This paper computes Poisson space noise functionals, $P^{\prime}(u)$, realised in a Gel'fand triple built from a L\'evy measure $\lambda_{\beta}(u)du$. We isolate three discretisation parameters: a small-amplitude cut-off, a Donsker delta truncation M, and a chaos order N. For a stable-type intensity $\lambda_{\beta}(u)=cu^{-1-\alpha}$ ($0<\alpha<2$), replacing discarded small amplitudes with matched Gaussian space noise improves the Wasserstein-1 error from $O(\epsilon^{1-\alpha/2})$ to $O(\epsilon)$. The residual is asymptotically normal at $O(\epsilon^{\alpha/2})$. This compensation reduces computational complexity from $O(\tau^{-2\alpha/(2-\alpha)})$ to $O(\tau^{-\alpha})$. We also evaluate the Gamma-type boundary ($\alpha=0$) and exponential tempering. Truncations converge algebraically (M) and super-geometrically (N). All predicted rates are tightly confirmed by deterministic numerical experiments via Gil-Pelaez inversion, eliminating Monte Carlo noise.
We study the periodic fractional stochastic Burgers equation $(\partial_t+\Lambda^\gamma)u=\partial_x(u^2)+|\partial_x|^{1-\alpha}\xi$, where $1<\gamma\leq 2$ and $\xi$ is space-time white noise. Under the condition $\alpha>\max{(7-4\gamma)/2,(15-8\gamma)/6}$, we establish pathwise $L^1$, energy, and Besov estimates fo...
We study $(2+1)$-dimensional oscillatory integral operators of the form \[ T_\lambda f(x,y)=\int_{\mathbb{R}}e^{i\lambda P(x,y)t^k}\psi(x,y,t)f(t)dt,\qquad k\geq 1, \] where the phase $P$ is a real-analytic function with a critical point at the origin. We establish the sharp $L^2\to L^2$ decay rate of $\frac12\min\{1/h...
We establish the existence of a stochastic flow on $L^{\infty} (\mathbb{T})$ for the stochastic Allen-Cahn equation with multiplicative noise \[ (\partial_t - \partial_x^2) u = u - u^3 + \sigma (u) \xi \quad \text{on} \quad \mathbb{R}_+ \times \mathbb{T}, \] where $\xi$ is space-time white noise and $\sigma : \mathbb{R...
We study bounded finite-interaction-range operators $H$ on $\ell^2(\Gamma)$, where $\Gamma$ is a uniformly discrete subset of a left-invariant doubling metric measure space $(X,d,\mu)$. Our goal is to approximate spectral information of $H$ from finite sections $H_{L,x}$ supported on balls $B_L(x)\cap\Gamma$. The main...
Let $\mathcal{L}_a=-\Delta_x-\Delta_y+\frac{a}{2}|x|^{-2}$ with $a>0$ denote the Schr\"odinger operator on $L^2(\mathbb{R}^2_x\times \mathbb{R}^n_y)$, which involves a singular partial inverse-square potential. The purpose of this manuscript is twofold. First, relying on the explicit representation for the spectral mea...
The Weyl calculus associates a function $a$ on phase space $\mathbb{R}^{2d}$ with the corresponding Weyl operator $L_a$ acting on $L^2(\mathbb{R}^d)$. At $p=2$, this correspondence is governed by an exact Hilbert--Schmidt identity. For $p\neq2$, two-sided $L^p$--Schatten estimates are known for Paley--Wiener type symbo...
Jie Liu· 0 citations
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