Papers

We establish nonasymptotic Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, for deterministic matrix-weighted sums of multivariate Poisson martingale differences generated by uniformly ergodic Markov chains. Under an $L^{(2+\eta)p}$-moment condition, $\eta>0$, we obtain the explicit bound $$O\left(p^3\|A\|_4^2+pd^{1/4}\|A\|_2^{1/2}\|A\|_4^2\right),$$ where $A\in\mathbb{R}^n$ collects the moment sizes of the weighted increments. When these sizes are of order $n^{-1/2}$, the bound achieves the first optimal $O(n^{-1/2})$ rate for fixed $p$ and $d$. Consequently, we also obtain the first optimal $O(n^{-1/2})$ $W_p$ Gaussian approximation rate for multivariate additive functionals of uniformly ergodic Markov chains. Our analysis develops two techniques for addressing the interplay between higher-order Wasserstein distance and temporal dependence. First, building on the Ornstein–Uhlenbeck relative-score approach of Fang and Koike (2023), we formulate the bound in terms of antisymmetric Stein couplings while retaining the conditional tensor structure. Second, we develop a refresh-then-maximal coupling that combines an independent first-step resampling, which preserves the desired Stein identity, with a subsequent maximal coupling that provides effective control of the coupling increment. We apply these bounds to nonlinear Markovian stochastic approximation by decomposing normalized errors into matrix-weighted Poisson martingale sums and remainders. This yields the first sharp $W_p$ Gaussian approximation rates for both last iterates and Polyak–Ruppert averages. In particular, the optimized rates are $O(n^{-1/3})$ for polynomial-stepsize last iterates and $O(n^{-1/4})$ for Polyak–Ruppert averages, with an $O(n^{-1/2})$ rate for harmonic-stepsize last iterates under an additional spectral condition.

We study sample covariance matrices $K = \frac{1}{N} \sum_{i=1}^N \mathbf{x}_i \mathbf{x}_i^* \in \mathbb{R}^{n \times n}$ in the proportional regime $n \asymp N$. The columns $ \mathbf{x}_1, \ldots, \mathbf{x}_N \in \mathbb{R}^n$ are independent and centered, with common covariance $\mathbb{E} \mathbf{x}_i \mathbf{x}_i^* = \Sigma$, but may otherwise have strongly and nonlinearly dependent coordinates. Assuming only that quadratic forms of the columns concentrate uniformly at the optimal rate $| \mathbf{x}_i^* A \mathbf{x}_i - \mathrm{Tr} \Sigma A | \prec \| A \|_F$, together with polynomial norm moments and a standard nondegeneracy condition on $\Sigma$, we prove the optimal anisotropic local law: on regular spectral domains, uniformly down to spectral scales $\eta:= \mathrm{Im}\, z \geq N^{-1 + \tau}$, \[ \big| \langle \mathbf{u} , \big( (K-z)^{-1} - (-zI_n-z\widetilde m_0(z)\Sigma \big)^{-1} \big) \mathbf{v} \> \big| \prec \sqrt{\frac{\mathrm{Im}\, \widetilde m_0 (z)}{N\eta}} + \frac{1}{N\eta} \] for all deterministic unit vectors $ \mathbf{u}, \mathbf{v} \in \mathbb{C}^n$, where $\widetilde m_0(z)$ is the Stieltjes transform of the deformed Marchenko-Pastur law. This removes the higher-cumulant tensor assumption of Fan, Ma, Paquette, and Wang (2026), thereby answering the question raised in their work. The result applies, among other examples, to every centered log-concave column distribution with bounded, nondegenerate covariance, nonlinear tilts of Gaussian vectors, deep random features, and a high-temperature spherical 4-spin model for which the cumulant assumption is known to fail.

We present a method for recovering the moral graph of a causal DAG from a probability distribution over discrete variables, using fully connected tensor networks (FCTNs) with nuclear-norm-regularized bond corrections. Each bond matrix is parameterized as a baseline all-ones matrix plus a low-rank correction $C_{ij} = U_{ij}V_{ij}^\top$, and the nuclear norm of the correction implemented via the variational Frobenius norm penalty on the factors drives unnecessary bonds to zero. We prove that under faithfulness, positivity, and a no-implicit-rerouting assumption on the local tensor architecture, every optimal FCTN with zero reconstruction error $\varepsilon = 0$ has effective graph exactly equal to the moral graph. For the approximate regime ($\varepsilon > 0$), we provide explicit recovery bounds using the Fannes-Audenaert continuity of conditional mutual information, and derive a sufficient condition on the regularization parameter $\beta$. The effective graph is read directly from the optimized bond matrices.

An infinitely divisible nonnegative matrix is an entry-wise nonnegative matrix that admits an entry-wise nonnegative $m$th root, with respect to usual matrix multiplication, for every positive integer $m$; it is called strongly infinitely divisible when it is, in addition, invertible. The study of such matrices has its origins in the theory of infinitely divisible probability distributions and the embedding problem for Markov matrices and is closely connected with continuous one-parameter semigroups; in the invertible case, this connection admits a natural description in terms of the exponential map. In this paper, we characterize the bijective linear maps on $M_n(\mathbb{R})$ that preserve strongly infinitely divisible matrices and infinitely divisible nonnegative matrices. In the former case, we combine the Inverse Function Theorem with Zariski-density techniques, using the Zariski-density approach recently developed in linear preserver theory by Fallat and Mondal [Proc. Amer. Math. Soc., 2026]. In the latter case, we first show that preserving infinite divisibility forces preservation of the cone of entry-wise nonnegative matrices and then exploit the additional structure afforded by infinite divisibility to complete the characterization.

We consider the stability problem of fork-join systems with redundancy (FJR) and heterogeneous servers under both static and dynamic capacity-allocation policies. In an $(n,k)$ FJR system, each arriving job is split into $n$ independent tasks, with one task assigned to each of $n$ parallel servers. Once $k \le n$ tasks have been processed, they are joined and the corresponding job departs the system; the remaining $n-k$ unprocessed tasks are then removed and are therefore termed redundant. We first identify the nominal traffic intensity and characterize the maximal stability region, defined as the set of traffic intensities for which there exists an admissible policy that stabilizes the system. We then establish conditions under which this maximal stability region is attained for two classes of policies: static and dynamic. Specifically, we show that for static allocation policies, in which service capacities remain fixed over time, maximality is achieved whenever the fastest server is allocated no more than $1/k$ of the total service capacity. For dynamic allocation policies, in which a fixed total service capacity may be repeatedly reallocated among the servers, we show that maximality is achieved whenever the cumulative capacity allocated to the $j$ shortest queues does not exceed $j/k$ of the total capacity for every $j=1,\ldots,k-1$. Our analysis is based on a projection of the $(n,k)$ FJR system onto a simpler $(k,k)$ system that has no redundancy, together with a novel sample-path comparison argument for multidimensional processes based on the generalized Schur-convex order.

We establish quantitative Parisi formulas and fluctuation bounds for the Sherrington-Kirkpatrick model at zero external field. In particular, using that the Parisi measure is supported on an interval, we show that for every fixed inverse temperature greater than one, the variance of the logarithmic partition function lies between orders $N^{4/15}$ and $N^{7/15}$. The same exponents hold for the ground-state variance. We also obtain upper and lower bounds on the finite-size bias of the mean free energy and ground-state energy, together with two-sided upper-tail estimates with exponent $6/5$ at both positive and zero temperature.

Synchronisation by noise for McKean–Vlasov stochastic differential equations is investigated. A transfer principle is introduced by which synchronisation by noise and diagonal mixing can be transferred from an associated limiting frozen-diffusion SDE to a genuinely law-dependent McKean–Vlasov SDE. The usefulness of this principle is demonstrated through an application to ensemble Kalman sampling, thereby providing its first application in a sampling context. Synchronisation for SDEs with multiplicative noise is then revisited from the perspective of sampling. Existing general frameworks providing sufficient conditions for synchronisation by noise are refined and extended to a control-oriented setting on noncompact state spaces, and coefficient-level conditions are derived by which these criteria can be verified. Motivated by extrapolation schemes in sampling, the freedom in the construction of couplings for a fixed sampler is emphasised, together with the advantage of choosing couplings that exhibit favourable synchronisation properties.

We provide a pathwise proof of global-in-time well-posedness for the generalised KPZ equation in the full subcritical regime by an adaptation of the strategy recently applied to the generalised Parabolic Anderson Model in [ES26]. Since this strategy relies crucially on the assumption that control of the supremum norm is sufficient to continue the solution, the main additional ingredient required is a treatment of the initial layer for gKPZ with merely $L^\infty$ initial data, rather than the more usual setting of $C^\theta$ initial data with $\theta > 0$. Our approach is based on an expansion around a deterministic profile followed by the introduction of an integrating factor in order to remove the terms which have critical scaling at time $0$. For pedagogical purposes, we first demonstrate the proof techniques in the case of the standard $(1+1)$-dimensional KPZ equation before turning to the more computationally involved case of generalised KPZ.

As an extension of the Point Estimate Method (PEM) to evaluate probabilistic moments of quantities of interest (QoI) in general $n$-dimensional spaces, the Quadratic Point Estimate Method (QPEM) has been recently developed. This new method is defined to fully represent up to fifth-order input moments in the Gaussian space, providing general analytical expressions for sample locations and weights, without requiring any numerical optimization. The QPEM can significantly improve the estimation accuracy of the output QoI moments, in relation to PEM-based methods whose numbers of sigma points grow linearly with the problem dimension, while at the same time having an affordable and competitive computational cost up to a considerable number of dimensions. The QPEM is further enhanced in this work by enabling copula integration into the framework, which enables effective modeling of the joint input probability density function by estimating marginals and the dependence structure of the involved random variables. The validity and efficient performance of the copula-based QPEM are showcased against numerous other sampling methods in various examples considering two practical scenarios: (i) when the joint dependence structure can be inferred from data, and (ii) when only marginal distributions and correlation matrices are known.

We study the number \(N_I(L)\) of zeros of the Hardy–Szegő zero process in the horizontal window \([0,L]\times I\) as \(L\to\infty\), where \(I=[\alpha,\beta]\Subset(0,\infty)\) is fixed. We obtain sharp point-probability asymptotics at the lower endpoint of the density scale. In the fixed-count regime, for every fixed integer \(k\geq0\), we determine a full asymptotic formula for \(\mathbb P\{N_I(L)=k\}\), identifying its exponential rate, order-one correction, and \(k\)-dependent polynomial prefactor; the case \(k=0\) gives the hole probability. In the vanishing-density regime, we prove a uniform local asymptotic formula for \(b_L\leq n\leq\varepsilon_L L\), where \(b_L\to\infty\), \(\varepsilon_L\downarrow0\), and \(b_L\leq\varepsilon_L L\), identifying the large-deviation exponent, endpoint correction, and Gaussian prefactor.

Ding-Song-Sun proved the remarkable correlation inequality that the truncated two-point correlation function of ferromagnetic Ising models with arbitrary mixed-sign external field is largest when the field vanishes. This inequality has various important consequences such as log-Sobolev inequalities for Ising and $\varphi^4$ models up to and at the critical point. We extend the DSS inequality to the class of single spin measures satisfying the GHS condition. This is an important ingredient in our proof of the log-Sobolev inequality for the massless sinh-Gordon model on $\mathbb{R}^2$ in a forthcoming article.

We study $p$-spread probability measures on the Boolean lattice. We show that if a family of sets $A$ is large under the product Bernoulli-$p$ measure, then no $p$-spread measure can be supported on sets that are not covered by the union of two members of $A$, answering the fractional version of Talagrand's discrete convexity problem. Consequently, we establish a coupling theorem between $p$-spread and product Bernoulli-$p$ measures.

We consider the weakly damped stochastic nonlinear Schrödinger (NLS) equation on the real line, driven by a noise that is white in time and smooth in space. Assuming that the noise is sufficiently non-degenerate, we prove that the equation has a unique stationary measure in the class of probability measures concentrated on $H^2$, and establish polynomial mixing in the dual-Lipschitz metric over $H^1$. We do not impose any restriction on the size of the damping. The proof is based on a coupling argument, whose key ingredient is a Foiaş-Prodi-type estimate in the $H^1$-norm, derived by means of a Lyapunov functional adapted to the linearized NLS dynamics. To compensate for the loss of compactness, we combine this estimate with a truncated Poincaré inequality and a space-time weight function quantifying the spatial decay of solutions.

Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body $K\subseteq\mathbb{R}^n$, let $\mu_{K,t} (\mathrm{d} x) \propto e^{-t\| x \| ^2} \mathbb{1}_K(x)\,\mathrm{d} x$, and let $Q_n$ be the supremum of $\|\operatorname{Cov}(\mu_{K,t})\|_{\mathrm{op}}$ over all such $K$ and all $t>0$. We prove the sharp bound $Q_n=\Theta(n^{2/5})$, closing the gap between the known $\Omega(n^{1/3})$ lower bound and the $O(\sqrt{n\log(en)})$ upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance $\Omega(n^{2/5})$.

We consider a random subgraph $G_p(n,r,<s)$ of the complete distance graph $G(n,r,<s)$ whose vertices are the $r$-element subsets of the set $\{1,\dots,n\}$ and whose edges join pairs of subsets that intersect in fewer than $s$ elements; each edge survives independently of the others with probability $p$. The independence number of the graph $G(n,r,<s)$ equals $C_{n-s}^{r-s}$ – this is the classical Erdos-Ko-Rado theorem. We prove that, for $r=r(n)\to\infty$, $s=s(n)\to\infty$, $s=o(r)$, $r^2=o(n)$ and $p\ge 16\,sr^2\ln(n/r)/n$, with probability tending to 1 the independence number of the random graph $G_p(n,r,<s)$ also equals $C_{n-s}^{r-s}$, i.e., the Erdos-Ko-Rado result is stable under random sparsification of the graph. Thereby, in the range of parameters $s\to\infty$, $s=o(r)$, a recent result of Raigorodskii and Karas is strengthened: the lower bound on the probability $p$ that guarantees stability is lowered by a factor of about $r/s$.

We consider the level sets of the Gaussian free field and the vacant set of random interlacements, both defined on a typical realization of the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z}^d$, $d \geq 3$. We prove that in the entire supercritical regime of Bernoulli bond percolation, both the level sets of the Gaussian free field and the vacant set of random interlacements undergo non-trivial percolation phase transitions at deterministic critical levels. A key aspect of the proof is the development of certain quenched controls over tree embeddings, permitting the application of a static renormalization scheme in the presence of spatial irregularities, which may be of independent interest.

Sparsity is a powerful structural resource in optimization and statistics. We develop frameworks for leveraging sparsity in sampling problems over the Hamming slice $\mathcal{X}_k^d:=\{\mathbf{x}\in\{\pm 1\}^d:|\{i:\mathbf{x}_i=1\}|=k\}$, in high-dimensional regimes where $k\ll d$ (i.e., where $\mathcal{X}_k^d$ is highly magnetized). We use our frameworks to design improved samplers for canonical problems in the study of Ising models and Bayesian sparse linear regression. Our first main result considers the Sherrington–Kirkpatrick (SK) model restricted to fixed-magnetization slices $\mathcal{X}_k^d$. We give a polynomial-time sampler for fixed-magnetization SK models at any inverse temperature $\beta>0$, under arbitrary external fields, provided that $k\le c_\beta d$ for an appropriate constant $c_\beta$. By combining this result with an annealing strategy for estimating normalizing constants, we obtain polynomial-time samplers for the SK model at arbitrarily low temperatures under a sufficiently strong external field of strength $h$. In the large-$\beta$ limit, our framework permits sampling at field strengths within constant factors of the Almeida–Thouless line delineating the replica-symmetric and replica-symmetry-breaking regions ([dAT78]), improving polynomially over the field strength $h(\beta)$ required by the recent work of [BAR26]. Our second main result concerns the measurement complexity of polynomial-time Bayesian sparse linear regression. Recent work by [KSTZ25] shows how to sample from the canonical Gaussian spike-and-slab posterior with expected sparsity $k$, at any signal-to-noise ratio, given $n\gtrsim k^3\log^3 d$ Gaussian measurements. We improve this requirement to $n\gtrsim k^{3/2}\log^2 d+k\log^3 d$, using a common sparsity-aware framework underlying both our results.

In this paper, we prove pathwise uniform convergence at rates up to $1/2$ for tamed exponential Euler schemes for semilinear hyperbolic stochastic evolution equations with superlinearly growing nonlinearities and multiplicative noise. We take the term hyperbolic to mean that the leading operator generates a contractive $C_0$-semigroup but no parabolic smoothing occurs. Under local Lipschitz, polynomial growth, coercivity, and monotonicity conditions on the nonlinearities, we establish pathwise uniform strong error estimates of the form \begin{equation*} \Big(\mathbb{E}\max_{0\le j \le N} \|U(t_j)-U^j\|_X^p\Big)^{1/p} \lesssim \sqrt{k} \end{equation*} on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $U^j$ is the tamed exponential Euler approximation at time $t_j=jk$ with step size $k>0$. This extends previous convergence results for non-parabolic SPDEs from globally to locally Lipschitz nonlinearities, allowing both drift and diffusion to grow polynomially. In a stochastic Kato framework, we further establish local and global well-posedness as well as uniform a priori estimates for the mild solution and its approximation. Applications to nonlinear stochastic transport, Airy, wave-type, and dissipatively damped nonlinear Schrödinger equations are included, covering different nonlinearities-stopped and fractionally tamed schemes. For the Klein-Gordon equation with cubic velocity damping, this complements previous results obtained for additive noise.

We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: \begin{itemize} \item There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. \item There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. \item If parametrizing the space of $d$-dimensional simplices by independent standard Gaussian vectors, then as $d$ increases, a random simplex degenerates asymptotically almost surely. \end{itemize} This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.

We prove a trajectory large deviation principle, in the low-temperature limit, for the two-dimensional Wick-ordered defocusing cubic nonlinear wave equation, with initial data drawn from the invariant Gibbs measure. We further show that asymptotically vanishing high-frequency perturbations of the Gibbs initial data, with the same static rate function, generate a continuum of distinct mass-shifted trajectory rate functions.

We study a moving-average process with (not necessarily symmetric) Laplace innovations under the constraint of positivity. In the three nondegenerate parameter regimes $0<\theta<1$, $\theta>1$, and $\theta<0$, we prove convergence of the conditioned finite-dimensional distributions and identify the limit as a Doob $h$-transform. The regimes lead to qualitatively different limiting dynamics: the invariant law is supported on the positive half-line for $0<\theta<1$, the conditioned chain is confined to the negative half-line for $\theta>1$, and the dynamics are genuinely two-sided and governed by $q$-trigonometric functions for $\theta<0$. In each case, the persistence exponent, sharp persistence asymptotics, the defining eigenfunction, and the unique invariant distribution are obtained explicitly.

In this work we study the maximal ratio $I$ between the white noise integrated over a set and the perimeter of the set, which can be seen as a random isoperimetric problem, and appears in the random-field Ising model (Ding-Wirth) and min-max optimal matching (Leighton-Shor). In the planar case considered here, such a ratio is scale-invariant and therefore the problem is critical. As such it requires an ultraviolet cutoff, which we impose by restricting to polygonal sets with side-length at least 1 contained in the ball $B_L$. Our main result establishes the leading-order asymptotics $\mathbb{E}I\approx i\ln^{3/4}L$ for some $i\in(0,\infty)$ and superconcentration at scale $O(\ln^{-1/4}L)$. This is done by relating to a simpler $(1+1)$-dimensional action, studied by the last two authors and C. Wagner. Such a connection is found by a classical geometric linearization of the perimeter to the Dirichlet energy, justified by Ried-Wagner large-scale regularity theory, and allows a coarse-graining and scale-by-scale iteration argument.

We consider the stochastic heat equation $\partial_t u(t,x) = \Delta u(t,x) + \dot{W}_\alpha(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\dot{W}_\alpha$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-\Delta)^{-\alpha}$ with $\alpha>0$. We prove that a unique pointwise defined mild solution exists if and only if $\alpha>d/2-1$. In this case, if in addition the domain is $C^2$, we also establish spatial and temporal Holder regularity of the solution. When $d/2-1<\alpha<d/2$, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.

We determine the exact first-order asymptotics of the expected optimal cost in two-dimensional random bipartite matching, for every finite power cost $q \ge 1$, on the flat torus. In the endpoint case $q=1$, this answers a question by Talagrand, in the periodic case. The argument involves the closely related asymptotics of the energy of the solution of the $p$-Poisson equation with a regularized white-noise source. In the limit of vanishing regularization parameter, we identify this energy as the solution of a Variational Martingale Problem on a limiting Gaussian filtration, whose value is characterized by a parabolic Monge-Ampère flow.

Let \(\{Z_n\}_{n\geq0}\) be a supercritical branching random walk with deterministic rooted \(b\)-ary tree, \(b\geq2\), and symmetric displacements satisfying \(\lim_{x\to+\infty}x^{-\alpha}\log\Pp(X>x)=-\lambda \) with \(\alpha,\lambda>0\). Set $\overline Z_n(\cdot):=Z_n(\cdot)/Z_n(\mathbb{R}),~n\geq0.$ For a finite union \(A\) of intervals, we establish the following full large-deviation principle: for every Borel set \(\Gamma\subset[0,1]\), \begin{align*} -\inf_{q\in\Gamma^\circ}Q_A(q) &\leq \liminf_{n\to\infty}n^{-\alpha/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\in\Gamma\right) &\leq \limsup_{n\to\infty}n^{-\alpha/2} \log\Pp\!\left(\overline Z_n(\sqrt n\,A)\in\Gamma\right) &\leq -\inf_{q\in\overline\Gamma}Q_A(q), \end{align*} where $Q_A$ is a good rate function on $[0,1]$. Meanwhile, we obtain large deviation probabilities for $\{\overline Z_n(\sqrt n\,A)\}_{n\geq1}.$ This strengthens the nonmatching upper and lower bounds obtained by Chen and He [Probab. Theory Related Fields 175 (2019) 255-307] for regular trees with Weibull displacements. Our method combines a large-deviation principle for rescaled displacement tree fields and exponential equivalence.

We prove that on the metric graph of $\mathbb{Z}^d$ for $d\in \{3,4,5\}$, clusters of the critical loop soup (or equivalently, clusters of the critical Gaussian free field level-set) admit a scaling limit. To the best of our knowledge, this presents the first scaling limit result for a percolation model on a three-dimensional lattice.

We consider the exclusion process with energy, a model with mass and energy conservation, and we put it in a boundary-driven setting. We are interested in its hydrodynamic and hydrostatic limits, together with the associated dynamical and stationary fluctuations. We provide a new proof for the derivation of dynamical fluctuations that does not require explicit estimates on the two-point correlation functions. While the hydrodynamic limit is given by two uncoupled heat equations, the dynamical fluctuations are described by a system of coupled generalized Ornstein-Uhlenbeck processes.

In this work we establish a connection between the min-max optimal matching in the critical dimension $d=2$ and a simpler, geometrically linear, action of random curves in a Brownian potential. This is done by following Leighton and Shor and working with the dual formulation, which we approximate by a problem of isoperimetric-type with a random volume term of white-noise character. This allows us to extract the leading-order term in the asymptotics of the expected cost, sharpening previous results.

We prove polynomial mixing for the defocusing damped cubic stochastic nonlinear Schrödinger equation on the three-dimensional torus under saturating smooth finite rank Brownian forcing. The mixing rate is measured in the $p$-Wasserstein metric induced by the $H^1$ distance for every $1\le p<\infty$. We also obtain sharp geometric characterizations of saturation. The proof is based on a polynomial mixing criterion built on a stable–compact decomposition of the exact solution differences with polynomial moment control of the logarithmic path amplification. Dense Malliavin range allows the compact defect to be compensated by finite-dimensional Cameron–Martin shifts, producing a block multiplier with negative mean logarithm. A logarithmic transportation gauge, combined with a renewal–reset coupling scheme, then yields mixing at every prescribed polynomial order in the weaker $L^2$ distance. A stationary regularity gain to $H^{2-}$ then enables us to upgrade the convergence to $H^1$.

In this paper, we consider an inverse random source problem for the stochastic Moore-Gibson-Thompson equation driven by fractional Brownian motion with Hurst index $H \in (0, 1)$ of the form $f_1(x)g_1(t)\dot{B}^H(t)+f_2(x)g_2(t)$. Given the random source, existence and uniqueness of mild solutions are verified. For the inverse problem, the uniqueness of recovering the strength $f_i(x)$ if the time functions $g_i$ are known and $g_i(t)$ if the spatial functions $f_i$ are known when $H \in(0,1)$ from the boundary flux on a special nonempty open subset is proved.