arXiv subjects

Jesse Gell-Redman

Publications and source records attributed to Jesse Gell-Redman.

At least 19 recordsLinked to original sources

arXiv2026

Mean first escape times of Brownian motion on asymptotically hyperbolic and gas giant metric surfaces

This paper deals with the mean first escape time of Brownian motion on asymptotically hyperbolic and gas giant surfaces. We show that for a boundary defining function $\rho$, the mean first escape time $u_\epsilon(x)$ from the truncated Riemannian surface with an asymptotically hyperbolic metric $(M_\epsilon,\bar{g}/\rho^2) = (\{x\in M:\rho(x)\geq \epsilon\},\bar{g}/\rho^2) \subset (M,\bar{g}/\rho^2)$ satisfies the asymptotic expansion $u_\epsilon(x) = -\log \epsilon + \mathcal{O}(1)$ as $\epsilon\to 0 $. Furthermore, we show that in the case of a gas giant metric $g = \bar{g}/\rho^\alpha$, where $\alpha\in (0,2)$, the mean first escape time from the surface $(M_\epsilon,\bar{g}/\rho^\alpha)$ satisfies $u_\epsilon(x) = \mathcal{O}(1)$ as $\epsilon\to 0 $. Using techniques from the theory of polyhomogeneous conormal functions we explain this difference between in the mean first escape time on gas giant metric surfaces and asymptotically hyperbolic surfaces on the unit disc. Finally, we confirm these results using Monte Carlo simulations and finite difference methods on the disc.

math.AP↗

arXiv2025

The Klein-Gordon equation on asymptotically Minkowski spacetimes: the Feynman propagator

We develop a theory of Feynman propagators for the massive Klein--Gordon equation with asymptotically static perturbations. Building on our previous work on the causal propagators, we employ a framework based on propagation of singularities estimates in Vasy's 3sc-calculus. We combine these estimates to prove global spacetime mapping properties for the Feynman propagator, and to show that it satisfies a microlocal Hadamard condition. We show that the Feynman propagator can be realized as the inverse of a mapping between appropriate $L^2$-based Sobolev spaces with additional regularity near the asymptotic sources of the Hamiltonian flow, realized as a family of radial points on a compactified spacetime.

math.AP↗

arXiv2025

The Klein-Gordon equation on asymptotically Minkowski spacetimes: causal propagators

We construct the causal (forward/backward) propagators for the massive Klein-Gordon equation perturbed by a first order operator which decays in space but not necessarily in time. In particular, we obtain global estimates for forward/backward solutions to the inhomogeneous, perturbed Klein-Gordon equation, including in the presence of bound states of the limiting spatial Hamiltonians. To this end, we prove propagation of singularities estimates in all regions of infinity (spatial, null, and causal) and use the estimates to prove that the Klein-Gordon operator is an invertible mapping between adapted weighted Sobolev spaces. This builds off work of Vasy in which inverses of hyperbolic PDEs are obtained via construction of a Fredholm mapping problem using radial points propagation estimates. To deal with the presence of a perturbation which persists in time, we employ a class of pseudodifferential operators first explored in Vasy's many-body work.

math.AP↗

arXiv2023

Propagation of singularities and Fredholm analysis for the time-dependent Schrödinger equation

We study the time-dependent Schrödinger operator $P = D_t + Δ_g + V$ acting on functions defined on $\mathbb{R}^{n+1}$, where, using coordinates $z \in \mathbb{R}^n$ and $t \in \mathbb{R}$, $D_t$ denotes $-i \partial_t$, $Δ_g$ is the positive Laplacian with respect to a time dependent family of non-trapping metrics $g_{ij}(z, t) dz^i dz^j$ on $\mathbb{R}^n$ which is equal to the Euclidean metric outside of a compact set in spacetime, and $V = V(z, t)$ is a potential function which is also compactly supported in spacetime. In this paper we introduce a new approach to studying $P$, by finding pairs of Hilbert spaces between which the operator acts invertibly. Using this invertibility it is straightforward to solve the `final state problem' for the time-dependent Schrödinger equation, that is, find a global solution $u(z, t)$ of $Pu = 0$ having prescribed asymptotics as $t \to \infty$. These asymptotics are of the form $$ u(z, t) \sim t^{-n/2} e^{i|z|^2/4t} f_+\big( \frac{z}{2t} \big), \quad t \to +\infty $$ where $f_+$, the `final state' or outgoing data, is an arbitrary element of a suitable function space $\mathcal{W}^k(\mathbb{R}^n)$; here $k$ is a regularity parameter simultaneously measuring smoothness and decay at infinity. We can of course equally well prescribe asymptotics as $t \to -\infty$; this leads to incoming data $f_-$. We consider the `Poisson operators' $\mathcal{P}_\pm : f_\pm \to u$ and precisely characterize the range of these operators on $\mathcal{W}^k(\mathbb{R}^n)$ spaces. Finally we show that the scattering matrix, mapping $f_-$ to $f_+$, preserves these spaces.

math.AP↗

arXiv2023

Asymptotics of the radiation field for the massless Dirac-Coulomb system

We consider the long-time behavior of the massless Dirac equation coupled to a Coulomb potential. For nice enough initial data, we find a joint asymptotic expansion for solutions near the null and future infinities and characterize explicitly the decay rates seen in the expansion. This paper can be viewed as a successor to previous work on asymptotic expansions for the radiation field. The key new elements are propagation estimates near the singularity of the potential, building on work of the first author with Wunsch and an explicit calculation with hypergeometric functions to determine the rates of decay.

math.AP↗

arXiv2023

Scattering regularity for small data solutions of the nonlinear Schrödinger equation

Using the Fredholm theory of the linear time-dependent Schrödinger equation set up in our previous article arXiv:2201.03140, we solve the final-state problem for the nonlinear Schrödinger problem $$ (D_t + Δ+ V) u = N[u], \quad u(z,t) \sim (4πit)^{-n/2} e^{i|z|^2/4t} f\big( \frac{z}{2t} \big), \quad t \to -\infty, $$ where $u : \mathbb{R}^{n+1} \to \mathbb{C}$ is the unknown and $f : \mathbb{R}^n \to \mathbb{C}$ is the asymptotic data. Here $D_t = -i \frac{\partial}{\partial t}$ and $Δ= \sum_{j=1}^n D_{z_j} D_{z_j}$ is the positive Laplacian, or more generally a compactly supported, nontrapping perturbation of this, $V$ is a smooth compactly supported potential function, and the nonlinear term $N$ is a (suitable) polynomial in $u$, $\partial_{z_j}u$ and their complex conjugates satisfying phase invariance. Our assumption on the asymptotic data $f$ is that it is small in a certain function space $\mathcal{W}^k$ constructed in arXiv:2201.03140, for sufficiently large $k \in \mathbb{N}$, where the index $k$ measures both regularity and decay at infinity (it is similar to, but not quite a standard weighted Sobolev space $H^{k, k}(\mathbb{R}^n)$). We find that for $N[u] = \pm |u|^{p-1} u$, $p$ odd, and $(n,p) \neq (1, 3)$ then if the asymptotic data as $t \to -\infty$ is small in $\mathcal{W}^k$, then the asymptotic data as $t \to +\infty$ is also in $\mathcal{W}^k$; that is, the nonlinear scattering map preserves these spaces of asymptotic data. For a more general nonlinearity involving derivatives of $u$, we show that if the asymptotic data as $t \to -\infty$ is small in $\langle ζ\rangle^{-1} \mathcal{W}^k_ζ$, then the asymptotic data as $t \to +\infty$ is also in this space (where $ζ$ is the argument of $f$).

math.AP↗

arXiv2023

Price's law on Minkowski space in the presence of an inverse square potential

We consider the pointwise decay of solutions to wave-type equations in two model singular settings. Our main result is a form of Price's law for solutions of the massless Dirac-Coulomb system in (3+1)-dimensions. Using identical techniques, we prove a similar theorem for the wave equation on Minkowski space with an inverse square potential. One novel feature of these singular models is that solutions exhibit two different leading decay rates at timelike infinity in two regimes, distinguished by whether the spatial momentum along a curve which approaches timelike infinity is zero or non-zero. An important feature of our analysis is that it yields a precise description of solutions at the interface of these two regions which comprise the whole of timelike infinity.

math.AP↗

arXiv2022

Regularity of the Scattering Matrix for Nonlinear Helmholtz Eigenfunctions

We study the nonlinear Helmholtz equation $(Δ- λ^2)u = \pm |u|^{p-1}u$ on $\mathbb{R}^n$, $λ> 0$, $p \in \mathbb{N}$ odd, and more generally $(Δ_g + V - λ^2)u = N[u]$, where $Δ_g$ is the (positive) Laplace-Beltrami operator on an asymptotically Euclidean or conic manifold, $V$ is a short range potential, and $N[u]$ is a more general polynomial nonlinearity. Under the conditions $(p-1)(n-1) > 4$ and $k > (n-1)/2$, for every $f \in H^k(S^{n-1}_ω)$ of sufficiently small norm, we show there is a nonlinear Helmholtz eigenfunction taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} b(ω) + O(r^{-ε}) \Big), \qquad \text{as } r \to \infty, \end{equation*} for some $b \in H^k(S_ω^{n-1})$ and $ε> 0$. That is, the scattering matrix $f \mapsto b$ preserves Sobolev regularity, which is an improvement over the authors' previous work with Zhang, that proved a similar result with a loss of four derivatives.

math.AP↗

arXiv2021

Riemann moduli spaces are quantum ergodic

In this note we show that the Riemann moduli spaces $M_{g, n}$ equipped with the Weil--Petersson metric are quantum ergodic for $3g+n \geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.

math.AP↗

arXiv2020

Deep Learning is Singular, and That's Good

In singular models, the optimal set of parameters forms an analytic set with singularities and classical statistical inference cannot be applied to such models. This is significant for deep learning as neural networks are singular and thus "dividing" by the determinant of the Hessian or employing the Laplace approximation are not appropriate. Despite its potential for addressing fundamental issues in deep learning, singular learning theory appears to have made little inroads into the developing canon of deep learning theory. Via a mix of theory and experiment, we present an invitation to singular learning theory as a vehicle for understanding deep learning and suggest important future work to make singular learning theory directly applicable to how deep learning is performed in practice.

cs.LG↗

arXiv2019

Existence and asymptotics of nonlinear Helmholtz eigenfunctions

We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (Δ- λ^2) u = N[u], \end{equation*} where $Δ= -\sum_j \partial^2_j$ is the Laplacian on $\mathbb{R}^n$ with sign convention that it is positive as an operator, $λ$ is a positive real number, and $N[u]$ is a nonlinear operator that is a sum of monomials of degree $\geq p$ in $u$, $\overline{u}$ and their derivatives of order up to two, for some $p \geq 2$. Nonlinear Helmholtz eigenfunctions with $N[u]= \pm |u|^{p-1} u$ were first considered by Gutiérrez. Such equations are of interest in part because, for certain nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution equations, that is, solutions that are time-harmonic. We show that, under the condition $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for every $f \in H^{k+2}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} g(ω) + O(r^{-ε}) \Big), \text{ as } r \to \infty, \quad ε> 0, \end{equation*} for some $g \in H^{k}(\mathbb{S}^{n-1})$. Moreover, we prove the result in the general setting of asymptotically conic manifolds.

math.AP↗

arXiv2019

Equidistribution of Phase Shifts in Obstacle Scattering

For scattering off a smooth, strictly convex obstacle $Ω\subset \mathbb{R}^d$ with positive curvature, we show that the eigenvalues of the scattering matrix -- the phase shifts -- equidistribute on the unit circle as the frequency $k \to \infty$ at a rate proportional to $k^{d - 1}$, under a standard condition on the set of closed orbits of the billiard map in the interior. Indeed, in any sector $S \subset \mathbb{S}^1$ not containing $1$, there are $c_d |S| \mathrm{Vol}(\partial Ω)\ k^{d - 1} + o(k^{d-1})$ eigenvalues for $k$ large, where $c_d$ is a constant depending only on the dimension. Using this result, the two term asymptotic expansion for the counting function of Dirichlet eigenvalues, and a spectral-duality result of Eckmann-Pillet, we then give an alternative proof of the two term asymptotic of the total scattering phase due to Majda-Ralston.

math.SP↗

arXiv2017

The index formula for families of Dirac type operators on pseudomanifolds

We study families of Dirac-type operators, with compatible perturbations, associated to wedge metrics on stratified spaces. We define a closed domain and, under an assumption of invertible boundary families, prove that the operators are self-adjoint and Fredholm with compact resolvents and trace-class heat kernels. We establish a formula for the Chern character of their index.

math.DG↗

arXiv2016

The Index of Dirac Operators on Incomplete Edge Spaces

We derive a formula for the index of a Dirac operator on a compact, even-dimensional incomplete edge space satisfying a "geometric Witt condition". We accomplish this by cutting off to a smooth manifold with boundary, applying the Atiyah-Patodi-Singer index theorem, and taking a limit. We deduce corollaries related to the existence of positive scalar curvature metrics on incomplete edge spaces.

math.DG↗

arXiv2016

The Feynman propagator on perturbations of Minkowski space

In this paper we analyze the Feynman wave equation on Lorentzian scattering spaces. We prove that the Feynman propagator exists as a map between certain Banach spaces defined by decay and microlocal Sobolev regularity properties. We go on to show that certain nonlinear wave equations arising in QFT are well-posed for small data in the Feynman setting.

math.AP↗

arXiv2015

Spectral and Hodge theory of `Witt' incomplete cusp edge spaces

Incomplete cusp edges model the behavior of the Weil-Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove that the Hodge-Laplacian on differential forms is essentially self-adjoint, with discrete spectrum satisfying Weyl asymptotics. We go on to prove bounds on the growth of $L^2$-harmonic forms at the singular set and to prove a Hodge theorem, namely that the space of $L^2$-harmonic forms is naturally isomorphic to the middle-perversity intersection cohomology. Moreover, we develop an asymptotic expansion for the heat trace near $t = 0$.

math.AP↗

arXiv2015

The distribution of phase shifts for semiclassical potentials with polynomial decay

This is the third paper in a series analyzing the asymptotic distribution of the phase shifts in the semiclassical limit. We analyze the distribution of phase shifts, or equivalently, eigenvalues of the scattering matrix, $S_h(E)$, for semiclassical Schrödinger operators on $\mathbb{R}^d$ which are perturbations of the free Hamiltonian by a potential $V$ with polynomial decay. Our assumption is that $V(x) \sim |x|^{-α} v(\hat x)$ as $x \to \infty$, for some $α> d$, with corresponding derivative estimates. In the semiclassical limit $h \to 0$, we show that the atomic measure on the unit circle defined by these eigenvalues, after suitable scaling in $h$, tends to a measure $μ$ on $\mathbb{S}^1$. Moreover, $μ$ is the pushforward from $\mathbb{R}$ to $\mathbb{R} / 2 π\mathbb{Z} = \mathbb{S}^1$ of a homogeneous distribution $ν$ of order $β$ depending on the dimension $d$ and the rate of decay $α$ of the potential function. As a corollary we obtain an asymptotic formula for the accumulation of phase shifts in a sector of $\mathbb{S}^1$. The proof relies on an extension of results of the second author and Wunsch on the classical Hamiltonian dynamics and semiclassical Poisson operator to the class of potentials under consideration here.

math.AP↗

arXiv2014

Hodge cohomology of some foliated boundary and foliated cusp metrics

For fibred boundary and fibred cusp metrics, Hausel, Hunsicker, and Mazzeo identified the space of $L^2$ harmonic forms of fixed degree with the images of maps between intersection cohomology groups of an associated stratified space obtained by collapsing the fibres of the fibration at infinity onto its base. In the present paper, we obtain a generalization of this result to situations where, rather than a fibration at infinity, there is a Riemannian foliation with compact leaves admitting a resolution by a fibration. If the associated stratified space (obtained now by collapsing the leaves of the foliation) is a Witt space and if the metric considered is a foliated cusp metric, then no such resolution is required.

math.DG↗