We study Gaussian fluctuations for a lower-triangular system of interacting diffusions in which particle $i$ interacts only with its predecessors. Although the empirical measure of this system converges to the same McKean--Vlasov limit as in the corresponding exchangeable mean-field system, the sequential structure remains visible at the $N^{-1/2}$ fluctuation scale. We introduce the logarithmically weighted fluctuation fields \[ Y_t^{N,n} = \frac{1}{\sqrt{N}} \sum_{i=1}^N \frac{\bigl(\log(N/i)\bigr)^n}{n!} (\delta_{X_t^i}-\bar\rho_t), \qquad n\ge 0, \] and prove joint convergence of the entire family in a countable product of weighted negative Sobolev path spaces to the unique probabilistically strong solution of a linear hierarchy in which $Y^n$ couples to $Y^{n+1}$. In particular, the limit of the empirical fluctuation field $\sqrt{N}(\mu_t^N-\bar\rho_t)=Y_t^{N,0}$ is not governed by the closed fluctuation SPDE arising in the classical exchangeable case. The proof combines conditional-measure replacement, deterministic estimates for the logarithmic weights, a martingale argument, and a weighted Volterra estimate.
In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-\rho(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{\rho(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random....
Under a fairly general condition on the underlying jump rates, we prove that the Exclusion Process with fixed particle density exhibits cutoff at time $t_{{mix}}\sim \frac{1}{2}{t}_{{rel}} \log N$, where $N$ is the volume and ${t}_{{rel}}$ the relaxation time of the single-particle dynamics. Our result covers, in parti...
We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous L\'evy--It\^o diffusions driven by a Brownian motion and an independent non-truncated pure-jump L\'evy process. Using Bismut's directional Malliavin calculus on the Wiener--Poisson space togeth...
For fermions in a slowly varying aperiodic potential with random interactions $V_{ij}/|i-j|^{\alpha}$, the potential fixes the resonance supply $N_0 \propto L^{2-n}/h$. On crossing ${\alpha}-2n$, wherever Hartree fragmentation is operative, the resonant-object ensemble reconstructs: turning-point runs fragment into clu...
We establish quantitative propagation of chaos in total variation for the unregularized three-dimensional repulsive Vlasov--Poisson--Fokker--Planck (VPFP) particle system. For tensorized initial data satisfying weighted Sobolev regularity and a Gaussian moment, we prove that for every $00$ and $C_b\ge1$, independent of...
We give a rigorous realization of the dilute instanton picture for a semiclassical Schr\"odinger operator with a symmetric double-well potential on $\mathbb{R}^n$. Using a localized Feynman--Kac representation, we decompose the heat-kernel trace according to the number of passages made by a Brownian bridge between shri...
Jacob Shapiro· 0 citations
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