Ben Seeger

I study analysis and partial differential equations (PDE). My research interests include nonlinear stochastic partial differential equations, stochastic homogenization and other asymptotic problems, numerical methods, mean field games, and rough path theory.


  1. A comparison principle for semilinear Hamilton-Jacobi-Bellman equations in the Wasserstein space (with Samuel Daudin) [arXiv, abstract]
  2. The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.
  3. Linear and nonlinear transport equations with coordinate-wise increasing velocity fieldss (with Pierre-Louis Lions) [arXiv, abstract]
  4. We consider linear and nonlinear transport equations with irregular velocity fields, motivated by models coming from mean field games. The velocity fields are assumed to increase in each coordinate, and the divergence therefore fails to be absolutely continuous with respect to the Lebesgue measure in general. For such velocity fields, the well-posedness of first- and second-order linear transport equations in Lebesgue spaces is established, as well as the existence and uniqueness of regular ODE and SDE Lagrangian flows. These results are then applied to the study of certain nonconservative, nonlinear systems of transport type, which are used to model mean field games in a finite state space. A notion of weak solution is identified for which a unique minimal and maximal solution exist, which do not coincide in general. A selection-by-noise result is established for a relevant example to demonstrate that different types of noise can select any of the admissible solutions in the vanishing noise limit.
  5. Transport equations and flows with one-sided Lipschitz velocity fields (with Pierre-Louis Lions) [arXiv, abstract]
  6. We study first- and second-order linear transport equations, as well as ODE and SDE flows, with velocity fields satisfying a one-sided Lipschitz condition. Depending on the time direction, the flows are either compressive or expansive. In the compressive regime, we characterize the stable continuous distributional solutions of both the first and second-order nonconservative transport equations as the unique viscosity solution. Our results in the expansive regime complement the theory of Bouchut, James, and Mancini, and we provide a complete theory for both the conservative and nonconservative equations in Lebesgue spaces, as well as proving the existence, uniqueness, and stability of the regular Lagrangian ODE flow. We also provide analogous results in this context for second order equations and SDEs with degenerate noise coefficients that are constant in the spatial variable.
  7. Mean field games with common noise and degenerate idiosyncratic noise (with Pierre Cardaliaguet and Panagiotis Souganidis) [arXiv, abstract]
  8. We study the forward-backward system of stochastic partial differential equations describing a mean field game for a large population of small players subject to both idiosyncratic and common noise. The unique feature of the problem is that the idiosyncratic noise coefficient may be degenerate, so that the system does not admit smooth solutions in general. We develop a new notion of weak solutions for backward stochastic Hamilton-Jacobi-Bellman equations, and use this to build probabilistically weak solutions of the mean field game system.


  1. The Neumann problem for fully nonlinear SPDE (with Paul Gassiat). To appear in Ann. Appl. Probab. [arXiv, abstract]
  2. We generalize the notion of pathwise viscosity solutions, put forward by Lions and Souganidis to study fully nonlinear stochastic partial differential equations, to equations set on a sub-domain with Neumann boundary conditions. Under a convexity assumption on the domain, we obtain a comparison theorem which yields existence and uniqueness of solutions as well as continuity with respect to the driving noise. As an application, we study the long time behaviour of a stochastically perturbed mean-curvature flow in a cylinder-like domain with right angle contact boundary condition.
  3. Interpolation results for pathwise Hamilton-Jacobi equations (with Pierre-Louis Lions and Panagiotis Souganidis). Indiana Univ. Math. J., 2022 [link, arXiv, abstract]
  4. We study the interplay between the regularity of paths and Hamiltonians in the theory of pathwise Hamilton-Jacobi equations with the use of interpolation methods. The regularity of the paths is measured with respect to Sobolev, Besov, Hölder, and variation norms, and criteria for the Hamiltonians are presented in terms of both regularity and structure. We also explore various properties of functions that are representable as the difference of convex functions, the largest space of Hamiltonians for which the equation is well-posed for all continuous paths. Finally, we discuss some open problems and conjectures.
  5. Besov rough path analysis (with Peter Friz). J. Differential Equations, 2022 [link, arXiv, abstract]
  6. Rough path analysis is developed in the full Besov scale. This extends, and essentially concludes, an investigation started by [Prömel--Trabs, Rough differential equations driven by signals in {B}esov spaces. J. Diff. Equ. 2016], further studied in a series of papers by Liu, Prömel and Teichmann. A new Besov sewing lemma, a real-analysis result of interest in its own right, plays a key role, and the flexibility in the choice of Besov parameters allows for the treatment of equations not available in the Hölder or variation settings. Important classes of stochastic processes fit in the present framework.
  7. Dimension reduction techniques in deterministic mean field games (with Jean-Michel Lasry and Pierre-Louis Lions). Communications in Partial Differential Equations [link, arXiv, abstract]
  8. We present examples of equations arising in the theory of mean field games that can be reduced to a system in smaller dimensions. Such examples come up in certain applications, and they can be used as modeling tools to numerically approximate more complicated problems. General conditions that bring about reduction phenomena are presented in both the finite and infinite state-space cases. We also compare solutions of equations with noise with their reduced versions in a small-noise expansion.
  9. Hölder regularity of Hamilton-Jacobi equations with stochastic forcing (with Pierre Cardaliaguet). Trans. Amer. Math. Soc., 2021 [link, arXiv, abstract]
  10. We obtain space-time Hölder regularity estimates for solutions of first- and second-order Haimlton-Jacobi equations perturbed with an additive stochastic forcing term. The bounds depend only on the growth of the Hamiltonian in the gradient and on the regularity of the stochastic coefficients, in a way that is invariant with respect to a hyperbolic scaling.
  11. Homogenization of a stochastically forced Hamilton-Jacobi equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2021 [link, arXiv, abstract]
  12. We study the homogenization of a Hamilton-Jacobi equation forced by rapidly oscillating noise that is colored in space and white in time. It is shown that the homogenized equation is deterministic, and, in general, the noise has an enhancement effect, for which we provide a quantitative estimate. As an application, we perform a noise sensitivity analysis for Hamilton-Jacobi equations forced by a noise term with small amplitude, and identify the scaling at which the macroscopic enhancement effect is felt. The results depend on new, probabilistic estimates for the large scale Hölder regularity of the solutions, which are of independent interest.
  13. Scaling limits and homogenization of mixing Hamilton-Jacobi equations. Communications in Partial Differential Equations, 2020 [link, arXiv, abstract]
  14. We study the homogenization of nonlinear, first-order equations with highly oscillatory mixing spatio-temporal dependence. It is shown in a variety of settings that the homogenized equations are stochastic Hamilton-Jacobi equations with deterministic, spatially homogenous Hamiltonians driven by white noise in time. The paper also contains proofs of some general regularity and path stability results for stochastic Hamilton-Jacobi equations, which are needed to prove some of the homogenization results and are of independent interest.
  15. Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations. Annals of Applied Probability, 2020 [link, arXiv, abstract]
  16. The aim of this paper is to develop a general method for constructing approximation schemes for viscosity solutions of fully nonlinear pathwise stochastic partial differential equations, and for proving their convergence. Our results apply to approximations such as explicit finite difference schemes and Trotter-Kato type mixing formulas. The irregular time dependence disrupts the usual methods from the classical viscosity theory for creating schemes that are both monotone and convergent, an obstacle that cannot be overcome by incorporating higher order correction terms, as is done for numerical approximations of stochastic or rough ordinary differential equations. The novelty here is to regularize those driving paths with non-trivial quadratic variation in order to guarantee both monotonicity and convergence.

    We present qualitative and quantitative results, the former covering a wide variety of schemes for second-order equations. An error estimate is established in the Hamilton-Jacobi case, its merit being that it depends on the path only through the modulus of continuity, and not on the derivatives or total variation. As a result, it is possible to choose a regularization of the path so as to obtain efficient rates of convergence. This is demonstrated in the specific setting of equations with multiplicative white noise in time, in which case the convergence holds with probability one. We also present an example using scaled random walks that exhibits convergence in distribution.
  17. Perron's method for pathwise viscosity solutions. Communications in Partial Differential Equations, 2018 [link, arXiv, abstract]
  18. We use Perron's method to construct viscosity solutions of fully nonlinear degenerate parabolic pathwise (rough) partial differential equations. This provides an intrinsic method for proving the existence of solutions that relies only on a comparison principle, rather than considering equations driven by smooth approximating paths. The result covers the case of multidimensional geometric rough path noise, where the noise coefficients depend nontrivially on space and on the gradient of the solution. Also included in this note is a discussion of the comparison principle and a summary of the pathwise equations for which one has been proved.
  19. Homogenization of pathwise Hamilton-Jacobi equations. Journal de Mathématiques Pures et Appliquées, 2018 [link, arXiv, abstract]
  20. We present qualitative and quantitative homogenization results for pathwise Hamilton-Jacobi equations with “rough” multiplicative driving signals. When there is only one such signal and the Hamiltonian is convex, we show that the equation, as well as equations with smooth approximating paths, homogenize. In the multi-signal setting, we demonstrate that blow-up or homogenization may take place. The paper also includes a new well-posedness result, which gives explicit estimates for the continuity of the solution map and the equicontinuity of solutions in the spatial variable.