arXiv subjects

Thomas Hughes

Publications and source records attributed to Thomas Hughes.

13 recordsLinked to original sources

arXiv2026

A compact support property for infinite-dimensional SDEs with H\"older continuous coefficients

We consider non-negative solutions to some infinite-dimensional SDEs on $\mathbb{Z}^d$ with H\"older continuous noise coefficients. We prove that if the H\"older exponent is less than $1/2$, solutions are compactly supported for almost all times, a variant of the classical compact support property for SPDEs. Our results imply that the instantaneous propagation of supports for superprocesses associated to discontinuous spatial motions is effectively sharp. We also show in a special case that the support is unbounded on a dense set of times. The proof uses a general approach which we expect can be applied to prove similar results for non-local SPDEs. It is based on an analysis of the excursions and zero sets of semimartingales whose quadratic variation satisfies a certain lower bound. As a corollary of our method, we show that the zero sets of non-negative solutions to some simple one-dimensional SDEs have positive Lebesgue measure, despite the absence of "sticky" dynamics.

math.PR↗

arXiv2026

Fluctuations for fully pushed stochastic fronts

We study the asymptotic behaviour, in the small noise limit, of stochastic travelling wave solutions to reaction-diffusion equations perturbed by Wright-Fisher noise. Such equations are predicted to display three distinct responses to noise in three parametric regimes: fully pushed, semi-pushed, and pulled. We prove, for the entire fully pushed regime, that solutions are asymptotically close to a stochastic shift of the deterministic travelling wave, and characterize the limiting shift process as a Brownian motion with drift. This gives the first full fluctuation theorem demonstrating fully pushed phenomenology for a non-linear stochastic reaction-diffusion equation and verifies a physical conjecture of Birzu, Hallatschek and Korolev [BHK18]. The proof uses an infinite-dimensional version of a method introduced by Katzenberger [Kat91], as pioneered by Funaki [Fun95]. This approach views the dynamics as a stochastic perturbation of a dynamical system (the PDE) with strong drift towards an invariant manifold, in our case the set of shifts of the travelling wave profile, and gives an expression for the stochastic motion "along" this manifold. Implementing this method in our setting requires many ingredients, including a close analysis of the dynamics of the corresponding PDE, integrability and regularity properties of solutions to the SPDE, and sharp control of the position of the right endpoint of the solution's support.

math.PR↗

arXiv2026

Evaluating Causal Discovery Algorithms for Path-Specific Fairness and Utility in Healthcare

Causal discovery in health data faces evaluation challenges when ground truth is unknown. We address this by collaborating with experts to construct proxy ground-truth graphs, establishing benchmarks for synthetic Alzheimer's disease and heart failure clinical records data. We evaluate the Peter-Clark, Greedy Equivalence Search, and Fast Causal Inference algorithms on structural recovery and path-specific fairness decomposition, going beyond composite fairness scores. On synthetic data, Peter-Clark achieved the best structural recovery. On heart failure data, Fast Causal Inference achieved the highest utility. For path-specific effects, ejection fraction contributed 3.37 percentage points to the indirect effect in the ground truth. These differences drove variations in the fairness-utility ratio across algorithms. Our results highlight the need for graph-aware fairness evaluation and fine-grained path-specific analysis when deploying causal discovery in clinical applications.

cs.LG↗

arXiv2025

Systematic assessment of the Hubble tension via Bayesian jackknife testing

Statistically-significant differences in the value of the Hubble parameter are found depending on the measurement method that is used, a result known as the Hubble tension. A variety of ways of comparing, grouping, and excluding measurements have been used to try to explain this, either in terms of physical effects or systematic errors. We present a systematic 'Bayesian jackknife' analysis of 16 independent measurements of the Hubble parameter in an attempt to identify whether the measurements fall into meaningful clusters that would help explain the origin of the tension. After evaluating evidence ratios for the commonly-used split into early- vs late-time measurements, we then study a range of simplified alternative physical scenarios that reflect different physical origins of an apparent bias or shift in the value of $H_0$, assigning phenomenological population parameters to each subset. These include scenarios where specific subsets are biased (e.g. due to unrecognised experimental systematics in the local distance ladder or cosmic microwave background measurements), as well as more cosmologically-motivated cases involving modifications to the expansion history. Many of these scenarios have similar marginal likelihood, but the model where no measurements are biased is strongly disfavoured. Finally, we marginalise over all these scenarios to estimate the 'model agnostic' posterior distribution of $H_0$. The resulting distribution is mildly multi-modal, but modestly favours values near $H_0=68$ km/s/Mpc, with a 95\% credible region of $66.7 < H_0 < 72.7$ km/s/Mpc.

astro-ph.CO↗

arXiv2024

Generalized front propagation for spatial stochastic population models

We present a general framework which can be used to prove that, in an annealed sense, rescaled spatial stochastic population models converge to generalized propagating fronts. Our work is motivated by recent results of Etheridge, Freeman, and Penington [EFP2017] and Huang and Durrett [HD2021], who proved convergence to classical mean curvature flow (MCF) for certain spatial stochastic processes, up until the first time when singularities of MCF form. Our arguments rely on the level-set method and the abstract approach to front propagation introduced by Barles and Souganidis [BS1998]. This approach is amenable to stochastic models equipped with moment duals which satisfy certain general and verifiable properties. Our main results improve the existing results in several ways, first by removing regularity conditions on the initial data, and second by establishing convergence beyond the formation of singularities of MCF. In particular, we obtain a general convergence theorem which holds globally in time. This is then applied to all of the models considered in [EFP2017] and [HD2021].

math.PR↗

arXiv2022

The compact support property for solutions to stochastic heat equations with stable noise

We consider weak non-negative solutions to the stochastic partial differential equation \[ \partial_t Y(t,x) = \Delta Y(t,x) + Y(t,x)^\gamma \dot{L}(t,x), \] for $(t,x) \in \mathbb{R}_+ \times \mathbb{R}^d$, where $\gamma > 0$ and $\dot{L}$ is a one-sided stable noise of index $\alpha \in (1,2)$. We prove that solutions with compactly supported initial data have compact support for all times if $\gamma \in (2-\alpha, 1)$ for $d=1$, and if $\gamma \in [1/\alpha,1)$ in dimensions $d \in [2,2/(\alpha-1)) \cap \mathbb{N}$. This complements known results on solutions to the equation with Gaussian noise. We also establish a stochastic integral formula for the density of a solution and associated moment bounds which hold in all dimensions for which solutions are defined.

math.PR↗

arXiv2022

Instantaneous support propagation for $Λ$-Fleming-Viot processes

For a probability-measure-valued neutral Fleming-Viot process $Z_t$ with Lévy mutation and resampling mechanism associated to a general $Λ$-coalescent with multiple collisions, we prove the instantaneous propagation of supports. That is, at any fixed time $t>0$, with probability one the closed support $S(Z_t)$ of the Fleming-Viot process satisfies $S(ν* Z_t) \subseteq S(Z_t)$, where $ν$ is the Lévy measure of the mutation process. To show this result, we apply Donnelly-Kurtz's lookdown particle representation for Fleming-Viot process.

math.PR↗

arXiv2020

The density of the $(α,β)$-superprocess and singular solutions to a fractional non-linear PDE

We consider the density $X_t(x)$ of the critical $(α,β)$-superprocess in $R^d$ with $α\in (0,2)$ and $β<\frac αd$. A recent result from PDE implies a dichotomy for the density: for fixed $x$, $X_t(x)>0$ a.s. on $\{X_t\neq 0\}$ if and only if $β\leq β^*(α) = \fracα{d+α}$. We strengthen this and show that in the continuous density regime, $β< β^*(α)$ implies that the density function is strictly positive a.s. on $\{X_t\neq 0\}$. We then give close to sharp conditions on a measure $μ$ such that $μ(X_t):=\int X_t(x)μ(dx)>0$ a.s. on $\{X_t\neq 0 \}$. Our characterization is based on the size of $supp(μ)$, in the sense of Hausdorff measure and dimension. For $s \in [0,d]$, if $β\leq β^*(α,s)=\fracα{d-s+α}$ and $supp(μ)$ has positive $x^s$-Hausdorff measure, then $μ(X_t)>0$ a.s. on $\{X_t\neq 0\}$; and when $β> β^*(α,s)$, if $μ$ satisfies a uniform lower density condition which implies $dim(supp(μ)) < s$, then $P(μ(X_t)=0|X_t\neq 0)>0$. We also give new result for the fractional PDE $\partial_t u(t,x) = -(-Δ)^{α/2}u(t,x)-u(t,x)^{1+β}$ with domain $(t,x)\in (0,\infty)\times R^d$. The initial trace of a solution $u_t(x)$ is a pair $(S,ν)$, where the singular set $S$ is a closed set around which local integrals of $u_t(x)$ diverge as $t \to 0$, and $ν$ is a Radon measure which gives the limiting behaviour of $u_t(x)$ on $S^c$ as $t\to 0$. We characterize the existence of solutions with initial trace $(S,0)$ in terms of a parameter called the saturation dimension, $d_{sat}=d+α(1-β^{-1})$. For $S\neq R^d$ with $dim(S)> d_{sat}$ (and in some cases $dim(S)=d_{sat}$) we prove that no such solution exists. When $dim(S)<d_{sat}$ and $S$ is the compact support of a measure satisfying a uniform lower density condition, we prove that a solution exists.

math.PR↗

arXiv2019

Polynomial spline spaces of non-uniform bi-degree on T-meshes: Combinatorial bounds on the dimension

Polynomial splines are ubiquitous in the fields of computer aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation requires the construction of a suitable spanning set of linearly independent splines, and a theoretical understanding of the spline space dimension can be a useful tool when assessing possible approaches for building such splines. Here, we provide such a tool. Focusing on T-meshes, we study the dimension of the space of bivariate polynomial splines, and we discuss the general setting where local mesh adaptivity is combined with local polynomial degree adaptivity. The latter allows for the flexibility of choosing non-uniform bi-degrees for the splines, i.e., different bi-degrees on different faces of the T-mesh. In particular, approaching the problem using tools from homo-logical algebra, we generalize the framework and the discourse presented by Mourrain (2014) for uniform bi-degree splines. We derive combinatorial lower and upper bounds on the spline space dimension and subsequently outline sufficient conditions for the bounds to coincide.

math.AG↗

arXiv2018

A Boundary Local Time For One-Dimensional Super-Brownian Motion And Applications

For a one-dimensional super-Brownian motion with density $X(t,x)$, we construct a random measure $L_t$ called the boundary local time which is supported on $\partial \{x:X(t,x) = 0\} =: BZ_t$, thus confirming a conjecture of Mueller, Mytnik and Perkins (2017). $L_t$ is analogous to the local time at $0$ of solutions to an SDE. We establish first and second moment formulas for $L_t$, some basic properties, and a representation in terms of a cluster decomposition. Via the moment measures and the energy method we give a more direct proof that $\text{dim}(BZ_t) = 2-2λ_0> 0$ with positive probability, a recent result of Mueller, Mytnik and Perkins (2017), where $-λ_0$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck operator that characterizes the left tail of $X(t,x)$. In a companion work, the author and Perkins use the boundary local time and some of its properties proved here to show that $\text{dim}(BZ_t) = 2-2λ_0$ a.s. on $\{X_t(\mathbb{R}) > 0 \}$.

math.PR↗

arXiv2018

On the boundary of the zero set of super-Brownian motion and its local time

If $X(t,x)$ is the density of one-dimensional super-Brownian motion, we prove that $\text{dim}(\partial\{x:X(t,x)>0\})=2-2λ_0\in(0,1)$ a.s. on $\{X_t\neq 0\}$, where $-λ_0\in(-1,-1/2)$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck process. This confirms a conjecture of Mueller, Mytnik and Perkins who proved the above with positive probability. To establish this result we derive some new basic properties of a recently introduced boundary local time and analyze the behaviour of $X(t,\cdot)$ near the upper edge of its support. Numerical estimates of $λ_0$ suggest that the above Hausdorff dimension is approximately $.224$.

math.PR↗

arXiv2014

The Herschel Virgo Cluster Survey. XVII. SPIRE point-source catalogs and number counts

We present three independent catalogs of point-sources extracted from SPIRE images at 250, 350, and 500 micron as a part of the Herschel Virgo Cluster Survey (HeViCS). The source positions are determined by estimating the likelihood to be a real source for each peak on the maps and the flux densities are estimated using the sourceExtractorTimeline, a timeline-based point source fitter. Afterwards, each source is subtracted from the maps, removing a Gaussian function in every position with the full width half maximum equal to that estimated in sourceExtractorTimeline. This procedure improves the robustness of our algorithm in terms of source identification. The HeViCS catalogs contain about 52000, 42200, and 18700 sources selected at 250, 350, and 500 micron above 3sigma and are ~ 75%, 62%, and 50% complete at flux densities of 20 mJy at 250, 350, 500 micron, respectively. We then measured source number counts at 250, 350, and 500 micron and we also cross-correlated the catalogs with the Sloan Digital Sky Survey to investigate the redshift distribution of the nearby sources. From this cross-correlation, we select ~2000 sources with reliable fluxes and a high signal-to-noise ratio, finding an average redshift z~0.3+/-0.22. The number counts at 250, 350, and 500 micron show an increase in the slope below 200 mJy, indicating a strong evolution in number of density for galaxies at these fluxes. In general, models tend to overpredict the counts at brighter flux densities, underlying the importance of studying the Rayleigh-Jeans part of the spectral energy distribution to refine the theoretical recipes of the models. Our iterative method for source identification allowed the detection of a family of 500 micron sources that are not foreground objects belonging to Virgo and not found in other catalogs.

astro-ph.GA↗

arXiv2011

The Herschel Virgo Cluster Survey. IX. Dust-to-gas mass ratio and metallicity gradients in four Virgo spiral galaxies

Using Herschel data from the Open Time Key Project the Herschel Virgo Cluster Survey (HeViCS), we investigated the relationship between the metallicity gradients expressed by metal abundances in the gas phase as traced by the chemical composition of HII regions, and in the solid phase, as traced by the dust-to-gas mass ratio. We derived the radial gradient of the dust-to-gas mass ratio for all galaxies observed by HeViCS whose metallicity gradients are available in the literature. They are all late type Sbc galaxies, namely NGC4254, NGC4303, NGC4321, and NGC4501. We examined different dependencies on metallicity of the CO-to-H$_2$ conversion factor (\xco), used to transform the $^{12}$CO observations into the amount of molecular hydrogen. We found that in these galaxies the dust-to-gas mass ratio radial profile is extremely sensitive to choice of the \xco\ value, since the molecular gas is the dominant component in the inner parts. We found that for three galaxies of our sample, namely NGC4254, NGC4321, and NGC4501, the slopes of the oxygen and of the dust-to-gas radial gradients agree up to $\sim$0.6-0.7R$_{25}$ using \xco\ values in the range 1/3-1/2 Galactic \xco. For NGC4303 a lower value of \xco$\sim0.1\times$ 10$^{20}$ is necessary. We suggest that such low \xco\ values might be due to a metallicity dependence of \xco (from close to linear for NGC4254, NGC4321, and NGC4501 to superlinear for NGC4303), especially in the radial regions R$_G<$0.6-0.7R$_{25}$ where the molecular gas dominates. On the other hand, the outer regions, where the atomic gas component is dominant, are less affected by the choice of \xco, and thus we cannot put constraints on its value.

astro-ph.CO↗