My research is concerned with stochastic analysis, and in particular studying the effects that rough or stochastic noise has on models coming from the physical or social sicneces. The objects of my research include ordinary and partial differential equations driven by rough or stochastic noise, stochastic homogenization and other asymptotic problems, numerical analysis, mean field games and mean field control, and rough path theory.
Currently, my research activities are supported by the NSF grant DMS-2307610.
Preprints
- Well-posedness of Hamilton-Jacobi equations in the Wasserstein space: non-convex Hamiltonians and common noise (with Samuel Daudin and Joe Jackson) [arXiv, abstract]
- Linear and nonlinear transport equations with coordinate-wise increasing velocity fieldss (with Pierre-Louis Lions) [arXiv, abstract]
- Transport equations and flows with one-sided Lipschitz velocity fields (with Pierre-Louis Lions) [arXiv, abstract]
- Mean field games with common noise and degenerate idiosyncratic noise (with Pierre Cardaliaguet and Panagiotis Souganidis) [arXiv, abstract]
We establish the well-posedness of viscosity solutions for a class of semi-linear Hamilton-Jacobi
equations set on the space of probability measures on the torus. In particular, we focus on
equations with both common and idiosyncratic noise, and with Hamiltonians which are not
necessarily convex in the momentum variable. Our main results show (i) existence, (ii) the
comparison principle (and hence uniqueness), and (iii) the convergence of finite-dimensional
approximations for such equations. Our proof strategy for the comparison principle is to first
use a mix of existing techniques (especially a change of variables inspired by Bayraktar, Ekren, and
Zhang to deal with the common noise) to prove a ``partial comparison" result, i.e. a comparison
principle which holds when either the subsolution or the supersolution is Lipschitz with respect
to a certain very weak metric. Our main innovation is then to develop a strategy for removing this
regularity assumption. In particular, we use some delicate estimates for a sequence of
finite-dimensional PDEs to show that under certain conditions there exists a viscosity
solution which is Lipschitz with respect to the relevant weak metric. We then use this existence
result together with a mollification procedure to establish a full comparison principle, i.e. a
comparison principle which holds even when the subsolution under consideration is just upper
semi-continuous (with respect to the weak topology) and the supersolution is just lower
semi-continuous. We then apply these results to mean-field control problems with common
noise and zero-sum games over the Wasserstein space.
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.
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.
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.
Publications
- A comparison principle for semilinear Hamilton-Jacobi-Bellman equations in the Wasserstein space (with Samuel Daudin). To appear in Calc. Var. Partial Differential Equations. [arXiv, abstract]
- The Neumann problem for fully nonlinear SPDE (with Paul Gassiat). To appear in Ann. Appl. Probab. [arXiv, abstract]
- Interpolation results for pathwise Hamilton-Jacobi equations (with Pierre-Louis Lions and Panagiotis Souganidis). Indiana Univ. Math. J., 2022 [link, arXiv, abstract]
- Besov rough path analysis (with Peter Friz). J. Differential Equations, 2022 [link, arXiv, abstract]
- Dimension reduction techniques in deterministic mean field games (with Jean-Michel Lasry and Pierre-Louis Lions). Communications in Partial Differential Equations [link, arXiv, abstract]
- Hölder regularity of Hamilton-Jacobi equations with stochastic forcing (with Pierre Cardaliaguet). Trans. Amer. Math. Soc., 2021 [link, arXiv, abstract]
- Homogenization of a stochastically forced Hamilton-Jacobi equation. Ann. Inst. H. Poincaré Anal. Non Linéaire, 2021 [link, arXiv, abstract]
- Scaling limits and homogenization of mixing Hamilton-Jacobi equations. Communications in Partial Differential Equations, 2020 [link, arXiv, abstract]
- Approximation schemes for viscosity solutions of fully nonlinear stochastic partial differential equations. Annals of Applied Probability, 2020 [link, arXiv, abstract]
- Perron's method for pathwise viscosity solutions. Communications in Partial Differential Equations, 2018 [link, arXiv, abstract]
- Homogenization of pathwise Hamilton-Jacobi equations. Journal de Mathématiques Pures et Appliquées, 2018 [link, arXiv, abstract]
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.