arXiv subjects

Obayda Julien Assaad

Publications and source records attributed to Obayda Julien Assaad.

4 recordsLinked to original sources

arXiv2026

Weak limits of multiple Wiener integrals: representation, extraction, and realization

We study weak limits of multiple Wiener integrals together with the limits of their kernel contractions. After passage to a subsequence, bounded kernel sequences admit a Gaussian Hermite representation whose coordinates carry weights determined by the original chaos orders. In the limits of kernel contractions, only pairings of whole Gaussian coordinates of equal weight contribute. We construct kernel projections with explicit bounds for Gaussian approximation and asymptotic independence. For homogeneous inputs, spectral extraction converges to the decomposable and primitive projections in the limiting kernel spaces. Conversely, prescribed weighted Hermite expansions are realized by smooth compactly supported kernels away from the diagonals, with exact inner products for finite expansions and asymptotically prescribed contractions. We then classify fixed-chaos limits with an absolute exponential moment. For a standard Gaussian vector $X$ and a symmetric matrix $\mathsf A$, a quadratic target $X^{\mathsf T}\mathsf A X-Tr\mathsf A+b^{\mathsf T}X$ is such a limit if and only if $\mathsf A b=0$. We also give a finite cumulant criterion and a joint version. Finally, we identify limits under stationary Ornstein--Uhlenbeck noise: a coordinate of weight $s$ has relaxation rate $s$. Two fourth-chaos examples have identical one-time laws and covariance functions but different asymptotic variances for time averages of their squares. For kernels with nonnegative coordinates, contractions of the weight-one component along simple regular graphs are nonnegative; this excludes an explicit cubic representation that is attained by signed kernels.

math.PR↗

arXiv2026

Heat Geometry from a Universal Scalar Process

We prove that the law of one smooth scalar process in a finite Wiener chaos completely determines finite families of symmetric tensors on arbitrary real separable Hilbert spaces, up to simultaneous orthogonal equivalence. Gaussian graph characters recover all contractions, while an intrinsic trace class Gram operator reduces the problem to finite dimensional invariant theory. Applied at a single fixed positive time to the canonical quadratic and quartic heat packet, this principle reconstructs the heat generator and forces every reconstructed unitary to be spatial. Consequently, one universal scalar process determines closed Riemannian manifolds, compact $\operatorname{RCD}$ spaces of finite dimension, and Euclidean bundles with metric connection and self adjoint potential. The heat packet functor is fully faithful, and its tensor automorphisms are precisely the geometric ones.

math.PR↗

arXiv2026

Quadratic variations of rough generalized Hermite processes: hidden long memory and a universal Gaussian boundary field

We study centered quadratic variations of fractionally filtered generalized Hermite processes. For every chaos order $q\ge2$, every admissible anisotropic monomial kernel $g_{\boldsymbolγ}$, and every fixed $0<h\le1/2$, we prove that $N^{2h+θ-1}V_N$ converges to a Rosenblatt process with self-similarity exponent $1-θ$, without subtracting any chaos projection. Thus the visible roughness $h$ determines the normalization, whereas the latent singularity exponent $θ$ determines the memory of the limit. The proof uses a sharp paired-filter threshold and charged diagram power counting. At $h=0$, the bare kernel has logarithmically divergent energy. Variance-normalized $h$-regularization and a hard endpoint cutoff have the same residue and converge in finite-dimensional distributions to the universal Gaussian field $G_t=(ζ_t-ζ_0)/\sqrt2$. Although $G$ has no stochastically continuous modification, the iterated boundary-first quadratic energies converge to $\sqrt3B$. In the isotropic case, simultaneous limits $h_N\downarrow0$ are governed by $λ_N=h_NN^{1/2-θ}$: $λ_N\to0$ yields a Brownian limit, $λ_N\toλ\in(0,\infty)$ yields an independent Brownian-Rosenblatt sum, and $λ_N\to\infty$ yields a Rosenblatt limit after division by $λ_N$. We also obtain an explicit polynomial rate in fixed Malliavin-Sobolev norms and functional convergence in little-Hölder spaces.

math.PR↗

arXiv2026

Finite Gaussian Reconstruction of Polynomial Orbits: From Correlated Moments to Oscillatory Periods

Let $P$ be a real polynomial of degree at most $m$ on $\mathbb{R}^d$, and let $X$ be standard Gaussian. Because Gaussian observations are invariant under $O(d)$, the natural inverse problem is to recover the orthogonal orbit of $P$; the law of $P(X)$ alone is generally insufficient. We prove that a prescribed finite family of mixed moments of correlated Gaussian replicas, $$ M_{P,r}(Σ)=\mathbb{E}\prod_{a=1}^r P(X_a), $$ separates $O(d)$-orbits. We construct an explicit replica cutoff and rational covariance grids satisfying $$ \frac{1}{2}I_r\preceqΣ\preceq\frac{3}{2}I_r. $$ Finite differences recover all complete Wick contractions needed by invariant theory, giving an exact finite decoder. The resulting probe map is bi-H"older equivalent to orbit distance on coefficient balls, with an effective exponent. We then identify the same certificate in an irregular period system. Replicated characteristic functions are polynomial oscillatory periods, and their mixed derivatives at zero are the moments above. If the leading homogeneous part of $P$ has an isolated critical point, the active-replica face indexed by $I$ has twisted de Rham rank $(m-1)^{d|I|}$; zero coupling is therefore a rank-changing boundary. The forced scaling $$ τ_a=ρ^{m-2}λ_a,\qquad x_a=ρ^{-1}u_a $$ produces compatible Rees--Jacobi lattices and, under central nonresonance, a canonical rank-one Gaussian branch. On admissible tame Morse chambers, the period matrix factors into algebraic Jacobi, sectorial thimble, and integral Betti components. Projecting the assembled real-contour period onto the Gaussian branch recovers exactly the finite orbit certificate.

math.PR↗