AI 中文总结
该研究针对黎曼流形上由α/2稳定次支配子从属布朗运动得到的过程X^(α)，证明了积分条件∫^∞ dt/V(o,t^(1/α))=∞可推出X^(α)回归，解决了Grigor'yan问题26的充分性部分。
AI 中文摘要
设M为连通、测地完备且无边界的黎曼流形，记μ为黎曼体积，V(o,r)为以o为中心的测地球B(o,r)的体积。对0<α<2，令X^(α)为通过独立α/2稳定次支配子从属布朗运动得到的过程，其L²生成元为-(-Δ)^(α/2)。我们证明积分∫^∞ dt/V(o,t^(1/α))=∞蕴含X^(α)是回归的。证明利用了谱迹估计与M×(0,∞)上的径向截断，辅助测度为y^(1-α)dμdy，解决了Grigor'yan问题26中的充分性蕴含。
英文摘要
Let $M$ be a connected geodesically complete Riemannian manifold without boundary, write $μ$ for Riemannian volume, and set $V(o,r)=μ(B(o,r))$ for geodesic balls centered at $o$. For $0<α<2$, let $X^{(α)}$ be the process obtained by subordinating Brownian motion with an independent $α/2$-stable subordinator; its $L^2$-generator is $-(-Δ)^{α/2}$. We prove that \[ \int^\infty\frac{dt}{V(o,t^{1/α})}=\infty \] implies that \(X^{(α)}\) is recurrent. The proof uses a spectral trace estimate and radial cutoffs on $M\times(0,\infty)$, where the auxiliary measure is $y^{1-α}\,dμ\,dy$. It proves the sufficient implication in Grigor'yan's Problem~26.
Comments14 pages
