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Chunhao Cai

Publications and source records attributed to Chunhao Cai.

At least 19 recordsLinked to original sources
arXiv2026

Weak uniqueness and autonomous approximation for the SLE-reversible Weil-Petersson response diffusion

We study the self-consistent common-noise equation associated with the Weil--Petersson response differential of the rotationally marked SLE welding probability. For $0<κ\leq4$, we prove joint weak uniqueness among continuous adapted weak solutions whose law at each deterministic time is the welding probability, and obtain the Markov property in the original filtration. A conditional Gaussian lift and capacity localization give the required $L^1$ generator-range density. Identification with the canonical law gives stationarity and reversibility. We also construct autonomous Fejér and Poisson regularizations with pathwise-unique global strong solutions. Their joint state-noise laws converge to the canonical law for every initial probability absolutely continuous with respect to the welding probability.

math.PR↗
arXiv2026

Spectral Response Geometry for Liouville Brownian Motion

For Dirichlet Liouville Brownian motion associated with a subcritical Gaussian multiplicative chaos measure on a bounded planar domain, we study the response of finite spectral blocks under Cameron--Martin perturbations of the underlying Gaussian free field. For each fixed $0<γ<2$, we prove that, almost surely, all unordered eigenfunction-product measures in every finite simple spectral block are linearly independent. This strengthens the corresponding diagonal independence statement and implies surjectivity of the joint eigenvalue and eigenframe response map. We use this transversality to construct real-analytic local deformations that preserve a prescribed finite block of eigenvalues while varying its eigenframe. We then identify the least Cameron--Martin energy required to realize a given infinitesimal angular response, expressing it through a positive Schur complement of the response operator. Finally, on finite-dimensional Gaussian slices, we derive the conditional law associated with the local isospectral leaf and compare the induced slice metric with the full Cameron--Martin minimum. The results provide a local geometric description of spectral flexibility for Liouville Brownian motion.

math.PR↗
arXiv2026

Optimal information loss for Liouville spectral responses

We study Fisher information about the amplitude of a known smooth mean shift from the first Dirichlet eigenvalue and spatial responses of a Liouville random medium. The complete response record determines the speed measure and the Gaussian free field, whereas every finite record strictly loses information; an optimal design exists in each finite-dimensional probe space. We quantify this gap as the known Liouville parameter $γ$ tends to zero. If the probe space contains an explicit matching probe, the minimum loss equals $γ^2$ times a positive compression error for a deterministic second-chaos operator, up to $o(γ^2)$, with designs allowed to depend on $γ$. After enlarging the probe dictionary, the optimal $m$-response coefficient is asymptotic to $\|Lg\|_{L^1(D)}^2/(8m)$, where $L=-(2π)^{-1}Δ_D$ on the unit disk $D$. A quadratic correction of the readouts attains the same fixed-design loss to second order.

math.PR↗
arXiv2026

Single-graph inference for fractal Gaussian networks

We study inference on the strength of Gaussian multiplicative chaos from one geometric graph with unobserved vertex positions. In the planar model, the edge-count statistic has a piecewise deterministic limit with a transition at $γ=1$, whereas degree quantiles consistently estimate $ν=γ^2/2$ throughout the subcritical range. Fractional moments give uniform finite-sample risk bounds. At finite resolution, common graph envelopes calibrate tests over continuous parameter boxes with unknown Poisson intensity. They give simultaneous coverage under adaptive refinement for a fixed statistic, and conditional coverage for an independently piloted size--degree residual on a fixed partition. For the exact periodic FFT--pixel approximation, we prove explicit total-variation rates for the annealed spatial Cox law and the observed graph law, uniformly on every compact subcritical range $0\leγ\leΓ<2$. The rates allow intensity to grow with resolution and imply uniform asymptotic coverage after recalibration at each resolution. Numerical studies examine parameter refinement, residual calibration, and multiresolution stability.

math.ST↗
arXiv2026

The canonical SLE-reversible Weil--Petersson response diffusion

The rough response calculus for the rotationally marked SLE welding law gives a closed Weil--Petersson energy but does not by itself give a process on rough weldings. For $0<κ\leq4$, we prove that this form is quasi-regular in the natural topology of circle homeomorphisms and therefore generates a canonical diffusion reversible with respect to the SLE welding probability. The proof constructs a compact form nest from inverse arc masses. We identify the generator as $-D^*D$ and prove absolute $L^2$ convergence of its Fourier expansion on the response cylinder core, together with the Fukushima decomposition. A common Fourier noise and planar estimates preventing boundary collapse yield a second quasi-regular realization by normalized Jordan welding triples. The diffusion is irreducible and converges strongly in $L^2$ to equilibrium. For $0<κ<4$, the complement of the conformally removable welding locus has zero capacity.

math.PR↗
arXiv2026

Closed Response Calculus for SLE Weldings and Weil--Petersson Kähler Geometry

For $0<κ\leq4$, we construct a closed response calculus for $\mathrm{SLE}_κ$ weldings and a canonical Dirichlet form. The divergence covariance combines the Weil--Petersson and Velling--Kirillov forms. The integrated response gives exact changes of measure and canonical Liouville--capacity increments on conformally removable weldings.

math.PR↗
arXiv2026

Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric

We give a general criterion for transferring first-order responses on an underlying probability space to a closed differential calculus on the law of an observable. An integration-by-parts identity removes presentation ambiguity and yields a closable gradient, its divergence, and a closed Markov form. We apply this scheme to the subcritical Liouville quantum gravity metric throughout the range $0<γ<2$. Sequential compactness of Weyl-perturbed geodesics identifies the derivative of a logarithmic distance ratio with the Sobolev Riesz representative of the difference of two normalized geodesic occupation measures. The induced form on the projective metric law is the image of a Gaussian directional form and has the energy-image-density property. Finally, fixed-target confluence and a leaf-elimination argument make the response Gram matrix positive definite for every finite forest of marked pairs. The corresponding vector of logarithmic distance ratios therefore has a Lebesgue density.

math.PR↗
arXiv2026

Spectral Simplicity and Joint Eigenvalue Densities for a Non-Gaussian Brownian Time Change

We study a Brownian time change on the unit square whose speed measure is constructed from a Dirichlet eigenfunction expansion with independent non-Gaussian coefficients. For $0<γ<\sqrt2$, the measure is obtained by a second-moment martingale argument. Finite coefficient translations induce coherent exponential tilts of the speed measure, and conditioning on the complementary coefficients gives positive Lebesgue densities on every finite-dimensional orbit. Unitary transport along these orbits gives a common-domain analytic family and explicit first-order cluster derivatives. A first-order splitting argument proves almost-sure simplicity, while the local eigenfunction-square identity and a Vandermonde argument give joint densities for all finite vectors of ordered positive eigenvalues.

math.PR↗
arXiv2026

Response Calculus for Spectral Simplicity and Joint Eigenvalue Densities

We develop a perturbative response calculus for spectral problems obtained by changing the speed measure of a fixed symmetric energy form in a Gaussian environment. If Cameron--Martin translation acts by $μ_{x+f}=e^{κf}μ_x$, unitary transport identifies the varying $L^2$ spaces and produces a common-domain analytic family. For a positive eigenvalue $Λ$, the first-order operator is $-κΛ$ times the compression of multiplication by $f$ to the $Λ$-eigenspace; its eigenvalues are the derivatives of the analytic branches issuing from $Λ$. A countable separation condition and finite-dimensional Gaussian disintegration then give almost-sure simplicity; a response-transversality condition and the inverse function theorem give joint densities for all finite vectors of positive ordered eigenvalues. We verify these hypotheses for every $0<γ<2$ in two models: Dirichlet Liouville Brownian motion on an arbitrary bounded connected planar domain, and the Liouville--Cauchy operator on the circle. In the first model the whole spectrum is almost surely simple; in the second the constants form the deterministic zero mode and the positive spectrum is almost surely simple. Transversality follows from a local eigenfunction-square identity in the Brownian case and its nonlocal jump-form analogue in the Cauchy case.

math.PR↗
arXiv2026

Local Asymptotic Normality for Mixed Fractional Brownian Motion with $0<H<3/4$

This paper establishes the Local Asymptotic Normality (LAN) property for the mixed fractional Brownian motion under high-frequency observations with Hurst index $H \in (0, 3/4)$. The simultaneous estimation of the volatility and the Hurst index encounters a degeneracy problem in the Fisher information matrix.

math.ST↗
arXiv2026

A GPH-Filtered Hannan--Rissanen Information Criterion for ARFIMA Order Selection

We propose a simple two-stage order selector for finite-order ARFIMA models. First, a preliminary log-periodogram estimate of the memory parameter is used to fractionally filter the data. Second, a Hannan--Rissanen residual construction is applied to the filtered series, and the autoregressive and moving-average orders are selected by a generalized information criterion over a growing candidate rectangle. The search bounds are allowed to satisfy \(P_n,Q_n\to\infty\), whereas the true orders remain fixed and finite. The penalty is allowed to be larger than the ordinary BIC penalty so that it dominates the error introduced by preliminary long-memory estimation and by the Hannan--Rissanen residual approximation. We prove a uniform residual-variance approximation over the growing rectangle and combine it with a population separation argument between the true finite ARMA representation and underfitted alternatives. The resulting generalized information-criterion selector is consistent.

math.ST↗
arXiv2026

Second-order PACF asymptotics and discrimination between fractional Gaussian noise and $\FARIMA(0,d,0)$

Fractional Gaussian noise and $\FARIMA(0,d,0)$ have the same long-memory pole $|θ|^{-2d}$ and hence the same leading PACF law $α(n)\sim d/n$. We show that this agreement breaks at the first non-universal order. For $0<d<1/2$, the pure fGn PACF satisfies $$ α_{\fGn}(n)=\frac d n+\frac{C_{\fGn}(d)}{n^2}+o(n^{-2}), \qquad C_{\fGn}(d)<d^2, $$ The proof uses the Bingham--Inoue--Kasahara representation, a phase-coefficient expansion for fGn, and a Hankel-operator perturbation argument. Thus the fGn spectral envelope is invisible at first order but visible in second-order finite prediction, explaining why short-memory order selection can differ when fGn data are fitted by FARIMA-type models.

math.ST↗
arXiv2026

Truncated Signature Information for Mixed Fractional Brownian Paths

We study finite expected-signature information for mixed-fBm paths with Hurst indices above $1/4$. Up to level three, the only parameter-dependent expected features are the variance transform $q_θ$ and the time-ordered transform $R_θ$. We prove the scale tradeoff $2K$ level-two scales versus $K$ selected level-two/three scales, together with separation and local inverse bounds.

math.PR↗
arXiv2026

Boundary Inference for Mixed Fractional Models under High-Frequency Observation Critical LAN and Score Tests at $H=3/4$

We study boundary inference at $H=3/4$ for mixed fractional Brownian motion and mixed fractional Ornstein--Uhlenbeck models under high-frequency observation. This boundary is economically important because it separates the critical and supercritical regimes of mixed fractional dynamics. We make three contributions. First, we identify the exact critical first-order scaling and show that, after removing the explicit linear component in the $H$-score, the transformed $(σ,H)$ block is already non-degenerate. Second, we establish critical score central limit theorems (CLT) and derive local asymptotic normality (LAN) with fully explicit leading information constants for both models. Third, we construct boundary-calibrated one-sided score tests for detecting entry into the supercritical region $H>3/4$ and discuss feasible implementation through restricted nuisance estimation. Monte Carlo evidence shows that the feasible statistic has the correct directional power but conservative null calibration. Finally, an intraday illustration on one-minute SPY data finds no persistent evidence in favor of $H>3/4$.

math.ST↗
arXiv2026

A mixed fractional CIR model: positivity and an implicit Euler scheme

We consider a Cox--Ingersoll--Ross (CIR) type short rate model driven by a mixed fractional Brownian motion. Let $M=B+B^H$ be a one-dimensional mixed fractional Brownian motion with Hurst index $H>1/2$, and let $\mathbf{M}=(M,\mathbb{M}^{\mathrm{It\hat{o}}})$ denote its canonical Itô rough path lift. We study the rough differential equation \begin{equation}\label{eqn1} \dd r_t = k(θ-r_t)\,\dd t + σ\sqrt{r_t}\,\dd\mathbf{M}_t,\qquad r_0>0, \end{equation} and prove that, under the Feller condition $2kθ>σ^2$, the unique rough path solution is almost surely strictly positive for all times. The proof relies on an Itô type formula for rough paths, together with refined pathwise estimates for the mixed fractional Brownian motion, including Lévy's modulus of continuity for the Brownian part and a law of the iterated logarithm for the fractional component. As a consequence, the positivity property of the classical CIR model extends to this non-Markovian rough path setting. We also establish the convergence of an implicit Euler scheme for the associated singular equation obtained by a square-root transformation.

math.PR↗
arXiv2026

Scaling limit of heavy tailed nearly unstable cumulative INAR($\infty$) processes and rough fractional diffusions

In this paper, we investigate the scaling limit of heavy-tailed nearly unstable cumulative INAR($\infty$) processes. These processes exhibit a power-law tail of the form $n^{-(1+α)}$ for $α\in (\frac{1}{2}, 1)$, and the $\ell^1$ norm of the kernel vector converges to 1. We demonstrate that the discrete-time scaling limit retains a long-memory property and can be viewed as an integrated fractional Cox-Ingersoll-Ross process. Moreover, we present an efficient method for simulating the fractional Cox-Ingersoll-Ross process. The simulation and Goodness-of-Fit Test code are available at https://github.com/gagawjbytw/INAR-rough-Heston.

math.PR↗
arXiv2026

Local Asymptotic Normality for Mixed Fractional Brownian Motion Under High-Frequency Observation

In this paper we will consider the LAN property for both the Hurst parameter $H>3/4$ and the variance of the fractional Brownian motion plus an independent standard Brownian motion (called mixed fractional Brownian motion) with high-frequency observation. We will first remove the $H$-score linear term and orthogonalize the remainder through two non-diagonal transformations, then we can construct the CLT for the quadratic form base on $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F\to0$. At last we obtain a diagonal Gaussian LAN expansion with an explicit information matrix. Beyond the case of $H>3/4$, we also present that the $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F\to0$ method is also useful for the case of $H<3/4$ and the proof will be concise compared with the Whittle translation method. We consider that this method can be applied to this type of problem, including the fractional Ornstein-Uhlenbeck model and mixed fractional O-U process.

math.PR↗