^\sigma=b\left(X_t^\sigma,\mu_t^\sigma\right){\rm d}t+\sigma~c\left(X_t^\sigma,\mu_t^\sigma\right){\rm d}B_t\,, \] where the drift term \(b\) and the diffusion term \(c\) are sufficiently regular for the equation to admit a unique strong solution on \(\mathbb{R}_+\) for any \(\sigma>0\). For instance, we assume that \(b\) and \(c\) are \({\rm L}-\)differentiable. Besides, I will also treat the case of diffusions of the following form: \[ {\rm d}X_t^\sigma=b\left(X_t^\sigma,\frac{1}{t}\int_0^t\mu_s^\sigma{\rm d}s\right){\rm d}t+\sigma c\left(X_t^\sigma,\frac{1}{t}\int_0^t\mu_s^\sigma{\rm d}s\right){\rm d}B_t\,. \] Such results will notably have impact in machine learning, in mean-field games and in molecular dynamics.

    To establish these results, my strategy will be to use the techniques that I developed on the first exit-times. In particular, I will prove some Kramers'type laws, I will get an Arrhenius law and I will show that the exit-times have an exponential behaviour in the small-noise limit (this result has first been obtained by Day for Itō diffusions). Then, it will be possible to precisely identify \(\displaystyle\lim_{\sigma\to0}\mu_{t(\sigma)}^\sigma\) where \(t(\sigma):=\exp\left[\frac{2H}{\sigma^2}\right]\), for any \(H>0\). For instance, it is not difficult to imagine that if \(\displaystyle\lim_{t\to+\infty}\mu_t^0=p\delta_{a_1(p)}+(1-p)\delta_{a_2(p)}\) then the characteristic time of convergence towards the invariant probability measure will be such a \(t(\sigma)\) where \(H:=H_0(p)\) does correspond to the minimum between the exit-cost from attractor \(a_1(p)\) and the one of attractor \(a_2(p)\).

Competition between common noise and idiosyncratic noise

    The classical McKean-Vlasov models, namely the ones that I have previously considered on this page and on the one of the results that I obtained (see here), are suffering from a failure related to the model itself. A classical McKean-Vlasov diffusion may be seen as the interpretation of the mean-field limit from an interacting particles system of the form: \[ {\rm d}X_t^{i,N}=b\left(X_t^{i,N},\frac{1}{N}\sum_{j=1}^N\delta_{X_t^{j,N}}\right){\rm d}t+c\left(X_t^{i,N},\frac{1}{N}\sum_{j=1}^N\delta_{X_t^{j,N}}\right){\rm d}B_t^i\,, \] for any \(i\in[\![1;N]\!]\), the number of particles \(N\) being intended to be large and where \(b\) and \(c\) satisfy suitable properties like the \({\rm L}-\)differentiability. We stress that \(\left(B^i\right)_{i\in\mathbb{N}^*}\) is a family of independent Brownian motions. A classical propagation of chaos result yields the solution to the McKean-Vlasov equation \[ {\rm d}X_t^{1,\infty}=b\left(X_t^{1,\infty},\mathcal{L}\left(X_t^{1,\infty}\right)\right){\rm d}t+c\left(X_t^{1,\infty},\mathcal{L}\left(X_t^{1,\infty}\right)\right){\rm d}B_t^1 \] to be a good approximation of the diffusion \(X^{1,N}\) as \(N\) is large. More precisely, we can write \[ \lim_{N\to+\infty}\mathbb{E}\left[\sup_{t\in[0;T]}\left|X_t^{1,\infty}-X_t^{1,N}\right|^2\right]=0\,, \] for any \(T>0\). Nevertheless, the hypothesis previously used namely that the gaussian noise assigned to the particle \(i\) is independent from the one assigned to the particle \(j\) if \(i\neq j\) is not really acceptable in a physical viewpoint. Indeed, it is difficult to believe that the cause of the idiosyncratic noise \(B^i\) does not generate any perturbation, does not have any effect on the particle \(j\).

    It is why a more realistic model, with a good physical interpretation hence a better model has been introduced. It is the model with a common noise. Then, the equation satisfied by the particle \(i\) is the following: \[ {\rm d}X_t^{i,N}=b\left(X_t^{i,N},\frac{1}{N}\sum_{j=1}^N\delta_{X_t^{j,N}}\right){\rm d}t+c\left(X_t^{i,N},\frac{1}{N}\sum_{j=1}^N\delta_{X_t^{j,N}}\right){\rm d}B_t^i+f\left(X_t^{i,N},\frac{1}{N}\sum_{j=1}^N\delta_{X_t^{j,N}}\right){\rm d}W_t\,, \] where \(W\) is a Wiener process independent from the idiosyncratic noises. Besides, \(f\) satisfies suitable assumptions so that the equation in dimension \(dN\) is well-posed.

    Of course, the propagation of chaos phenomenon differently behaves in this case. In particular, despite the compensation of the idiosyncratic noise leading to a Laplacian term, the common noise being common, the limit in law of the empirical measure \(\displaystyle\frac{1}{N}\sum_{\ell=1}^N\delta_{X_t^{\ell,N}}\) is not deterministic. In fact, it will be the conditional law of \(X_t^{1,\infty}\) with respect to the \(\sigma\)-algebra \(\sigma\left\{W_s\,\,:\,\,0\leq s\leq t\right\}\). Consequently, the introduction of common noise completely changes the paradigm since we naturally obtain a stochastic partial differential equation. The literature on this subject is huge. Indeed, the applications in mean-field games are extremely rich due to the realism of these models. As a consequence, we can find these models in economy, in finance...

    What I am the most interested in concerning these models is the understanding of the influence from each type of noise, in the regime with small temperature. Namely, what are the obervable phenomena due to the common noise? What are those due to the idiosyncratic noises? Paul-Éric Chaudru de Raynal and me will tackle this problem by the large deviations principles. Which noise do what when the system is slightly perturbated? To answer this question, we establish large deviations principles on the particles system, on the associated McKean-Vlasov diffusion; we prove the limit as \(N\) tends to infinity of the good rate function \(\mathcal{I}^N\) driving the typical behaviour of the first particle towards the good rate function \(\mathcal{I}^\infty\) driving the typical behaviour of the McKean-Vlasov diffusion.

    Besides, we will obtain some results on the asymptotics of the first exit-times and of the first collision-times. Let us stress that some first achievements in a relatively simple case allow us to see a separation between the behaviours that is to say we can precisely see the cooperation or the competition between the common noise and the idiosyncratic noises in the weakly perturbed regime. It is worth noticing that in this simple case, the exit-location does depend on the interplay between the common noise and the idiosyncratic ones. Furthermore, to obtain the exit-time is a non-trivial consequence of the work done by Jean-François Jabir and me on the collision-times. This strongly suggests that we may be able to proceed similarly in more general cases that are not of Ornstein-Uhlenbeck type.

Constrained nonlinear diffusions

    The perspective that I describe here is protean. Indeed, it consists sometimes to add a reflection on the boundary of a domain that is compact or not, convex or not, to consider several types of reflection (normal one, oblique one...), to enforce the positiveness of \(\mathbb{E}[h(X_t)]\) where \(h\) is a potential on \(\mathbb{R}^d\), to add jumps which modify the different vector fields and also to add temporal perturbations, which may be used to model the cathod charge in a lithium battery.


    Firstly, I have already obtained first results on the case of reflected nonlinear diffusions, published in Stochastic Processes and their Applications in collaboration with Daniel Adams, Gonçalo Dos Reis, Romain Ravaille and William Salkeld. We assumed that the phase space was convex. This hypothesis could easily be replaced by the stability of this domain with respect to the vector field. As a possible extension, I aim to look at more general domains which could possibly break this stability assumption.

    To establish the well-posedness, the long-time behaviour or the propagation of chaos in the case of reflected particles systems is not a simple task. Nevertheless to obtain a large deviations principle is a real challenge. Indeed, consider the following equation: \[ {\rm d}X_t=b(X_t){\rm d}t+\sigma{\rm d}W_t-{\rm d}k_t\,, \] where \(b\) is a vector field, \(W\) is a Wiener process in \(\mathbb{R}^d\) and \(k\) is the bounded variations process which models the reflection on the boundary of the domain \(\mathcal{D}\). If the domain \(\mathcal{D}\) is stable with respect to the vector field, it will be easy to establish a large deviations principle: everything is exactly like if the reflection does not supplement any difficulty. Nonetheless, as soon as this reflection conflicts with the good rate function, we know - up to my knowledge - absolutely nothing. Then, we should rebuild the Freidlin and Wentzell theory by incorporating the reflection.

    Besides, to prove an Arrhenius law, a Kramers'type law or a Day's type result (exponential behaviour of the first exit-time in the small-noise limit) is not obvious, even though the domain \(\mathcal{D}\) is stable by \(b\). In any case, it is a real challenge, especially when considering interacting particles system or nonlinear, in the sense of McKean, diffusions. Moreover, there exist several type of reflections. For instance, in the oblique reflection case, the behaviour is even less classical... And yet, it is thrilling, as well as the weak reflection.

    The weak reflection consists in adding a constraint of the following form to a diffusion \(X\): \[ \mathbb{E}\left[h(X_t)\right]\geq0\,. \] Here, \(h\) is a potential on \(\mathbb{R}^d\). This type of constraint has notably been studied by Paul-Éric Chaudru de Raynal, Arnaud Guillin, Céline Labart...

    The equation driving the proportion of lithium atoms in a lithium battery is as follows: \[ {\rm d}X_t=\sigma {\rm d}B_t-V'\left(X_t\right){\rm d}t+\mathbb{E}\left[V'\left(X_t\right)\right]{\rm d}t+\dot{q}(t){\rm d}t\,, \] where \(q\) is the charge of the battery. Typically, we are interested in a slow charge, of the form \(\dot{q}(t)=\exp\left\{-\frac{2\mu}{\sigma^2}\right\}\). We may generalize to a charge of the type \(\dot{q}_\sigma(t)\) such that \[ \int_0^{\exp\left(\frac{2\mu}{\sigma^2}\right)}\dot{q}_\sigma(t){\rm d}t=1\,. \] Here, \(\mu\) is a positive constant. Let us stress that, even if the charge is constant, this system never reaches an equilibrium since we have \(\displaystyle\mathbb{E}\left[X_t\right]=q(t)=t\exp\left\{-\frac{2\mu}{\sigma^2}\right\}\).

    The interpretation with particles system is like so: \[ {\rm d}X_t^i=-V'\left(X_t^i\right){\rm d}t+\frac{1}{N}\sum_{j=1}^NV'\left(X_t^j\right){\rm d}t+\sigma {\rm d}B_t^i-\frac{\sigma}{N}\sum_{j=1}^N{\rm d}B_t^j\,. \]     We remark that \(\frac{1}{N}\sum_{j=1}^NX_t^j=q(t)\) for any \(t\) and for any \(N\). Let us stress that to stick to the real model, we should add a reflection on the boundary of \([0;1]\) (such a reflection for each particle in fact).


    Another type of constraints that I am interested in is the one where the regimes are varying themselves with respect to the time. The philosophy of this kind of models is as follows: \[ {\rm d}X_t=b_{I_t}(X_t,\mu_t){\rm d}t+\sigma_{I_t}\left(X_t,\mu_t\right){\rm d}W_t\,, \] where \(\mu_t:=\mathcal{L}(X_t)\). Here, \(\left(I_t\right)_{t\geq0}\) is a jump process with values in \([\![1;q]\!]\) and it defines the regime assigned to the diffusion. Hence, we have \(q\) vector fields \(b_1,\cdots,b_q\) and \(q\) diffusion coefficients \(\sigma_1,\cdots,\sigma_q\). We assume that the generator of the jump process \(I\) does depend on the position of \(X\) at time \(t\) but also on the law \(\mu_t\).


    In some cases introduced in this section, the well-posedness itself is problematic. So, firstly, I will have to show that these equations, highly nonlinear for some of them, admit a unique strong solution on \(\mathbb{R}_+\). Then, the results related to the propagation of chaos (quantitative or not, uniform with respect to the time or not) should be established; in particular if we mix such constraints. In a second step, I will find the now classical results on the invariant probability measures and their possible non-uniqueness. Afterwards, I will establish long-time convergence of such diffusions.

    Last but not the least is strongly related to the previous perspectives: it consists in finding the small-noise asymptotics of the first exit-times in order to deduce the rate of convergence of the solution to the associated partial differential equations as well as the characterization of the basins of attraction of each steady state.

Other perspectives

    A thrilling research perspective that I am interested in is the concrete application of my results. To do so, it is worth noticing that in general, in the sciences (molecular dynamics, econophysics, biology, machine learning...), the numerical simulations are of crucial interest. Consequently, the main objects that are studied are not the diffusions themselves but their numerical schemes; namely some random walks. Furthermore, the way that the diffusions are approximated should be thought in a way that the numerical tools can simulate in a reasonable time. In particular, I aim to study the Euler-Maruyama method for the Itō diffusions, the nonlinear ones, the self-interacting ones and the mean-field interacting particles system. More precisely, I am interested in the time required by the different processes to exit from a given domain. This will require to rely on the Freidlin-Wentzell theory. I see two ways to tackle these questions: on the one hand, to use a coupling and a control of the laws to show that the exit-time satisfies an Arrhenius law or a Kramers'type law of the same form as the one of the associated diffusion; on the other hand, to directly tackle the exit-time by remaking the Freidlin and Wentzell theory from its foundations.


    Besides, in the direct continuity of this perspective, it is worth noticing that all the results on exit-times in the small-noise limit are asymptotic. When it is about Arrhenius law, or about Kramers'type law, the Kramers law itself, everything holds when \(\sigma\longrightarrow0\). Hence, very few is known about the critical diffusion coefficient under which our results may be applied. Nevertheless, again, in view of practical application of the results, these estimates should be non-asymptotic. According to me, this will require to use the Onsager-Machlup theory of which extension to nonlinear diffusions is of great interest to me. Moreover, I aim to establish such kind of results for the associated numerical schemes: then, it will be necessary to control the exit-time with respect to \(\sigma\) (the diffusion coefficient), with respect to \(h\) (the time-discretization step), with respect to \(\delta\) (the space-discretization step) and with respect to \(d\) (the dimension of the space in which lives the process).


    A third axis of research with application in the concrete world is the study of explosive kernels. It can be Keller-Segel type in chemotaxis, Navier-Stokes type in fluid mechanics or Biot-Savart type in electrostatic, in the physical world, the interaction forces are explosive namely two particles can not occupy the same position in the same time. To deal with such kernels is not easy since taking a huge number of particles yields that the collision is not an unlikely event. Worst, such nonlinear diffusions satisfy equations suffering from an inherent possible ill-posedness. Hence, we have to deal with the collisions. To my mind, the most natural and simplest idea consists in putting a threshold above which the interaction force can not go beyond. With such a cut-off, it is then easy to prove the existence of a unique strong solution, to prove the existence of an invariant probability measure, to characterize the set of the steady states, to establish the long-time convergence then to obtain the small-noise asymptotics of the first exit-times and of the first collision-time. It is worth noticing that it will be necessary to do so in a Markov case as well as in a self-interacting one. Indeed, the parabolic-parabolic Keller-Segel equation is the mean-field limit of a system of interacting particles involving a self-interacting term linked to the past paths of each particle. The associated partial differential equation is as follows: \[ \left\{\begin{array}{l} \frac{\partial}{\partial t}\rho(t,x)={\rm div}\left\{\frac{\sigma^2}{2}\nabla\rho(t,x)-\chi\rho(t,x)\nabla c(t,x)\right\}\,,\\ \frac{\partial}{\partial t}c(t,x)=\frac{\nu^2}{2}\Delta c(t,x)-\lambda c(t,x)+\rho(t,x)\,, \end{array}\right. \] where \(\chi\) is the transmission coefficient and \(\lambda\) corresponds to a repelling force. \(\sigma\) and \(\nu\) are the diffusion coefficients. Here, \(\rho\) does correspond to the density of probability of the biological specie whereas \(c\) is the concentration of the chemical component. I am interested in, after some Hubert Curien program (Sakura, that is to say with Japan) to the blow-up phenomenon in the two Keller-Segel models (parabolic-elliptic as well as parabolic-parabolic). The french team used a probabilistic viewpoint while the japanese team used the viewpoint of partial differential equations. More precisely, we aim to see the blow-up phenomenon directly from the associated system of particles. Also, we want to understand what happens after the blow-up.


    A fourth axis that I aim to accomplish in my work is the study of a model widely used in biology to explain the phenomenon of neuronal conduction. In November 2021, I had the honor and the pleasure to to go to Oslo further to the selection of my Åsgard project. I met David Ruiz Baños (Assistant Professor) and a subject emerged from our discussions: the application of the McKean-Vlasov diffusions (of type "integrate-and-fire model") to the finance with a common noise and a stochastic volatility. The main difference between this model and the classical self-stabilizing diffusion is that the law intervenes in the drift of the stochastic differential equation through the exit-times of the diffusion, which makes the study extremely strenuous. In the finance context, the drift driving the volatility (which is itself stochastic) increases when an asset reaches zero (which naturally explains from where come the exit-times). In the opposite, when an asset reaches zero, the drifts driving the other assets decrease. We aim to prove a phenomonenon of propagation of chaos for this model when the number of assets tends to infinity. Also, we will precisely study the associated limiting nonlinear model.