1. Study of the stable states of the granular media equation


2. Long-time convergence of self-stabilizing diffusions


3. Large deviations principle for the McKean-Vlasov diffusions


4. Exit-problem for nonlinear diffusions


5. Collision problem for independent diffusions


6. Exit-problem for non Markov diffusions


7. Kalman Ensembles

Study of the stable states of the granular media equation

    My first article, "Non uniqueness of stationary measures for self-stabilizing diffusions", written with Samuel Herrmann (who was my PhD advisor) and published in Stochastic Processes and their Applications, goes against the intuition since it shows that the granular media equation may admit several stable states with total mass equal to \(1\) and furthermore it may admit exactly three. It is essential to precise that the set of the invariant probability measures has no reason to be a convex set. Consequently, my study is about the invariant probability measures, not about the extremal ones; the latter notion having no sense here.

    Let us describe precisely the result. We consider two potentials on \(\mathbb{R}\), \(V\) denoted as the confining potential and \(F\) denoted as the interacting one. We also assume a uniform ellipticity condition that is to say we have a positive parameter \(\sigma\) as diffusion coefficient. Then, the granular media equation is \[ \tag{I} \frac{\partial}{\partial t}\mu_t=\frac{\sigma^2}{2}\Delta\mu_t+{\rm div}\left\{\mu_t\left(\nabla V+\nabla F\ast\mu_t\right)\right\}\,, \] where \(\ast\) is the convolution product. This partial differential equation is nonlinear. There were plenty of results in the case where the two potentials were convex even uniformly convex. One particular result was the uniqueness of the invariant measure with total mass equal to \(1\). This uniqueness condition was a necessary condition in order to find out classical results in linear partial differential equations.

    It is why the article, that I am the main author of, mentioned above is a pioneering one. The used techniques are simple which makes its strength since it is easy to extend to more general cases. The global idea is as follows. An invariant probability measure of the granular media equation admits the following implicit form: \[ \mu^\sigma({\rm d}x)=\frac{\exp\left\{-\frac{2}{\sigma^2}\Big[V(x)+F\ast\mu^\sigma(x)\Big]\right\}}{\int_{\mathbb{R}}\exp\left\{-\frac{2}{\sigma^2}\Big[V(y)+F\ast\mu^\sigma(y)\Big]\right\}{\rm d}y}{\rm d}x\,. \]     Then, if the potential \(F\) is quadratic and uniformly convex that is to say if \(F(x)=\frac{\alpha}{2}x^2\) with \(\alpha>0\), the implicit equation becomes \[ \mu^\sigma({\rm d}x)=\frac{\exp\left\{-\frac{2}{\sigma^2}\left[V(x)+\frac{\alpha}{2}x^2-\alpha m^\sigma x\right]\right\}}{\int_{\mathbb{R}}\exp\left\{-\frac{2}{\sigma^2}\left[V(y)+\frac{\alpha}{2}y^2-\alpha m^\sigma y\right]\right\}{\rm d}y}{\rm d}x\,, \] where \(m^\sigma:=\int_{\mathbb{R}}x\mu^\sigma(x){\rm d}x\). Thus, it is sufficient to find a zero to the following function: \[ \chi_\sigma(m):=\frac{\int_{\mathbb{R}}x\exp\left\{-\frac{2}{\sigma^2}\left[V(x)+\frac{\alpha}{2}x^2-\alpha mx\right]\right\}dx}{\int_{\mathbb{R}}\exp\left\{-\frac{2}{\sigma^2}\left[V(x)+\frac{\alpha}{2}x^2-\alpha mx\right]\right\}dx}-m\,. \]     The study of this function with small coefficient \(\sigma\) has been done by using the Laplace method. I have gone beyond in this article, dealing also with the case where \(F'\) is not linear. In particular, under simple assumptions, we have proved that there are exactly three such zeros to the function \(\chi_\sigma\) providing that \(\sigma\) is small enough. The main hypotheses are the uniform positiveness of \(V^{(4)}\) and of \(F^{(4)}\). This work opened an avenue to mainy researchers as Carrillo, Duong and Pavliotis.

    Also, this paper has been followed by some works of which I am the main author. In particular, I proved in the paper "Phase transitions of McKean-Vlasov processes in double-wells landscape" that if the interacting potential \(F\) is quadratic and uniformly convex and if moreover the confining potential \(V\) is of the form \[ V(x):=-\frac{\theta}{2}x^2+\sum_{k=2}^{n}\frac{\theta_{2k}}{(2k)!}x^{2k}\,, \] where the coefficient \(\theta_i\) is positive for any \(i\in[\![1;n]\!]\) and where \(n\geq2\), there exists a critical value \(\sigma_c(\alpha)\) at which we observe a bifurcation. Namely, if \(\sigma\geq\sigma_c(\alpha)\), there is a unique invariant probability measure. Otherwise, if \(\sigma<\sigma_c(\alpha)\), there are exactly three. This critical value has been characterized with Bessel functions by using the free-energy of the unique symmetrical invariant probability (this measure always exists providing that the potentials are even). Let us mention that this paper also contains some simulations done with C++.

    It is interesting, albeit not pleasant, to note that some colleagues have taken this paper, published in Stochastics, in order to make another publication in Journal of Statistical Physics.

    In a subsequent paper, entitled "Self-stabilizing processes in multi-wells landscape in \(\mathbb{R}^d\)- Invariant probabilities" and published in Journal of Theoretical Probability, I have also proved, that under simple assumptions, we can associate to each minimizer \(a\) of the confining potential \(V\) at least one invariant probability measure \(\mu_a^\sigma\) which is close, in some sense, to \(\delta_a\). The two assumptions allowing the existence of such a stable state are, on the one hand, \(\sigma<\sigma_c(a)\) where \(\sigma_c(a)\) is a positive parameter and on the other hand an hypothesis strongly related to the Laplace method. Namely, for any \(x\in\mathbb{R}^d\) with \(x\neq a\), we have \(W_a(x)>W_a(a)\). Here, the potential \(W_a\), denoted as the effective one, corresponds to what sees the system when the diffusion coefficient \(\sigma\) is small: \(W_a:=V+F\ast\delta_a\).

    Let us also note that in the case of the dimension \(1\), in collaboration with Samuel Herrmann, then in dimension \(d\), I have studied the convergence of the set of invariant probability measures when \(\sigma\) tends to \(0\). This leads to the two papers "Stationary measures for self-stabilizing diffusions: asymptotic analysis in the small noise limit" and "Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit" respectively published in Electronic Journal of Probability and in ESAIM: Probability and Statistics. I have also published a note to Comptes Rendus Mathématiques from the Académie des Sciences concerning the approximation between the synchronized case and the asynchronized one. The title is: "McKean-Vlasov diffusions: from the synchronization to the asynchronization".

    Then, I have written a paper with Hong Duong in Applied Mathematics Letters about the stable states of the Vlasov-Fokker-Planck partial differential equation, which corresponds to the case of a nonlinear Langevin diffusion. More precisely, we have obtained a bijective correspondence between the steady states of the overdamped system and the ones of the kinetic one. Again with Hong Duong, we have written a paper published in Stochastics concerning the stable states of the coupled nonlinear diffusions.

    It is important to understand that this lack of uniqueness for the invariant probability measures is very unpleasant for whoever aims to obtain a long-time convergence. Indeed, there is no ergodicity property that can be proved since the multiple stationnary measures are all related to Lebesgue measure. In the same vein, to obtain a uniform in time propagation of chaos is impossible. Namely, we can note approximate a McKean-Vlasov diffusion by a system of particles on \(\mathbb{R}_+\). Furthermore, the free-energy functional is not convex. Thus, some questions of metastability naturally occur. For example, what is the time (with respect to the number of particles) to go from a metastable set to another one ?

    Another metastability problem which happens is the following. Let us consider a double-wells confining potential \(V\): one well is at \(a_-<0\) and the other is at \(a_+>0\). Also, \(V(a_-)>V(a_+)\). Assume that the interacting potential \(F\) is uniformly convex and quadratic but such that \(W_{a_-}\) is not minimal in \(a_-\). Then, we can show easily by using my work that there is a unique invariant probability measure and this measure converges in the small-noise limit towards \(\delta_{a_+}\). However, starting from the initial state \(\mu_0=\delta_{a_-}\), we expect to stay a long time close to \(\mu_0\). Then, it is a metastable state, not a stable one. This question has been studied in an article that I published in Communications on Stochastic Analysis.

    A last question strongly related to this first result is whether or not we recover the uniqueness of the invariant probability measure by using stochastic partial differential equations; these equations naturally appearing if we add a common noise to the idiosyncratic one.

Long-time convergence of self-stabilizing diffusions

    The self-stabilizing diffusion is a particular instance of the McKean-Vlasov diffusions. It is naturally linked to the granular media equation. Indeed, like Cattiaux, Guillin and Malrieu have done in the convex case, the partial differential equation of granular media may be interpreted, in the microscopical viewpoint, by a stochastic differential equation: \[ \tag{II} X_t=X_0+\sigma B_t-\int_0^t\nabla V(X_s){\rm d}s-\int_0^t\Big[\nabla F\ast\left(\mathcal{L}(X_s)\right)\Big](X_s){\rm d}s\,. \]     This stochastic differential equation, denoted as self-stabilizing, admits a unique strong solution \(X\) and for any \(t>0\), \(\mathcal{L}(X_t)=:\mu_t\) is absolutely continuous with respect to the Lebesgue measure (with a density that I denote as \(\mu_t\) to simplify) and it satisfies Equation (I).

    An intensively studied question (notably by Carrillo, McCann, Villani, Benedetto, Pulvirenti, Guillin, Bolley, Malrieu, Gentil...) is the long-time convergence of the law of \(X_t\) towards the unique invariant probability measure. Nevertheless, like written in the previous paragraph, I have proved that this uniqueness of the invariant probability measure is not true in general. Thus, the convergence question remains. Do we have long-time convergence? If yes, can we obtain an explicit rate of convergence? And, what are the basins of attraction? Namely, can we determine - with simple hypotheses on the initial measure - the steady state that is the long-time limit of the solution to the granular media equation?

    I have positively answered to the question of the convergence in the paper "Convergence to the equilibria for self-stabilizing processes in double-well landscape" published in The Annals of Probability and in the paper "Self-stabilizing processes in multi-wells landscape in \(\mathbb{R}^d\) - Convergence" published in Stochastic Processes and their Applications as the dimension \(d\) is general.

    Indeed, adapting the remarkable work of Benedetto, Caglioti, Carrillo and Pulvirenti in 1998, I succeeded to show that \(\mu_t\) weakly converges towards one of the three invariant probability measures; if we restrict ourselves to the thirdness case.

    The main idea is to consider the Helmholtz free-energy: \[ \Upsilon_\sigma(\mu):=\frac{\sigma^2}{2}\int_{\mathbb{R}^d}\mu(x)\log(\mu(x)){\rm d}x+\int_{\mathbb{R}^d}V(x)\mu(x){\rm d}x+\frac{1}{2}\int_{\mathbb{R}^d}F\ast\mu(x)\mu(x){\rm d}x\,. \]     This Lyapunov functional is nonincreasing since we have \[ \frac{{\rm d}}{{\rm d}t}\Upsilon_\sigma(\mu_t)\leq-\int_{\mathbb{R}^d}\left|\frac{\sigma^2}{2}\frac{\nabla\mu_t(x)}{\mu_t(x)}+\nabla V(x)+\nabla F\ast\mu_t(x)\right|^2\mu_t(x){\rm d}x\,. \]     It is worth noticing that if \(V\) and \(F\) are both convex, we have a logarithmic Sobolev inequality and so there is an exponential entropic convergence. However, as soon as there are several invariant probability measures, we do not have such an inequality. It is what makes the question so hard and, in fine, so exciting. In the same way, Cattiaux, Guillin and Malrieu have obtained a convergence by using the uniform propagation of chaos due to the convexity of the potentials. Nevertheless, as soon as there are several invariant probability measures, such a uniform propagation of chaos can not occur. In fact, there is one case and only one where we can simply characterize the basin of attraction and get the convergence: when the confining potential \(V\) is constant. In this case, the set of the invariant probability measures is infinite and moreover, the first moment of \(\mu_t\) is a constant with respect to \(t\). Thus, everything is similar to the uniqueness case, see the paper of Benachour, Roynette, Talay and Vallois in 1998 then the one of Benachour, Roynette and Vallois in 1998. Also, assuming that the first moment is time-invariant, Carrillo, McCann and Villani have been able to extend the work of Benedetto, Caglioti, Carrillo and Pulvirenti to the case where the potential \(V\) is not convex providing that the convexity of \(F\) is sufficiently strong to compensate. However, I do not know any example (except the one mentioned above) where this invariance is satisfied; when the phase space is \(\mathbb{R}^d\).

    This convergence result that I obtained opens a lot of questions. Indeed, even when there is a unique invariant probability measure, that is when \(\sigma\geq\sigma_c\), to obtain a rate of convergence is highly non trivial. Despite this fact, Pierre del Moral and me succeeded to establish a rate of convergence of exponential type. However, the coefficient driving this rate is not tractable. This work, published in Stochastic Analysis and its Applications, allows us to establish a uniform with respect to the time propagation of chaos, using the exponential rate via a \(WJ\) type inequality, see the works of Bolley, Gentil and Guillin. In the same vein, Bartłomiej Dyda and me have established an exponential rate of convergence independent from the number of particles if \(\sigma\) is large enough in a paper published in Probability and Mathematical Physics.

    Contrary to what one can believe due to what is written above, one of the immediate consequence of the non-uniqueness of the invariant probability measures (which corresponds to the case where \(\sigma\) is small enough whereas what is above concerns the large enough diffusion coefficient case) is the dependency, with respect to the number of particles, of the time spent by the mean-field system of particles, associated to the self-stabilizing diffusion, in a metastable set.

    Besides, to come back on the metastability, I succeeeded to establish that the time spent by the self-stabilizing diffusion in a local minimizer which is not associate to an invariant probability measure is at most exponential of the form \(\exp\left\{\frac{2\Delta}{\sigma^2}\right\}\) where \(\Delta\) grosso modo corresponds to the exit-cost (in the large deviations regime) of the McKean-Vlasov diffusion. This result has been published in Communications On Stochastic Analysis.

    About the rate of convergence, I published a first note in Comptes Rendus Mathématiques from the Académie des Sciences. Then, I submitted a work in July 2023 in which I established a local stability of the invariant probabilities, albeit in dimension \(1\). Furthermore, I established a rate of convergence of the form \(O\left(e^{-C\sqrt{t}}\right)\); which is, I must confess, quite disappointing. Indeed, it is natural to imagine that the local stability is related to an exponential convergence. As an example, see the recent preprint of Quentin Cormier on this subject. In the same way, the work of Guillin, Le Bris and Monmarché seems to confirm this assertion since they obtain a uniform in time propagation of chaos with asynchronous coupling and an exponential rate of convergence; in the case where there is the uniqueness of the invariant probability measure.

    For my part, it is the small-noise case that I am interested in. Once, I have been asked a question on this subject: "if the initial measure is compactly supported in a basin of attraction of the noiseless system, can't we establish, in the small noise case, that the noisy system will converge towards the invariant probability measure associated to the local minimizer?" Despite this question may seem trivial, it is not. To answer it, I made some circonvolutions by Freidlin and Wentzell theory. I succeeded to positively answer the question. This has been written in a paper published in Kinetic and Related Models. About the characterization of the limiting steady state, I obtained a result in the case of dimension \(1\).

    Let us mention that the convergence in the kinetic case, that is to say about Vlasov-Fokker-Planck diffusion has been solved by Hong Duong and me, in a work published in Electronic Communications in Probability. Despite this extension to the underdamped case seems trivial, it was not and we had to read a lot of works of Carrillo on the nonlinear partial differential equations.

    Finally, it is essential to understand that the knowledge of the basins of attraction is of crucial interest to obtain the good rate function in the large deviations principles for self-stabilizing diffusions. Indeed, knowing the limit of \(\mu_t\) when \(t\to+\infty\) would allow us to determine the form of the limit of the good rate function. It is, grosso modo, what Herrmann, Imkeller and Peithmann did in their remarkable paper concerning the Kramers'type law for the first exit-time of a self-stabilizing diffusion when the potentials are uniformly convex. As soon as there is a lack of convexity, such results are not immediate. Let us point out that the limit is not sufficient: we also need a rate of convergence such that the stabilization time towards the limit is at most \(\exp\left\{\frac{2H}{\sigma^2}\right\}\) where \(H\) is the exit-cost of the domain.

Large deviations principle for the McKean-Vlasov diffusions

    In their remarkable article published in The Annals of Applied Probability, Herrmann, Imkeller and Peithmann have established a large deviations principle for the McKean-Vlasov diffusion without assuming neither boundedness nor Lipschitz condition on the coefficients. The norm on which is established this large deviations principle is the usual one, that is to say the uniform norm.

    In collaboration with William Salkeld and Gonçalo dos Reis, we have extended this result to the case of the Hölder norm. Hence, we can deduce exit-time for a different norm. Let us mention that our method is robust and may be applied to any norm as soon as it is a classical one. This result has been published in The Annals of Applied Probability. Besides, it has lead us to a proof of the iterated logarithm law.

    Furthermore, in collaboration with Daniel Adams, William Salkeld, Gonçalo dos Reis and Romain Ravaille, we have published a paper in Stochastic Processes and their Applications concerning the large deviations principle for general reflected McKean-Vlasov diffusion of the form \[ {\rm d}X_t=b(X_t,\mathcal{L}(X_t),t){\rm d}t+\sigma(X_t,\mathcal{L}(X_t),t){\rm d}W_t+{\rm d}k_t\,, \] where \(k\) is a bounded variation process which corresponds to the reflection on the boundary of a domain \(\mathcal{D}\). Let us mention that the latter domain is neither assumed to be bounded nor convex. However, a simple assumption has been taken: the phase space in which lives the diffusion is positively invariant by the drift \(b\). This work allows us to obtain a Kramers'type law concerning the first exit-time from an open domain \(\mathcal{G}\) providing that the drift is of the form \(b(x,\mu,t)=-\nabla V(x)-\nabla F\ast\mu(x)\) where \(V\) and \(F\) are both convex potentials and that \(\overline{\mathcal{G}}\subset\mathcal{D}\).

    Also, in a paper published in Electronic Journal of Probability, I succeeded to show a commutativity result on the limits of the good rate functions in the large deviations principles. This is not the heart of the paper but a corollary. It is crucial to understand that getting this result was requiring the diffusion coefficient to be constant and the drift to be of the form \(b(x,\mu,t):=-\nabla V(x)-\nabla F\ast\mu(x)\) where \(V\) and \(F\) are convex potentials. In collaboration with Samuel Herrmann, we have gone further. We consider the mean-field interacting system of particles associated to the McKean-Vlasov diffusion. By \(\mathcal{I}^N\), we denote the good rate function associated to the first particle. Similarly, by \(\mathcal{I}^\infty\), we denote the one of the associated McKean-Vlasov diffusion. Then, without any convexity nor reversibility assumptions, we proved that \(\mathcal{I}^N\) tends towards \(\mathcal{I}^\infty\) as \(N\) goes to infinity. Besides, the kind of considered interaction is general since in this article published in Communications on Stochastic Analysis, we can also deal with the diffusion involved in the modelling of lithium batteries, that is to say when the interaction is of scalar type. Let us stress that, in this work, we have more than just the convergence but we also obtained the optimal path (with respect to the \(N-1\) other particles). This is interesting and important since we deduce that the small-noise behavior of the first particle is close to the one of the McKean-Vlasov diffusion. To sum up, we have established a commutativity between the two limits \(N\longrightarrow+\infty\) and \(\sigma\longrightarrow0\). Thus, it goes much further than the propagation of chaos. Nevertheless, there is no uniformity with respect to the time.

    In a current work, Ashot Aleksian has established under my supervision a large deviations principle by assuming that the initial law is not a Dirac measure. Indeed, in the case of a McKean-Vlasov diffusion of the form \[ {\rm d}X_t=-\nabla V(X_t){\rm d}t-\nabla F\ast\mu_t(X_t){\rm d}t+\sigma{\rm d}W_t\,, \] where \(\mu_t:=\mathcal{L}(X_t)\) is the density of probability of the random variable \(X_t\), if \(\mu_0=\delta_{x_0}\), it is well-known that the good rate function of the latter diffusion is derived from the noiseless dynamic \[ \frac{{\rm d}}{{\rm d}t}\gamma_t(x_0)=-\nabla V\left(\gamma_t(x_0)\right)-\nabla F\ast\delta_{\gamma_t(x_0)}(\gamma_t(x_0))\,. \] However, a classically used assumption is the fact that \(F\) is rotationaly invariant. Thus, \(-\nabla F\ast\delta_{\gamma_t(x_0)}(\gamma_t(x_0))=-\nabla F(0)=0\). This implies that the interacting potential does not have any influence on the good rate function. Nevertheless, Ashot Aleksian and me are interested in the case without rotational invariance and where the initial law \(\mu_0\) is not a Dirac measure.

    In the case of non Markov diffusion of self-interacting form (that is to say the nonlinear diffusions where the convolution in the drift is with the occupation measure of the process), Ashot Aleksian, Aline Kurtzmann and me have also established some large deviations principles. This has been submitted.

Exit-problem for nonlinear diffusions

    A question that I have been wondering for years is to get an Arrhenius law or a Kramers'type law on the exit-times for nonlinear diffusions. Up to now, I worked on a particular instance of such diffusion: the self-stabilizing diffusion, solution to a stochastic differential equation of the form: \[ \tag{III} {\rm d}X_t=\sigma {\rm d}B_t-\nabla V(X_t){\rm d}t-\alpha(X_t-\mathbb{E}(X_t)){\rm d}t\,, \] where \(\alpha,\sigma>0\) and \(V\) is a potential on \(\mathbb{R}^d\).

    The study deals with the first exit-time of such a diffusion in the small-noise limit, that is to say \(\tau(\sigma):=\inf\left\{t\geq0\,\,:\,\,X_t\notin\mathcal{D}\right\}\) where \(\mathcal{D}\) is an open domain satisfying simple assumptions so that the asymptotics of \(\tau(\sigma)\) with small \(\sigma\) are not trivial. Here, to get the Arrhenius law means the limit \(\lim_{\sigma\to0}\frac{\sigma^2}{2}\log\left(\mathbb{E}\left[\tau(\sigma)\right]\right)=H\) where \(H>0\) is denoted as exit-cost. The Kramers'law goes beyond since it furnishes the prefactor: \(\mathbb{E}\left[\tau(\sigma)\right]=C(\sigma)\exp\left\{\frac{2H}{\sigma^2}\right\}\left(1+o_\sigma(1)\right)\) where \(C(\sigma)\) is explicit. What we denote as Kramers'type law is a limit of the form: \[ \lim_{\sigma\to0}\mathbb{P}\left\{\exp\left(\frac{2}{\sigma^2}(H-\delta)\right)<\tau(\sigma)<\exp\left(\frac{2}{\sigma^2}(H+\delta)\right)\right\}=1\,, \] where \(\delta>0\) is arbitrarily small.

    It is worth noticing that in the above case, the interacting potential \(F\) which represents the nonlocal term is quadratic (\(F(x):=\frac{\alpha}{2}x^2\)). Nevertheless, here, we take this hypothesis only to introduce in a simpler way the results. Hence, in their remarkable article entitled "Large deviations and a Kramers'type law for self-stabilizing diffusions" published in The Annals of Applied Probability, Herrmann, Imkeller and Peithmann have established a Kramers'type law in a more general case where the drift is not necessarily of gradient form: \[ {\rm d}X_t=\sigma{\rm d}B_t+b(X_t){\rm d}t+c\ast\mu_t(X_t){\rm d}t\,, \] where \(b\) and \(c\) are vector fields on \(\mathbb{R}^d\). However, their work is reduced to the case of contractions: the Jacobian matrices of \(b\) and \(c\) are definite negative, uniformly with respect to the space variable. Namely, in the gradient case (that is to say when \(b:=-\nabla V\) and \(c:=-\nabla F\)), the related potentials are uniformly convex.

    The main objective of my work on the past decade consists in the extension of this work to the non-convex case. Indeed, the appeal of the metastability (the question being related to metastability) is to study multi-modal case.


    In a first step, I proved the result of Herrmann, Imkeller and Peithmann on the exit-time under convexity assumptions. This corresponds to the papers "Exit problem of McKean-Vlasov diffusions in convex landscape" published in Electronic Journal of Probability and "A simple proof of a Kramers'type law for self-stabilizing diffusions" published in Electronic Communications in Probability. In this framework, \(V\) is uniformly convex: \(\nabla^2V\geq\theta{\rm Id}\) with \(\theta>0\). By \(a\), we denote its unique minimizer and we consider an open domain \(\mathcal{D}\subset\mathbb{R}^d\) which contains \(a\) and is positively invariant by the dynamic \(x\mapsto-\nabla V(x)-\alpha(x-a)\).

    Then, we recover the Kramers'type law with the exit-cost \(H:=\inf_{\partial\mathcal{D}}(V+F\ast\delta_a-V(a))\). In order to obtain this result, I have developed two methods strongly different from the one of Herrmann, Imkeller and Peithmann. Indeed, their method mainly consists in rebuilding Freidlin and Wentzell theory (with the large deviations principles viewpoint of the type Dembo and Zeitouni). In my first paper on the subject, I took advantage of the system of interacting particles which is the microscopical interpretation of the self-stabilizing diffusion. Despite this system does correspond to a high dimensional diffusion process (with dimension \(Nd\) if \(N\) is the number of particles), it is just a classical Itô diffusion. Thus, by adapting the Freidlin and Wentzell theory and by using it as a black box, I obtained the classical asymptotics, when the potentials are uniformly convex, on the first exit-time of the first particle. Then, extending the classical coupling to a uniform with respect to a time interval of the form \(\left[t_0;t_0+\exp\left\{\frac{2K}{\sigma^2}\right\}\right]\) where \(K>H\) is arbitrarily large, I have been able to recover the result to the self-stabilizing diffusion in convex landscape.

    In my second paper on the subject, I used an auxiliary diffusion process: \[ Y_t=X_{T_0}+\sigma\left(B_t-B_{T_0}\right)-\int_{T_0}^t\nabla V(Y_s){\rm d}s-\int_{T_0}^t\alpha(Y_s-a){\rm d}s\,, \] the time \(T_0\) being deterministic and related to the precision that we wish to achieve in the coupling between the diffusions \(X\) and \(Y\). By taking advantage of the convexity, I have been able to obtain a uniform, with respect to the time, coupling. And, the diffusion \(Y\) being a classical Kolmogorov one, its exit-time is well-known. Finally, by adapting the domain from which exits the diffusion \(Y\), it is possible to find the exit-time of the diffusion \(X\) from the domain \(\mathcal{D}\). Also, by a totally new method method, I have found again: \[ \lim_{\sigma\to0}\mathbb{P}\left\{X_{\tau(\sigma)}\in\mathcal{N}\right\}=0\,, \] if \(\mathcal{N}\subset\partial\mathcal{D}\) is such that \(\inf_{\mathcal{N}}(V+F\ast\delta_a-V(a))>H\). It is worth noticing that the latter result does not require any convexity assumption to be true. More exactly, the only thing that we need in order to know the exit-location \(X_{\tau(\sigma)}\) is the asymptotic in the small-noise limit of the exit-time \(\tau(\sigma)\). Consequently, the only goal in the non-convex case is to establish the Kramers'type law.


    In a second step, I have been interested in the case where \(V\) is not convex and I obtained partial results. Hence, in the one-dimensional case, the problem is solved, see "A simple proof of a Kramers'type law for self-stabilizing diffusions in double-wells landscape" published in Alea. In the same vein, in the general dimensional case, under synchronization hypothesis that is when \(F(x)=\frac{\alpha}{2}|x|^2\) with \(\alpha>0\) and \(\alpha{\rm Id}>\sup_{\mathbb{R}^d}-\nabla^2V\), then the problem is also solved: see "Exit problem of McKean-Vlasov diffusion in double-wells landscape" published in Journal of Theoretical Probability.

    The idea to obtain such a result is to control \(\mathcal{L}(X_t)\) with respect to the exit-time \(\tau(\sigma)\). More precisely, the keystone to go beyond the convexity assumption is the following inequality: \[ \frac{d}{dt}\mathbb{E}\left[|X_t-a|^2\right]\leq-2\rho\mathbb{E}\left[|X_t-a|^2\right]+K\sqrt{\mathbb{P}\left(\tau(\sigma)\leq t\right)}+C\sigma^2\,, \] where \(K\) and \(C\) are positive constant. Then, the idea is to couple the inhomogeneous diffusion (III) with the diffusion \(Y\) which evolution equation is written above.


    It is worth noticing that, up to now, the interacting potential \(F\) is convex. And, we can easily remark that the exit-cost is larger with the McKean-Vlasov diffusion than with the linear diffusion without interaction. Thus, a natural question appears: if \(F\) is repulsive, will the exit-cost be reduced? The answer is yes, providing that the linear part of the drift is a contraction. This corresponds to the article "Reducing exit-times of diffusions with repulsive interactions" with Paul-Eric Chaudru de Raynal, Hong Duong, Pierre Monmarché and Milica Tomašević; the latter article having been published in ESAIM Probability and Statistics.

    In this work, we show, in a simple case opening an avenue, that if \(V\) is convex we can reduce the exit-time to escape from the minimizer. It is worth noticing that we did not restrict ourselves neither to the gradient case nor to the overdamped one. Furthermore, the techniques that we use are robust with respect to the parameters.


    With Daniel Adams, Gonçalo dos Reis, Romain Ravaille and William Salkeld, we have established a Kramers'type result in the case where the phase space is not \(\mathbb{R}^d\) but a subdomain in which we put a reflection on the boundary. However, to obtain this result, we had to assume that the drift is of gradient type and that the two potentials (confining and interacting) are uniformly convex. Indeed, the used method strongly relies on similar techniques than the ones that I had previously developed. This paper entitled "Large Deviations and Exit-times for reflected McKean-Vlasov equations with self-stabilizing terms and superlinear drifts" has been published in Stochastic Processes and their Applications.

    Previously, I have written that I used the system of particles to understand the asymptotics of the exit-time for the self-stabilizing diffusion, in the convex case. Nevertheless, one can wonder what happens to the interacting particles system when \(V\) is not convex. This corresponds to a paper published in ESAIM Probability and Statistics: "Exit-time of mean-field particles system".

    The general case is still open. I am currently finalizing a paper in collaboration with Ashot Aleksian on the self-stabilizing diffusion when \(V\) is not convex, when \(F\) is possibly nonconvex, in the case of dimension \(d\) and moreover, without assuming the usual hypothesis that has been driving my work these last years.


    In fine, the problem is almost solved, even in the non-gradient case. This corresponds to a succession of major results which applications in machine learning, in molecular dynamic, in econophysics are important. It is worth noticing that these different works also have a real impact on the long-time behavior of nonlinear (in the sense of McKean) diffusions. Let us mention that the last works have been obtained thanks to the ANR project METANOLIN.

Collision problem for independent diffusions

    The collision problem does correspond to a question that Jean-François Jabir and me asked ourselves some time ago. This question is completely new. It consists in estimating the first time that two processes touch themselves. We put ourself in the framework of Itô diffusions, self-stabilizing diffusions, self-interacting diffusions (modelling polymers) or interacting particles systems. Let us introduce the simplest possible case; the one which is containing in its hearth the whole difficulty related to the collision problem.

    Set two potentials \(\Psi_1\) and \(\Psi_2\) on \(\mathbb{R}\). We assume that they are uniformly convex with respective minimizers \(\lambda_1\) and \(\lambda_2\) with \(\lambda_1\neq\lambda_2\). Without any loss of generality, we take the hypothesis \(\lambda_1<\lambda_2\). We also take \(\sigma>0\). Then, we put the two following diffusions on \(\mathbb{R}\) : \[ x^1_t=x^1_0+\sigma B_t-\int_0^t\Psi_1'\left(x^1_s\right){\rm d}s\quad\mbox{and}\quad x^2_t=x^2_0+\sigma\widetilde{B_t}-\int_0^t\Psi_2'\left(x^2_s\right){\rm d}s\,, \] where \(x^1_0\neq x^2_0\) are two real numbers and where \(B\) and \(\widetilde{B}\) are two independent Brownian motions. We are interested in the asymptotics in small-noise of the first time of collision: \[ c(\sigma):=\inf\left\{t\geq0\,\,:\,\,x^1_t=x^2_t\right\}\,. \] This collision problem is wellposed due to the recurrent nature of the diffusions in \(\mathbb{R}\). Also, it is non-degenerate since \(x^1_0\neq x^2_0\). However, we need to enforce that the distance between the deterministic trajectories is positive. In other words, we put \(\varphi^1(t):=x^1_0-\int_0^t\Psi_1'\left(\varphi^1(s)\right){\rm d}s\) and \(\varphi^2(t):=x^2_0-\int_0^t\Psi_2'\left(\varphi^2(s)\right){\rm d}s\) then we assume that the following quantity is positive: \(\inf_{t\geq0}\left|\varphi^1(t)-\varphi^2(t)\right|\). By using large deviations principles for the stochastic processes, it is then easy to show the following limit: \[ \lim_{\sigma\to0}\mathbb{P}\left\{c(\sigma)\leq T\right\}=0\,, \] for any \(T>0\). Hence, it seems reasonable to try to establish a Kramers'type law: \(\mathbb{P}-\lim_{\sigma\to0}\frac{\sigma^2}{2}\log(c(\sigma))=\Delta>0\). In fact, it is exactly what will be done next.

    Instead of introducing the most general case, let us limit ourselves to a typical case which will allow the reader to understand the underlying idea which yields the Kramers'type law for this collision time. We assume \(x^1_0<\lambda_1\) and \(x^2_0>\lambda_2\). To begin, it is immediate that the distance between the deterministic trajectories is positive. First, \(x^1_t\) will approach \(\lambda_1\) whereas \(x^2_t\) will stabilize around \(\lambda_2\). Then, to obtain a collision requires that \(x^1\) goes to the right towards \(x^2\) and that \(x^2\) goes to the left towards \(x^1\). Indeed, here, the persistence of the collision-location is absolutely crucial to apprehend the collision-time. Let us assume, for example, that the collision is at the left of \(x^1\) then it is necessary for the process \(x^2\) to reach out \(\lambda_1\). However, the cost for the latter diffusion to go from \(\lambda_2\) to \(\lambda_1\) is exactly \(\Psi_2(\lambda_1)-\Psi_2(\lambda_2)\). Subsequently, we will see that such a cost is larger than \(\Delta\), the collision-cost. In the same vein, if the collision-location occurs at the right of \(\lambda_2\), then it is necessary for the process \(x^1\) to reach out \(\lambda_2\). The cost yielding to this event is \(\Psi_1(\lambda_2)-\Psi_1(\lambda_1)\). Nevertheless, we have the following inequality: \[ \Delta:=\inf_{z\in\mathbb{R}}\left(\Psi_1(z)+\Psi_2(z)-\Psi_1(\lambda_1)-\Psi_2(\lambda_2)\right)<\min\left\{\Psi_2(\lambda_1)-\Psi_2(\lambda_2);\Psi_1(\lambda_2)-\Psi_1(\lambda_1)\right\}\,, \] this inequality being trivial due to the convexity of the potential \(\Psi_1+\Psi_2\), since \(\Psi_1\) and \(\Psi_2\) are, by hypothesis, both convex. The idea guideleading our intuition is to consider the following times for any \(z\in\mathbb{R}\): \[ c_z(\sigma):=\inf\left\{t\geq0\,\,:\,\,x^1_t=x^2_t=z\right\}\,. \] Indeed, we remark \(c(\sigma)=\inf_{z\in\mathbb{R}}c_z(\sigma)\). From now on, we will establish the Kramers'type law for any \(c_z(\sigma)\).

    To obtain such a law for \(c_z(\sigma)\) with \(z\in[\lambda_1;\lambda_2]\) is quite obvious. We directly get \(\mathbb{P}-\lim_{\sigma\to0}\frac{\sigma^2}{2}\log\left(c_z(\sigma)\right)=\Delta(z):=\Psi_1(z)+\Psi_2(z)-\Psi_1(\lambda_1)-\Psi_2(\lambda_2)\). Thus, it is quite simple to prove that the probability for \(c(\sigma)\) to be smaller than \(e^{\frac{2}{\sigma^2}\left(\Delta-\delta\right)}\) goes to \(0\) when the diffusion coefficient \(\sigma\) tends to \(0\), for any arbitrarily small \(\delta>0\). Then, the collision-location is necessarily close to the unique \(\lambda_0\) minimizing the potential \(\Psi_1+\Psi_2\). Finally, we are able to prove that the collision-time does not exceed \(e^{\frac{2}{\sigma^2}\left(\Delta+\delta\right)}\) when \(\sigma\) goes to \(0\), for any arbitrarily small \(\delta>0\).

    The extension to the general dimensional case is not immediate, even with uniformly convex potentials. Indeed, the diffusions do not exhibit a recurrent behavior anymore so that the collision-problem is ill-posed. Thus, we have established similar results albeit for the following collision-time: \[ c_\epsilon(\sigma):=\inf\left\{t\geq0\,\,:\,\,|x^1_t-x^2_t|<\epsilon\right\}\,, \] where \(\epsilon>0\) is intended to go to \(0\); after the small-noise limit. With this new formulation, there is no more problem. We then proceed to a very technical generalization of the inuition established in the one-dimensional case. Indeed, despite the two diffusions are independent, and despite it could have looked like an idea for solving the collision-problem, we can not use the results of Day on the exponential behavior of the exit-time. Indeed, some hypotheses related to the domains from which we search the exit of the diffusions are necessary for applying the large deviations principles for the processes. Thus, the whole idea is to construct domains of \(\mathbb{R}^d\times\mathbb{R}^d\) satisfying all the assumptions of the classical Freidlin and Wentzell theory. In particular, these domains must be positively invariant with respect to the noiseless dynamic.

    Consequently, we obtain a similar result to the one of the above case. Namely, for any \(\delta>0\), we have: \[ \lim_{\epsilon\to0}\lim_{\sigma\to0}\mathbb{P}\left\{\frac{\sigma^2}{2}\log\left(c_\epsilon(\sigma)\right)\in\left[\Delta-\delta;\Delta+\delta\right]\right\}=1\,, \] where \(\Delta:=\inf_{z\in\mathbb{R}^d}\left(\Psi_1(z)+\Psi_2(z)-\Psi_1(\lambda_1)-\Psi_2(\lambda_2)\right)\).

    The convexity of the potential \(\Psi_1+\Psi_2\) immediately ensures that the collision location is near \(\lambda_0\), the unique minimizer of the latter potential. In a next step, we have used the techniques that I had previously developed about the exit-time of the self-stabilizing diffusion and we succeeded to show similar results to McKean-Vlasov diffusions (of self-stabilizing type) and for the associated particles systems. We have not dealt with the self-interacting diffusions. Nevertheless, it will not add any difficulty with respect to what we have already done.

    It is worth noticing that in the case of McKean-Vlasov diffusions, we do not assume the convexity of the potentials. Typically, the equations that we are studying are of this form: \[ {\rm d}X_t^1=\sigma {\rm d}B_t^1-\nabla V(X_t^1){\rm d}t-\alpha(X_t^1-\mathbb{E}(X_t^1)){\rm d}t\,, \] and \[ {\rm d}X_t^2=\sigma {\rm d}B_t^2-\nabla V(X_t^2){\rm d}t-\alpha(X_t^2-\mathbb{E}(X_t^2)){\rm d}t\,, \] where the confining potential \(V\) is double-wells, hence is not convex. However, in order to be able to apply our results on the collision-problem for independent Itô diffusions, we assume that \(\alpha{\rm Id}+\nabla^2V\geq\rho{\rm Id}\) where \(\rho>0\). In other words, we assume that we are in the so-called strictly synchronized case. \(B^1\) and \(B^2\) are two independent Brownian motions. We also assume that \(X_0^1\) and \(X_0^2\) are two points of \(\mathbb{R}^d\); each one belonging to a different basin of attraction. Then, the coupling techniques that I developed allow us to obtain a Kramers'type law on the collision-time. Moreover, the persistence of the collision-location is, of course, also established.


    This work has been extended to the case where neither the confining part of the drift nor its interaction part are deriving from gradients. This corresponds to a preprint of around fourty pages.


    This first paper opens an avenue to a serie of works where we will extend to Langevin diffusions and to asynchronized cases. To remove the synchronization consists in fact in establishing the results on the collision-time of time-homogeneous diffusions with drift non-contractive. This implies to deeply study the Freidlin and Wentzell theory and the metastability results; as the stable cycles hierarchy which will require a deep understanding of the so-called quasi-potential.

    In order to succeed in this task, Jean-François Jabir and me have recruited a very good PhD student, Hetranso AHNI. He will work on the collision problem for Langevin diffusions from November 2023. After, we will deal with case where there is no independency between the diffusions. Besides, some applications of these works concern the population dynamics through the stochastic Cucker-Smale model. Another natural application is related to the multi-species models.

Exit-problem for non Markov diffusions

    The Freidlin and Wentzell theory for the weakly perturbed random dynamical system (in particular by a Wiener process when we are interested in the case of diffusions with small temperature) is very popular and widely used. However, the classical version of this theory is interested in linear and Markov diffusions. As mentioned above, the nonlinear case has been intensively studied these last years. Nonetheless, the non Markov case is more recent. To establish a general theory on the systems which do not satisfy the Markov property is obviously left out. Indeed, this Markov property is at the hearth of the proof like it is given in the book of Dembo and Zeitouni. Hence, we deal case by case.

    In this paragraph, we introduce the obtained results for a particular case of non Markov diffusions: the so-called self-interacting diffusions. These diffusions look like the McKean-Vlasov ones of self-stabilizing type. The choice of such a diffusion is not harmless: it does correspond to a model studied by Aline Kurtzmann. Furthermore, this diffusion has some similarities with the self-stabilizing diffusion. Whereas the self-stabilizing diffusion \(X\) exhibits a nonlinearity in its drift through a convolution product between the gradient of the interacting potential \(F\) and the law at time \(t\) of the process, \(\mathcal{L}(X_t)\), the self-interacting diffusion \(Y\) contains in its drift a convolution between the gradient of the interacting potential and its empirical measure at time \(t\) that is to say \(\nu_t:=\frac{1}{t}\int_0^t\delta_{Y_s}{\rm d}s\): \[ Y_t=Y_0+\sigma B_t-\int_0^t\nabla V(Y_s){\rm d}s-\int_0^t\frac{1}{s}\int_0^s\nabla F\left(Y_s-Y_r\right){\rm d}r{\rm d}s\,, \] where the diffusion coefficient \(\sigma\) is, like often in my works, intended to be small. It is worth noticing that this equation is difficult and costly to simulate since it does not admit an interpretation in terms of interacting particles system. Then, the whole trajectory should be kept in the memory and it makes the computations lengthy and difficult.

    Despite this, the self-interacting diffusion is commonly used in the stochastic gradient descent method and in the one of the simulated annealing method. It is crucial to understand that its long-time behavior is not so different from the one of the self-stabilizing diffusion (with the same confining and interacting potentials). Indeed, we can show that any invariant probability measure \(\nu_\infty^\sigma\) of the self-interacting diffusion satisfies the following implicit equation: \[ \nu_\infty^\sigma({\rm d}x)=\frac{\exp\left\{-\frac{2}{\sigma^2}\Big[V(x)+F\ast\nu_\infty^\sigma(x)\Big]\right\}}{\int_{\mathbb{R}^d}\exp\left\{-\frac{2}{\sigma^2}\Big[V(y)+F\ast\nu_\infty^\sigma(y)\Big]\right\}{\rm d}y}\,{\rm d}x\,. \] According to the latter fact, the steady states of the self-interacting diffusion are exactly the same as the ones of the self-stabilizing diffusion. These stationary measures have been intensively studied by me since these states do correspond to the stable states of the granular media equation. The long-time convergence has been established by Aline Kurtzmann.

    These self-interacting diffusions belong to the more general class of reinforced diffusions, which is used to study the polymers. Hence, if \(F\) is convex, the polymer tends to conform itself to its past whereas it tends to escape it if the interacting potential is concave. This family of diffusions has been studied by Durrett, Rogers, Benaïm, Raimond... In any case, these diffusions are not Markov. It may be because the occupation measure \(\int_0^t\delta_{Y_s}{\rm d}s\) intervenes in the drift or because the empirical probability measure intervenes in the drift. It may also be because the drift admits the following form: \(f\left(Y_t-\frac{1}{t}\int_0^tY_s{\rm d}s\right)\)... The results that we have obtained are in the case of the empirical probability measure but we easily imagine that we can extend it for the occupation measure and for other more general settings.

    The first result on this subject that we have obtained concerns the asymptotics in the small-noise limit of the first exit-time \(\tau(\sigma):=\inf\left\{t\geq0\,\,:\,\,Y_t\notin\mathcal{D}\right\}\) where \(\mathcal{D}\) satisfies classical hypotheses in Freidlin and Wentzell theory. In this paper, in minor revision for ESAIM Probability and Statistics, we have assumed that both \(V\) and \(F\) are uniformly convex. With these hypotheses, Ashot Aleksian, Pierre Del Moral, Aline Kurtzmann and me succeeded to obtain a Kramers'type law that is to say: \(\mathbb{P}-\lim_{\sigma\to0}\frac{\sigma^2}{2}\log\left(\tau(\sigma)\right)=H\) where the exit-cost \(H\) is defined as follows: \(H:=\inf_{z\in\partial\mathcal{D}}\left(V+F\ast\delta_a-V(a)-F(0)\right)\), the point \(a\) being the unique minimizer of the potential \(V\). To obtain this result, the idea is similar to the one that I had developed for the self-stabilizing diffusion in the paper entitled "A simple proof of a Kramers'type law for self-stabilizing diffusions". First, we use the results of Aline Kurtzmann on the convergence in the convex framework. Moreover, we take advantage of the rate of convergence, which is independent from the diffusion coefficient. Then, we proceed a coupling between this diffusion and the time-homogeneous Markov diffusion: \[ Z_t=Y_{T_0}+\sigma\left(B_t-B_{T_0}\right)-\int_{T_0}^t\nabla V\left(Z_s\right){\rm d}s-\int_{T_0}^t\nabla F\left(Z_s-a\right){\rm d}s\,, \] where \(T_0\) is chosen such that \(\nu_{T_0+s}\) is close to \(\delta_a\) for any \(s\geq0\). From the classical aymptotics of the first exit-time of the diffusion \(Z\), the asymptotics of the time \(\tau(\sigma)\) yield.

    The second result mainly consists in obtaining the same asymptotics without convexity assumptions neither on \(V\) nor on \(F\). The main difficulty is that we do not know the empirical measure \(\nu_t\). In particular, it does not satisfy a simple partial differential equation since this measure is non-deterministic. The idea that we had was to establish more general large deviations principles controlling this empirical measure. This result does correspond to one of the two main objectives of the ANR project METANOLIN; this project allowed us (Aline Kurtzmann and me) to recruit Ashot Aleksian as PhD student. We went further than what was planned and the robustness of our methods will permit to Ashot Aleksian to extend to non-gradient cases and to more general non Markov settings. The used method is so technical that I will not give any detail. The main idea consists in controlling the time spent in the neighborhood of \(a\), one of the minimizers of \(V\) and the time spent far from \(a\). Hence, if we succeed to (and we succeeded) to show that the time spent far from \(a\) is negligible with respect to the one spent near \(a\) then \(\nu_t\) is close to \(\delta_a\), at least before the time \(\tau(\sigma)\). These very difficult controls have been done by Ashot. This allowed us (Ashot Aleksian, Aline Kurtzmann and me) to obtain a paper in minor revision for Probability Theory and Related Fields.

    The next step will be to study the convergence of \(\nu_t\) towards one of the invariant probability, to establish the characteristic times for which the measure \(\nu_{t(\sigma)}\) will be, for small diffusion coefficient \(\sigma\), close to \(\delta_a\). Another step will be to generalize to other non Markov settings. It is worth noticing that the applications in machine learning with the stochastic gradient descent method or with the simulated annealing method are not harmless. Finally, despite it was not the objective during the first discussions on this project, the self-interacting diffusions are nothing else than the interpretation in terms of particles system for memorial McKean-Vlasov diffusion. An example of such diffusion with memory is the probabilistic interpretation of the parabolic-parabolic Keller-Segel partial differential equation. Consequently, to regularize the kernels of this equation and to look at the associated interacting particles system remains to study a self-interacting diffusion in high dimension. As a consequence, we have here a promising and exciting avenue.

Kalman Ensembles

    Despite the Kalman Ensembles are not in the hearth of my works, I have been interested in for a while with Pierre Del Moral and Aline Kurtzmann. This lead us to three papers: "Uniform propagation of chaos properties of Ensemble Kalman-Bucy particle filters" in The Annals of Applied Probability, "On the stability and the uniform propagation of chaos of a class of extended ensemble Kalman-Bucy filters" in SIAM Journal on Control and Optimization and "On the stability and the concentration of extended Kalman-Bucy filters" in Electronic Journal of Probability.


    It is worth noticing that these models do correspond to nonlinear diffusions and to interacting particles system. We consider the following linear equation: \[ {\rm d}X_t=\left(AX_t+a\right){\rm d}t\,, \] where \(X_0\) is a gaussian random variable living in \(\mathbb{R}^{r_1}\). In concrete applications like the sciences of ocean and athmosphere, \(r_1\) may be equal to \(10~000\). Here, \(A\) is a square matrix of size \(r_1\) and \(a\) is a vector of \(\mathbb{R}^{r_1}\). The problem is that we do not have directly access to \(X_t\) despite it is the interesting signal. Nonetheless, we have access to the signal \(Y\) defined as follows: \[ {\rm d}Y_t=\left(CX_t+c\right){\rm d}t\,, \] where \(Y_0=0\). Here, \(Y_t\) is a random variable living in \(\mathbb{R}^{r_2}\). In practical situations, \(r_2\) is smaller than \(r_1\). How to find \(X\) knowning \(Y\)?

    We introduce the matrix \(M\in\mathcal{M}_{r_1\times r_2,r_1}\) defined as \[ M:=\left( \begin{array}{c} C\\ CA\\ \vdots\\ CA^{r_1-1} \end{array}\right)\,. \]     We assume that the rank of this matrix \(M\) is \(r_1\). Then, the knowledge of \(Y\) (and its increments) is sufficient to describe the process \(X\). Besides, this observability condition is necessary.

    In practical cases, some perturbations appear in the equation of the state variable and in the one of the observation variable. In fine, the system of equations is nothing else than: \[ \left\{ \begin{array}{l} {\rm d}X_t=(Ax_t+a){\rm d}t+R_1^{\frac{1}{2}}{\rm d}W_t\,,\\ {\rm d}Y_t=(CX_t+c){\rm d}t+R_2^{\frac{1}{2}}{\rm d}V_t\,. \end{array}\right. \]     We put \(\mathcal{F}_t:=\sigma\left(Y_s\,\,:\,\,0\leq s\leq t\right)\). And, \(\eta_t\) denotes the law of \(X_t\) knowing \(\mathcal{F}_t\). It is well-known that the conditional law \(\eta_t\) is gaussian and thus it is characterized by its expectation and its variance. We thus put \(\widehat{X}_t:=\mathbb{E}\left[X_t\,\left|\right.\,\mathcal{F}_t\right]\) and \(P_t:=\mathbb{E}\left\{\left(X_t-\widehat{X}_t\right)\left(X_t-\widehat{X}_t\right)^T\right\}\). Moreover, we have the following equation (Kalman-Bucy filter): \[ {\rm d}\widehat{X}_t=\left(A\widehat{X}_t+a\right){\rm d}t+P_tC^TR_2^{-1}\left({\rm d}Y_t-\left(C\widehat{X}_t+c\right){\rm d}t\right)\,, \] and the Riccati equation on \(P_t\): \[ \frac{{\rm d}}{{\rm d}t}P_t=AP_t+P_tA^T-P_tC^TR_2^{-1}CP_t+R_1\,. \]     The Riccati equation is very costly to simulate if \(r_1=10~000\) since multiplying two square matrices of this size requires \(3\times10^{12}\) elementary computations. Thus, we consider the following diffusion: \[ {\rm d}\overline{X}_t=\left(A\overline{X}_t+a\right){\rm d}t+R_1^{\frac{1}{2}}{\rm d}\overline{W}_t+\mathcal{P}_{\eta_t}C^TR_2^{-1}\left[{\rm d}Y_t-\left(C\overline{X}_t+c\right){\rm d}t+R_2^{\frac{1}{2}}{\rm d}\overline{V}_t\right]\,. \]     Here, \(\mathcal{P}_{\eta_t}:=\mathbb{E}\left\{\left(\overline{X}_t-\mathbb{E}\left[\overline{X}_t\,\left|\right.\,\mathcal{F}_t\right]\right)\left(\overline{X}_t-\mathbb{E}\left[\overline{X}_t\,\left|\right.\,\mathcal{F}_t\right]\right)^T\right\}\). It is a McKean-Vlasov type equation with a conditional law instead of the law of the process itself. Then, we can use the classical following approximation: \[ {\rm d}\xi_t^i=\left(A\xi_t^i+a\right){\rm d}t+R_1^{\frac{1}{2}}{\rm d}\overline{W}_t^i+\mathcal{P}_t^NC^TR_2^{-1}\left[{\rm d}Y_t-\left(C\xi_t^i+c\right){\rm d}t+R_2^{\frac{1}{2}}{\rm d}\overline{V}_t^i\right]\,. \] Here, \(\left(\overline{W}^i\right)_{i\in\mathbb{N}^*}\) is a family of independent Brownian motions in \(\mathbb{R}^{r_1}\) and \(\left(\overline{V}^i\right)_{i\in\mathbb{N}^*}\) is such a family in \(\mathbb{R}^{r_2}\).

    We have also been interested in extended Kalman equation. This corresponds to a nonlinear filter for which we have also proven a long-time convergence and a propagation of chaos.

    I want to stress that I do not work anymore on this subject.