Article | Armand Bernou

Mean-field approximation, Gibbs relaxation, and cross estimates

Thu, 31 Jul 2025 00:00:00 +0000

Mitia’s webpage can be found here.

Convex order and increasing convex order for McKean-Vlasov processes with common noise

Thu, 24 Apr 2025 00:00:00 +0000

Abstract: We establish results on the conditional and standard convex order, as well as the increasing convex order, for two processes $X = (X_t)_{t \in [0, T]}$ and $Y = (Y_t)_{t \in [0, T]}$, defined by the following McKean-Vlasov equations with common Brownian noise $B^0 = (B_t^0)_{t \in [0, T]}$:

$$\displaystyle \begin{align} dX_t &= b(t, X_t, \mathcal{L}^1(X_t))dt + \sigma(t, X_t, \mathcal{L}^1(X_t))dB_t + \sigma^0(t, \mathcal{L}^1(X_t))dB^0_t, \nonumber \\ dY_t &= \beta(t, Y_t, \mathcal{L}^1(Y_t))dt + \theta(t, Y_t, \mathcal{L}^1(Y_t))dB_t + \theta^0(t, \mathcal{L}^1(Y_t))dB^0_t, \nonumber \end{align} $$

where $\mathcal{L}^1(X_t)$ (respectively $\mathcal{L}^1(Y_t)$) denotes a version of the conditional distribution of $X_t$ (resp. $Y_t$) given $B^0$. These results extend those established for standard McKean-Vlasov equations in Liu et al. (2023) and Liu and Qiu (2021). Under suitable conditions, for a (non-decreasing) convex functional $F$ on the path space with polynomial growth, we show $\mathbb{E}[F(X) \mid B^0] \leq \mathbb{E}[F(Y) \mid B^0]$ almost surely. Moreover, for a (non-decreasing) convex functional $G$ defined on the product space of paths and their marginal distributions, we establish

$$\displaystyle \begin{align} \mathbb{E}\Big[G\big(X, (\mathcal{L}^1(X_t))_{t\in[0, T]}\big)\Big| B^0\Big] \leq \mathbb{E}\Big[G\big(Y, (\mathcal{L}^1(Y_t))_{t\in[0, T]}\big)\Big| B^0\Big] \quad \text{a.s.} \nonumber \end{align} $$

Similar convex order results are also established for the corresponding particle system. Finally, we explore applications of these results to stochastic control problems—deducing in particular an associated comparison principle for Hamilton-Jacobi-Bellman equations with different coefficients—and to the interbank systemic risk model introduced by Carmona, Fouque, and Sun (2015).

What’s the graph about ? On the figure, one can see the expected size of default in a systemic risk model for interbank exchanges. This is a convex functional. On the top, the processes have diffusion coefficients $\sigma = 4$ and common diffusion coefficient $\sigma^0 = 3$. On the bottom, the processes $(X^i)_{1 \le i \le N}$ have a diffusion coefficients $\theta(X^i) = 4 S(X^i)$ where $S$ is the sigmoid function, so that $\theta \le \sigma$, and common diffusion coefficient $\theta^0 = 2 \le \sigma^0$. As predicted by our result on the convex order for particle system, the expected size of default of the second model is consistently below the one of the first.

About my co-authors: You can find Yating’s webpage here and Théophile’s info here.

Creation of chaos for interacting Brownian particles

Mon, 14 Apr 2025 00:00:00 +0000

Mitia’s webpage can be found here.

Matthieu’s webpage can be found here.

Particle method for the numerical simulation of the path-dependent McKean-Vlasov equation

Sat, 01 Jun 2024 00:00:00 +0000

You can find Yating’s webpage here.

Homogenization of Active Suspensions and Reduction of Effective Viscosity

Wed, 01 Mar 2023 00:00:00 +0000

The image above is taken from Saintillan (Ann. Rev. in Fl. Mech. 2017), and shows model vs experimental results for the disturbance flow near a bacterium, with the two swimming mechanisms displayed: pusher on the left, puller on the right. Check the introduction of the paper for more details !

The slides are from a presentation I gave at the I2M in Marseille in January 2022.

You can find Mitia’s webpage here, and Antoine’s personal page here. This work was part of Antoine’s ERC Grant COR-RAND.

Invariance principle for the random walk in random environment

Tue, 01 Nov 2022 00:00:00 +0000

This report was written as a detailed abstract for the presentation I gave during the Arbeitsgemeinschaft: Quantitative Stochastic Homogenization, organized by A. Gloria and F. Otto in Oberwolfach in October 2022.

The picture above illustrates a key point in the proof of De Masi-Ferrari-Golstein-Wick and Kipnis-Varadhan: one should rather look at the environment as seen by the particle (thus the environment is itself a process changing with time) to derive invariance principles. For a particle starting from the center of the square at time 0, the left-hand-side image shows the position at time 1, while the right-hand-side image shows the environment seen by the particle at time 1, which is just the l.h.s. grid appropriately shifted.

Long-Time Behavior of Kinetic Equations with Boundary Effects

Fri, 18 Dec 2020 00:00:00 +0000

This is the manuscript of my PhD thesis. Some slides are also available, from which the image above, taken from the simulation of the evolution of 5000 particles in a star-shaped domain following a free-transport evolution with boundary effects, is extracted (I advise to open the pdf with Adobe, as embedded simulations work better there).