arXiv subjects

Guangqian Zhao

Publications and source records attributed to Guangqian Zhao.

10 recordsLinked to original sources

arXiv2026

Gaussian Operator Algebras: Observability, Reconstruction, and Microscopic Limits

We reconstruct Gaussian operator algebras from finite source-labelled observations, recovering coefficients, ordinary multiplication, and completion in independently specified Schatten norms. Gaussian duality and ordered contractions give a reconstruction theorem including the trace-class endpoint, while entire Mehler responses describe the resulting topology. In the Hilbert coefficient geometry, symmetric Rademacher systems realize the same algebra as a fluctuation limit: microscopic products converge at inverse site-number rate in the original norm scale, and finite decoding commutes with this limit. Relative Gram residuals determine target-dependent inverse costs. For bounded tuples, positive word observations close into completely positive maps, with contextual Schwarz defects controlling realization and multiplication. Positive mass and source innovations characterize compactness. Spectral invariants, Wick operators on closed manifolds, and Gaussian networks provide applications with explicit noise and tail bounds.

math.PR↗

arXiv2026

Infinite-Dimensional Levy Area and Spectral Ancestry

We relate primitive ancestry in free-Lie expansions of infinite-dimensional Lévy area to critical weak-Schatten spectra. For $L=\mathfrak L(V)$, $D=[L,L]$, and $J=\ker(T(V)\to S(V))$, we prove $L\cap J^r=γ_r(D)$ and identify the exact normal layers with $\mathfrak L_r(D/[D,D])$. Continuous Hilbert-martingale area has the sharp split endpoint $H\times\mathcal S^0_{1,\infty}(H)_{\rm skew}\times\mathcal S_1(H)_{\rm sa}$. The critical grade $r$ has singular profile $(\log(e+N))^{r-1}/N$. A Rees response kernel characterizes strictness, equality of ancestry and spectral depth, and recovery of the finite labelled ancestry flag. Pure-area Hall experiments give full-support kernels and exact critical laws. Revealed brackets and evolution families provide conditional covariance and second-order Lévy/SPDE interfaces.

math.PR↗

arXiv2026

Operational State Selection in Stochastic Analysis: The Least State for a Calibrated Quadratic Ito Program

We study the least stable state determined by a causal response program and its quantitative calibration. For nondominated families of continuous Hilbert-valued semimartingale laws with uniform drift and trace-clock bounds, the calibrated quadratic Itô program selects a state $\mathcal A_2$ with canonical coordinates $(X,A,Q)$, where $A$ is Hilbert--Schmidt skew area and $Q$ is trace-class covariance. Brownian experiments identify the Hilbert--Schmidt/operator coefficient gauges and hence the dual $\mathcal S_2/\mathcal S_1$ state geometry. A lossless encoder--decoder pair shows that every joint realization factors through $\mathcal A_2$. A single sequence of total Borel causal approximants constructs the common state, while the laws verify its classical semantics. A dimension-free defect--energy estimate gives independence from the finite-variation regularization, and spatial trace tightness yields restart-stable compact capacity cores in infinite dimensions. Each fixed higher signature level is a continuous readout of $\mathcal A_2$. Rough flows inherit corewise continuity, while Euler schemes construct a jointly Borel raw-causal Itô field outside one parameter-independent polar set. On compact covariance-envelope subclasses, covariance responses select a backward first jet which, under the stated completion and continuation hypotheses, has modelwise Galtchouk--Kunita--Watanabe semantics and agrees with the solver's martingale coordinate.

math.PR↗

arXiv2026

Color--Phase Separation for Mixed Random Operators in Two-Speed Stochastic Klein--Gordon Systems

We study a two-component stochastic fractional Klein--Gordon system on the three-dimensional torus with fractional dispersion exponent $α$, distinct propagation speeds, and a pure cross nonlinearity. The singular mixed terms combine a low--high paraproduct inside a Duhamel integral with an outer resonant product. Their structure is governed by two index pairs: the color pair determines the possible Gaussian contraction, whereas the propagator--input pair determines the Duhamel phase. In every same-color block generated by the interaction graph, the outer propagator has a different speed. The contracted term is therefore a Fourier-diagonal Volterra multiplier, and integration by parts in time yields a factor of order $N^{-α}$ at frequency scale $N$. For the centered remainder we give a finite Hilbert-space kernel normal form that preserves the Fourier incidence relation and reduces the operator estimate to oriented Gaussian-tensor flattenings. For independent noises we construct the enhanced stochastic data, prove local well-posedness in the prescribed enhanced paracontrolled/Galerkin class, and establish pathwise convergence of the Fourier--Galerkin equations for $12/13<α\le1$. We also prove a logically separate perturbative extension to Fourier-diagonal color covariances with frequency-decaying off-diagonal entries.

math.PR↗

arXiv2026

Random Tensor Estimates and Deterministic Diagonal Resolutions for Mixed Paracontrolled Operators

We prove an operator-level convergence theorem for the mixed paracontrolled blocks \[ T^{i;j,k}_Λ(w) =I_i\bigl(w<Ψ_{j,Λ}\bigr)\circΨ_{k,Λ} \] on $\mathbb T^d$, under Fourier-diagonal Gaussian covariance. At finite Galerkin cutoff, the Wick decomposition is \[ T^τ_Λ=D^τ_Λ+B^τ_Λ, \qquad τ=(i;j,k), \] where $D^τ_Λ$ is a deterministic Volterra multiplier on the external diagonal $n=q$, while $B^τ_Λ$ is a centered second homogeneous Gaussian chaos with incidence $n=q+\ell+r$. The centered coefficient has two Gaussian frequency legs and one input-output pair. Write $Δ_T=\{(t,s):0\le s\le t\le T\}$. An iterated rectangular non-commutative Khintchine argument, followed by the four associated oriented flattenings, yields \[ \bigl\|B^τ_{Λ,N,Q,M}\bigr\|_ {L^p\bigl(Ω;C(Δ_T;\mathcal L(\ell_q^2,\ell_n^2))\bigr)} \lesssim_{p,\varepsilon} N^{d/2-Γ_τ+\varepsilon} \bigl(M^{d/2}+Q^{d/2}\bigr), \qquad Γ_τ=λ_i+α_j+α_k. \] Under the resulting strict Sobolev--Besov summability conditions, the centered cutoffs converge in $L^p(Ω)$ and almost surely in operator norm. The covariance contraction is kept separate from the tensor estimate: retained raw branches, or prescribed finite combinations of branches, enter through a scalar Volterra criterion imposed after the relevant cancellation has been formed. We give a frequency-envelope variant and verify the diagonal criterion for distinct-speed wave and Klein--Gordon contractions.

math.PR↗

arXiv2026

Common Causal Itô Flows under Nondominated Martingale Laws: Capacity Cores and Robust Sensitivities

Let $Ω=C_0([0,T];\mathbb R)$, and let $\mathfrak M_Λ$ be the laws under which the coordinate process is a continuous square-integrable martingale satisfying $\mathrm d\langle X\rangle_t\leq Λ\,\mathrm d t$. Regularized dyadic square sums define a single Borel causal map $Q:Ω\to\mathcal A_Λ$ which equals the quadratic variation almost surely under every law in $\mathfrak M_Λ$. The approximants converge uniformly in $L^r$ at rate $2^{-n/2}$ and uniformly on common compact sets whose complementary upper capacities tend to zero. For uniformly elliptic scalar coefficients, the Lamperti transform applied to $Q$ produces a total Borel causal $C^1$ flow with the cocycle property. Every fixed section coincides almost surely with the classical strong Itô solution under every model. The flow and its initial-state Jacobian are continuous on the same capacity cores; for differentiable finite-dimensional coefficient families, so is the parameter tangent. A capacity-core $C^1$ transfer theorem yields class-uniform $L^p$ Fréchet expansions, $W_p$-continuity of the first-jet laws, and attainment for continuous field payoffs of polynomial growth. For robust terminal payoffs, the transferred first jet gives a joint Hadamard--Danskin formula in the coefficient parameter and initial state. An explicit constant-volatility family exhibits an active-model switch and a nondifferentiable robust value on an explicit switching surface. The bracket is also invariant under continuous finite-variation translations. The induced response kernel, multiplied by the deterministic volatility, agrees with the Malliavin derivative under Gaussian volatility models.

math.PR↗

arXiv2026

Quantitative Driver-Only Capacity Cores and Simultaneous Scalar Itô Flows

We construct quantitative compact raw-driver cores for the maximal nondominated class $M_Λ$ of laws under which the coordinate process is a scalar continuous local martingale satisfying $d[X]_t\leqΛ\,dt$. Fix $0<η<1/2$ and $0<γ<\min\{1/2,2η\}$. Explicit compact sets $K_{A,B}$ have sub-Gaussian tails in $A$ and subexponential tails in $B$, while continuous dyadic realized variation converges uniformly on each core at rate $O(BΛT2^{-γn})$. The intrinsic dyadic quadratic variation is Holder continuous in the raw uniform topology on the cores. The parabolic family $K_R=K_{R,R^2}$ has complement of capacity at most $Ce^{-cR^2}$ and is stable under stopping and controlled deterministic concatenation. For every compact family of sufficiently smooth autonomous coefficients with uniformly positive diffusion, a deterministic Lamperti--Follmer equation constructs Borel causal solution and integral fields jointly in the coefficient, starting time, and initial value. The fields satisfy the solution and integral cocycles. The solution field forms an orientation-preserving $C^1$ flow, and both fields have quantitative raw-driver Holder moduli on each $K_R$. Under every law in $M_Λ$, each fixed section agrees with the corresponding classical Ito solution and stochastic integral. We also place the Bichteler--Karandikar construction on a fixed Borel pair-domain, with projective causality and full-sequence semimartingale identification.

math.PR↗

arXiv2026

Algebraic Transfer for Operator-Valued Gaussian Chaoses:Oriented Schatten Profiles and Singular Wick Multipliers

We develop an algebraic transfer calculus for the oriented Schatten profiles of kernels underlying operator-valued Gaussian chaoses. A dimension-free link inequality propagates profile bounds through cut factorizations, tensor products, coefficient maps, and ordered contractions. Besides the constant-one strong Schatten theorem, we prove the sharp weak endpoint \[ \mathfrak{S}_{r,\infty}\times\mathfrak{S}_{r,\infty} \longrightarrow \mathfrak{S}_{r,\infty}(\log\mathfrak{S})^{-1/r}, \qquad 1<r<\infty. \] The estimate holds uniformly over all contracted Hilbert spaces, and the exponent $1/r$ cannot be decreased within the displayed $q=\infty$ Lorentz--Zygmund scale. A separate finite-complexity argument gives sharp effective-rank and finite-cut-rank logarithmic bridges from all-cut operator profiles to Gaussian operator norms. Combined with oriented-flattening Gaussian estimates, the calculus yields continuous multiplication on completed Wick chaoses with noncommuting coefficients, an associative algebra of factorially weighted analytic Wick series, and a local-to-global theorem for loop-free Peter--Weyl fusion trees. We then apply the method to singular Wick multipliers on groups of polynomial growth. For second-order multipliers we obtain sharp necessary and sufficient Schatten convergence thresholds; on $\mathbb{Z}^{D}$ we determine the full singular phase diagram at every order. Fourier transfer gives exact Sobolev, Schatten-class, compactness, and trace-class thresholds for sandwiched Wick multiplication operators on $\mathbb{T}^{D}$, together with sharp Fourier--Galerkin rates and approximation-number decay.

math.PR↗

arXiv2026

A Singular Stochastic Wave--Klein--Gordon System with Rough Gaussian Forcing and Distinct Propagation Speeds

We construct spatially local paracontrolled solutions on $\mathbb{R}^3$ for the stochastic wave--Klein--Gordon system \[ (\partial_t^2-c_{\mathrm{W}}^2Δ)u=uv+ξ_{\mathrm{W}},\qquad (\partial_t^2-c_{\mathrm{K}}^2Δ+m^2)v=uv+ξ_{\mathrm{K}}, \] with $c_{\mathrm{W}}\ne c_{\mathrm{K}}$. The independent forcings are white in time and stationary in space, generated by Fourier multipliers of respective orders $β_{\mathrm{W}},β_{\mathrm{K}}$, and we assume $β_*:=\max\{β_{\mathrm{W}},β_{\mathrm{K}}\}<1/8$. The main analytic step is the construction of the mixed random operators $I_a(w\precΨ_b)\circΨ_c$. At finite cutoff they decompose into a centered second Gaussian chaos and, when $b=c$, a deterministic covariance contraction. In the latter case the two propagation channels are different; the resulting low--high phase gap gives the diagonal shell bound $N^{-1+2β_b}$. The centered terms are controlled through four Schatten flattenings of a continuous-frequency kernel with two physical localizers. Weighted phase-layer estimates also construct the first Picard and cubic stochastic terms. A localized auxiliary system, a finite-cutoff Sobolev bootstrap, and finite propagation yield a compatible family of local solutions with compact-dependent measurable lifetimes. We prove uniqueness in the stated paracontrolled class, local Lipschitz dependence on the enhanced data and Cauchy data, independence of the localization pair, and convergence of the spectral approximations and nonlinear sources. Fixed cutoff profiles converge almost surely along the full sequence; arbitrary admissible cofinal cutoffs converge in probability up to the reference lifetime.

math.PR↗

arXiv2026

Localized Centered Second-Chaos Operator

We prove a localized continuous-frequency operator estimate for centered Gaussian chaoses of order two. The result applies to operator-valued centered second chaoses, including Wick-centered same-family variants, between Hilbert spaces. In the model, two Gaussian frequency legs at scale $N$, an input leg at scale $Q$, and an output leg at scale $M$ are coupled through a soft incidence kernel; non-orthogonal Gaussian profiles are represented by covariance synthesis maps. The proof combines four oriented flattenings, rectangular non-commutative Khintchine inequalities, soft-incidence Schatten bounds, and Sobolev--Besov dyadic summation. The time lift gives $L^p$ operator convergence, while a Galerkin stabilization hypothesis gives pathwise full-cutoff convergence by the first Borel--Cantelli lemma. Under $\mathcal G(N)\lesssim N^{-Γ}$ one obtains the window \[ Γ>\frac d2, \qquad s<λ+Γ-d, \qquad \max\{0,d-Γ\}<σ<λ+Γ-d. \] The theorem applies to the near-output Wick-centered branch of localized paracontrolled resonant products on $\mathbb R^d$.

math.PR↗