James-Michael Leahy - Academia.edu (original) (raw)

Papers by James-Michael Leahy

Journal of Functional Analysis

We consider the Euler equations for the incompressible flow of an ideal fluid with an additional ... more We consider the Euler equations for the incompressible flow of an ideal fluid with an additional rough-in-time, divergence-free, Lie-advecting vector field. In recent work, we have demonstrated that this system arises from Clebsch and Hamilton-Pontryagin variational principles with a perturbative geometric rough path Lie-advection constraint. In this paper, we prove local well-posedness of the system in 2-Sobolev spaces with integer regularity ≥ ⌊ /2⌋ + 2 and establish a Beale-Kato-Majda (BKM) blow-up criterion in terms of the 1 ∞-norm of the vorticity. In dimension two, we show that the-norms of the vorticity are conserved, which yields global well-posedness and a Wong-Zakai approximation theorem for the stochastic version of the equation.

Advances in Mathematics

In recent works, beginning with [Hol15], several stochastic geophysical fluid dynamics (SGFD) mod... more In recent works, beginning with [Hol15], several stochastic geophysical fluid dynamics (SGFD) models have been derived from variational principles. In this paper, we introduce a new framework for parametrization schemes (PS) in GFD. We derive a class of rough geophysical fluid dynamics (RGFD) models as critical points of rough action functionals using the theory of controlled rough paths. These RGFD models characterize Lagrangian trajectories in fluid dynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps. We formulate three constrained variational approaches for the derivation of these models. The first is the Clebsch formulation, in which the constraints are imposed as rough advection laws. The second is the Hamilton-Pontryagin formulation, in which the constraints are imposed as right-invariant rough vector fields. And the third is the Euler-Poincaré formulation, in which the variations are constrained. These constrained rough variational principles lead directly to the Lie-Poisson Hamiltonian formulation of fluid dynamics on GRP. The GRP framework preserves the geometric structure of fluid dynamics obtained by using Lie group reduction to pass from Lagrangian to Eulerian variational principles, yielding a rough formulation of the Kelvin circulation theorem. The rough formulation enhances its stochastic counterpart developed in [Hol15], and extended to semimartingales in [SC20]. For example, the rough-path variational approach includes non-Markovian perturbations of the Lagrangian fluid trajectories. In particular, memory effects can be introduced through a judicious choice of the rough path (e.g. a realization of a fractional Brownian motion). In the particular case when the rough path is a realization of a semimartingale, we recover the SGFD models in [Hol15, SC20]. However, by eliminating the need for stochastic variational tools, we retain a pathwise interpretation of the Lagrangian trajectories. In contrast, the Lagrangian trajectories in the stochastic framework are described by stochastic integrals, which do not have a pathwise interpretation. Thus, the rough path formulation restores this property. Contents

We study the global convergence of policy gradient for infinite-horizon, continuous state and act... more We study the global convergence of policy gradient for infinite-horizon, continuous state and action space, entropy-regularized Markov decision processes (MDPs). We consider a softmax policy with (one-hidden layer) neural network approximation in a mean-field regime. Additional entropic regularization in the associated mean-field probability measure is added, and the corresponding gradient flow is studied in the 2-Wasserstein metric. We show that the objective function is increasing along the gradient flow. Further, we prove that if the regularization in terms of the mean-field measure is sufficient, the gradient flow converges exponentially fast to the unique stationary solution, which is the unique maximizer of the regularized MDP objective. Lastly, we study the sensitivity of the value function along the gradient flow with respect to regularization parameters and the initial condition. Our results rely on the careful analysis of non-linear Fokker--Planck--Kolmogorov equation and ...

The Annals of Applied Probability, 2021

We introduce a rough perturbation of the Navier-Stokes system and justify its physical relevance ... more We introduce a rough perturbation of the Navier-Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong-Zakai result for Brownian driving paths, and show that for a large class of driving signals, the system generates a continuous random dynamical system. In dimension three, the noise is not enstrophy balanced, and we establish the existence of local in time solutions.

Journal of Statistical Physics, 2020

We formulate a class of stochastic partial differential equations based on Kelvin’s circulation t... more We formulate a class of stochastic partial differential equations based on Kelvin’s circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and non-local, in both physical and probability space. Before taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm (Proc R Soc A 471(2176):20140963, 2015). Remarkably, the introduction of the non-locality in probability space in the form of momentum transported by its own mean velocity gives rise to a closed equation for the expectation field which comprises Navier–Stokes equations with Lie–Laplacian ‘dissipation’. As such, this form of non-locality provides a regularization mechanism. The formalism we devel...

Journal of Evolution Equations, 2018

We consider the Navier-Stokes system in two and three space dimensions perturbed by transport noi... more We consider the Navier-Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space-time dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier-Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the solution is unique and stable with respect to the driving noise.

Stochastic Processes and their Applications, 2016

We study the rate of convergence of an explicit and an implicit-explicit finite difference scheme... more We study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear stochastic integro-differential equations of parabolic type arising in non-linear filtering of jump-diffusion processes. We show that the rate is of order one in space and order one-half in time. Let (Ω, F , F, P), F = (F t) t≥0 , be a complete filtered probability space such that the filtration is right continuous and F 0 contains all P-null sets of F. Let {w ̺ } ∞ ̺=1 be a sequence of independent real-valued F-adapted Wiener processes. Let π 1 (dz) and π 2 (dz) be a Borel sigma-finite measures on R d satisfying R d |z| 2 ∧ 1 π r (dz) < ∞, r ∈ {1, 2}. Let q(dt, dz) = p(dt, dz) − π 2 (dz)dt be a compensated F-adapted Poisson random measure on R + × R d. Let T > 0 be an arbitrary fixed constant. On [0, T ] × R d , we consider finite difference approximation of a linear second order SPDE driven by continuous martingale noise can be accelerated to any order by Richardson's extrapolation method. For the nondegenerate case, we refer to [10] and [11], and for the degenerate case, we refer to [7]. In [14] and [15], E. Hall proved that the same method of acceleration can be applied to implicit time-discretized SPDEs driven by continuous martingale noise. In the literature, finite element, spectral, and, more generally, Galerkin schemes have been studied for SPDEs driven by discontinuous martingale noise. One of the earliest

Stochastics and Partial Differential Equations: Analysis and Computations, 2016

We prove the existence of classical solutions to parabolic linear stochastic integrodifferential ... more We prove the existence of classical solutions to parabolic linear stochastic integrodifferential equations with adapted coefficients using Feynman-Kac transformations, conditioning, and the interlacing of space-inverses of stochastic flows associated with the equations. The equations are forward and the derivation of existence does not use the "general theory" of SPDEs. Uniqueness is proved in the class of classical solutions with polynomial growth. Contents 1 Introduction 2 Outline of main results 3 Proof of main theorems 3.1 Proof of uniqueness for Theorem 2.2 3.2 Small jump case 3.3 Adding free and zero-order terms 3.4 Adding uncorrelated part (Proof of Theorem 2.2) 3.5 Interlacing a sequence of large jumps (Proof of Theorem 2.5) 4 Appendix 4.1 Martingale and point measure measure moment estimates 4.2 Optional projection 4.3 Estimates of Hölder continuous functions 4.4 Stochastic Fubini thoerem 4.5 Itô-Wentzell formula D k |H k t (x, z)| α + |ρ k t (x, z)| 2 π k (dz) + E k |H k t (x, z)| 1∧α + |ρ k t (x, z)| π k (dz) < ∞.

Stochastic Partial Differential Equations: Analysis and Computations, 2015

We derive moment estimates and a strong limit theorem for space inverses of stochastic flows gene... more We derive moment estimates and a strong limit theorem for space inverses of stochastic flows generated by jump SDEs with adapted coefficients in weighted Hölder norms using the Sobolev embedding theorem and the change of variable formula. As an application of some basic properties of flows of continuous SDEs, we derive the existence and uniqueness of classical solutions of linear parabolic second order SPDEs by partitioning the time interval and passing to the limit. The methods we use allow us to improve on previously known results in the continuous case and to derive new ones in the jump case. Let (Ω, F , F = (F t) t≥0 , P) be a complete filtered probability space satisfying the usual conditions of right-continuity and completeness. Let (w ̺ t) ρ≥1 , t ≥ 0, ̺ ∈ N, be a sequence of independent one-dimensional F-adapted Wiener processes. For a (Z, Z, π) is a sigma-finite mea-u t (x) = x, t ≤ τ, (1.2) whereb i t (x) = b i t (x) − σ j̺ t (x)∂ j σ i̺ t (x). In [LM14], we use all of the properties of the flow X t (τ, x) that are established in this work in order to derive the existence and uniqueness of classical solutions of linear parabolic

Stochastic Processes and their Applications, 2015

We prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equa... more We prove the existence and uniqueness of solutions of degenerate linear stochastic evolution equations driven by jump processes in a Hilbert scale using the variational framework of stochastic evolution equations and the method of vanishing viscosity. As an application of this result, we derive the existence and uniqueness of solutions of degenerate parabolic linear stochastic integro-differential equations (SIDEs) in the Sobolev scale. The SIDEs that we consider arise in the theory of non-linear filtering as the equations governing the conditional density of a degenerate jump-diffusion signal given a jump-diffusion observation, possibly with correlated noise. t) ρ∈N , t ≥ 0, be a sequence of continuous local uncorrelated martingales such that d w ̺ t = dV t , for all ρ ∈ N. Let d 1 , d 2 ∈ N. For convenience, we set (Z 1 , Z 1) = (Z, Z) and π 1 t = π t. We consider the d 2-dimensional system of SIDEs on [0, T ]×R d 1