arXiv subjects

Aparna Rajput

Publications and source records attributed to Aparna Rajput.

5 recordsLinked to original sources

arXiv2026

Statistical Stability and Linear Response for Suspension Measures under Base and Roof Perturbations

We study the dependence of canonical invariant probability measures of suspension semiflows on simultaneous perturbations of the invariant base density and the roof function. We compare two natural representations of the suspension measure: physical-height coordinates, in which the support depends on the roof function, and normalised-height coordinates, in which all measures are defined on a fixed cylinder. In physical-height coordinates, we obtain an exact fibrewise formula for the $L^1$ discrepancy between suspension densities and derive quantitative total-variation estimates. In normalised-height coordinates, we establish corresponding stability estimates through the product of the base density and the roof function. These results transfer quantitative statistical-stability estimates for the base densities to the associated suspension measures. For one-parameter families whose base densities are differentiable in $L^1$ and whose roof functions are differentiable in $L^\infty$, we derive an $L^1$ linear-response formula on the fixed cylinder and a bounded-Lipschitz first-variation formula in physical height. The latter contains a signed measure supported on the graph of the unperturbed roof. We further compute the exact first-order total-variation speed and prove that total-variation differentiability holds if and only if the first-order roof velocity vanishes almost everywhere with respect to the unperturbed fibre density. The results are measure-theoretic and do not require spectral or mixing assumptions on the suspension semiflow.

math.DS↗

arXiv2026

Empirical Transfer Operators and Finite-Sample Change Detection for Noisy Expanding Interval Maps

We study finite-sample change detection for one-dimensional noisy dynamical systems using partition-based empirical approximations of stationary behaviour. Given observations from an interval-valued process, we partition the state space, estimate a finite transition matrix from observed transitions between partition elements, and apply a small Doeblin-type regularisation to ensure a unique stationary distribution. From an initial reference segment, we compute a baseline empirical stationary distribution $\widehatπ_{0,ρ}$. For each later sliding window, we compute $\widehatπ_{t,ρ}$ and define the score \[ S_t=\|\widehatπ_{t,ρ}-\widehatπ_{0,ρ}\|_1. \] Large values of $S_t$ indicate a change in stationary behaviour relative to the baseline. The statistic detects changes in invariant density or stationary law, but not all possible changes in transition dynamics. Under explicit assumptions on empirical transition concentration, finite-state stationary distribution stability, partition approximation, regularisation bias, and noise stability, we derive a finite-sample bound for the empirical stationary density. The bound separates sampling error, regularisation bias, partition approximation error, and noise bias. We then obtain a single-window false-alarm guarantee and a sufficient detection condition when the invariant density changes by more than the estimation error. We illustrate the method on synthetic noisy beta-map change-point experiments.

stat.ML↗

arXiv2026

Stochastic Stability of ACIMs for Piecewise Expanding $C^{1+\varepsilon}$ Maps

We prove stochastic stability of absolutely continuous invariant measures (ACIMs) for piecewise expanding $C^{1+\varepsilon}$ maps of the interval. For maps $τ$ in the class $\mathcal{T}([0,1]; s, \varepsilon)$, we consider perturbed Frobenius--Perron operators $P_δ= Q_δP_τ$, where $Q_δ$ is a Markov smoothing operator modeling noise of intensity $δ> 0$. In the generalized bounded variation space $BV_{1,1/p}$, we establish a Lasota--Yorke inequality uniform in $δ$. Consequently, each $P_δ$ admits an invariant density $h_δ\in BV_{1,1/p}$, and $h_δ\to h$ in $L^1$ as $δ\to 0$, where $h$ is the ACIM density of $P_τ$. Our proof combines the $BV_{1,1/p}$ framework, adapted from recent ACIM existence results, with uniform quasi-compactness and perturbation theory for transfer operators. This establishes stochastic stability under minimal $C^{1+\varepsilon}$ regularity ($\varepsilon > 0$), where the $C^1$ case is known to fail.

math.DS↗

arXiv2025

Existence of ACIM for Piecewise Expanding $C^{1+\varepsilon}$ maps

In this paper, we establish Lasota-Yorke inequality for the Frobenius-Perron Operator of a piecewise expanding $C^{1+\varepsilon}$ map of an interval. By adapting this inequality to satisfy the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem \cite{ionescu1950}, we demonstrate the existence of an absolutely continuous invariant measure (ACIM) for the map. Furthermore, we prove the quasi-compactness of the Frobenius-Perron operator induced by the map. Additionally, we explore significant properties of the system, including weak mixing and exponential decay of correlations.

math.DS↗

arXiv2025

Quasi-compactness of Frobenius-Perron Operator for Piecewise Convex Maps with Countable Branches

In this paper, we prove the quasi-compactness of the Frobenius-Perron operator for a piecewise convex map $τ$ with a countably infinite number of branches on the interval $I=[0,1]$. We establish that for high enough $n$ iterates of $τ$, $τ^n$ are piecewise expanding. Using the Lasota-Yorke Inequality derived from references \cite{hofbauer1982} and \cite{keller1985}, adapted to meet the assumptions of the Ionescu-Tulcea and Marinescu ergodic theorem, we demonstrate the existence of absolutely continuous invariant measure (ACIM) $μ$ for $τ$, the exactness of the dynamical system $(I, τ,μ)$ and the quasi-compactness of Frobenius-Perron operator $P_τ$ induced by $τ$. The last fact implies a multitude of strong ergodic properties of $τ$.

math.DS↗