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 for a transformed remainder. The argument combines a response decomposition with a conservative Zvonkin transformation. For $1<\gamma<2$, this produces a compensated nonlocal diffusion operator whose coercivity is coupled to a cubic-increment estimate adapted from the modified Karman--Howarth--Monin argument.
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 $\frac12\min\{1/h_{P}, 1/k\}$, where $h_{P}$ denotes Varchenko's Newton height of $P$. The two terms in the minimum reflect a natural competition between the spatial degeneracy of $P$ and the temporal degeneracy of $t^k$; their optimality is confirmed by a Knapp-type and a focusing example, respectively. A $TT^{*}$ reduction transforms the $L^2$ estimate into a scalar oscillatory integral, allowing Varchenko's theorem to apply directly. Building on this foundation, complex interpolation yields the sharp $L^2\to L^{2k+2}$ bound. Finally, in the regime $h_{P}\geq k$, we obtain sharp $L^2\to L^p$ decay estimates for all $p$.
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 $\sigma : \mathbb{R} \rightarrow \mathbb{R}$ is sufficiently smooth, bounded, and has bounded derivatives. Our strategy is to obtain pathwise a priori estimates via regularity structures. In fact, we consider a general singular multiplicative equation with superlinear damping, driven by noises of parabolic regularity $\alpha - 2$, for ${\alpha \in (0, 1)}$, which can be lifted to a weakly admissible model. We show that the required estimates hold whenever \[ m>\frac{2 - \alpha}{\alpha} \varepsilon_{\alpha}, \quad \text{where} \quad \varepsilon_{\alpha} = 1 - \alpha \left( 1 - \frac{2}{3 - \alpha} \right) \in (0, 1) . \] Thus the strength of the damping needs to be chosen only as a function of the regularity of the driving noise. Under an additional smoothness assumption on $\sigma$, we show that the stochastic flow is differentiable with respect to its initial condition.
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 technical input is a commutator estimate for Lipschitz''tent''localisations $W_{L,x}$, which depends only on geometric properties (doubling) and a uniform interaction degree. As a consequence, we obtain explicit two-sided pseudospectral inclusion bounds of the form \[ \gamma_{L,\varepsilon}(H)\subset \sigma_\varepsilon(H)\subset \gamma_{L,\varepsilon+C_0/L}(H), \] and Hausdorff convergence of window pseudospectra to the (global) pseudospectrum as $L\to\infty$. In the self-adjoint/ normal case this yields computable gap tests and spectral sampling schemes with rigorous $O(1/L)$ error control, while in the non-normal case it leads to corresponding approximation results for pseudospectra. The framework isolates the geometric core behind earlier approaches on $\mathbb{R}^n$ and on countable Abelian groups, and covers irregular geometries arising in quasicrystal models, as well as new cases such as discrete (non-Abelian) nilpotent groups (including for example the discrete Heisenberg group).
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 measure associated with the operator $\mathcal{L}_a$ established by Zhang-Zhang [J. Geom. Anal. \textbf{35}(3), Paper No. 71, 27pp (2025)], we investigate the decay estimate for a class of dispersive semigroups of the form $e^{it\phi(\sqrt{\mathcal{L}_a})}$, where $\phi: \mathbb{R}^+ \to \mathbb{R}$ is a smooth function. To handle the technical difficulty arising from the inhomogeneity of the phase function $\phi$, we adopt the frequency localization and the stationary phase method. In the second part of the paper, we first derive boundary Strichartz estimates for the fractional Schr\"odinger operator $e^{it\mathcal{L}_a^\nu}$, $0<\nu\neq\frac{1}{2}$. As applications of the established decay estimates, we further obtain Strichartz estimates for some concrete wave equations associated with the operator $\mathcal{L}_a$, which corresponds to $\phi(r)=r, r^2, r^2+r^4, \sqrt{1+r^2}, \sqrt{1+r^4}$, and $r^\mu,0<\mu\neq 1$. Most notably, our results unify and simplify existing dispersive estimates for the operator $\mathcal{L}_a$, while extending the relevant theory to more general scenarios.
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 symbols, with constants depending on the Fourier-support scale. We study this quantitative dependence, improve the known upper bounds, and show that in large ranges of $p$ no support-independent global comparison can hold. Let $F_{\sigma}$ denote the symplectic Fourier transform, and let $u\in E'(\mathbb{R}^{2d})$ satisfy $\operatorname{supp}u\subset\overline{B(z_0,R)}$, where $R\geq1$. Then, for every $1\leq p\leq\infty$ and $\varepsilon>0$, we prove \[ \|L_{F_{\sigma} u}\|_{S_p}\lesssim_{d,p,\varepsilon}R^{(2d+1+\varepsilon)|1-2/p|}\, \|F_{\sigma} u\|_{L^p(\mathbb{R}^{2d})}, \] together with the reverse estimate with the same power of $R$. This sharpens the exponential dependence $\mathrm{e}^{cR^2}$ obtained by Luef and Samuelsen and M\"uller's polynomial dependence $R^{(5d+2)|1-2/p|}$. The main ingredient is a radial trace-class estimate based on the Hermite--Laguerre correspondence $\rho(\varphi_k)=P_k$, which reduces the relevant Weyl operators to finite-rank Hermite projections. We also show that dependence on $R$ is unavoidable. Compactly supported examples obtained by truncating Laguerre functions yield polynomial lower bounds for the best comparison constants. These examples refine M\"uller's operator-norm example and give nontrivial lower bounds for a larger range of Schatten exponents, which can cover the full range $1\le p\le\infty$ except for the Hilbert--Schmidt point $p=2$ as $d\to\infty$. Moreover, for every fixed $p>2$, the exponent in the reverse comparison estimate is asymptotically optimal as $d\to\infty$.
Jie Liu· 0 citations
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