标题： 双曲空间中的抛物型安德森模型与相变 显示英文标题

标题： Parabolic Anderson Model in Hyperbolic Spaces and Phase Transition

摘要： 考虑一个抛物线Anderson模型（PAM），其噪声在时间上是白噪声，在空间上是彩色噪声，其中空间相关性以多项式阶次 $\alpha$衰减。 在维度大于 $2$的欧几里得空间中，已知 $\alpha$的临界值为 $2$。 具体来说，对于 $\alpha<2$，解的二阶矩随时间呈指数增长，而对于 $\alpha>2$，当反温度增加时，二阶矩从时间上一致有界转变为时间上的指数增长，存在相变。 这种临界行为源于在欧几里得空间中，布朗运动以 $\sqrt{t}$的速度趋向于无穷远。 本研究探讨了在双曲空间上的PAM。 已知在双曲空间中的布朗运动以$t$的速度移动，可以预期$\alpha=1$会是上述现象的关键值。我们确认这种直觉确实是正确的。此外，我们发现$\alpha<1$中存在一种新的相，在该相中二阶矩以次指数方式爆炸，这与欧几里得空间中观察到的行为不同。 显示英文摘要

摘要： Consider a Parabolic Anderson model (PAM) with Gaussian noise that is white in time and colored in space, where the spatial correlation decays polynomially with order $\alpha$. In Euclidean spaces with dimension greater than $2$, it is well-understood that the critical value for $\alpha$ is $2$. Specifically, for $\alpha<2$, the second moment of the solution grows exponentially over time, while for $\alpha>2$, there is a phase transition, from the second moment being uniformly bounded in time to exhibiting exponential growth in time when the inverse temperature increases. This critical behavior arises from the fact that in Euclidean space, Brownian motion tends to infinity at a speed of $\sqrt{t}$. The present work explores the PAM on a hyperbolic space. Given that Brownian motion in a hyperbolic space travels at a speed of $t$, one expects that $\alpha=1$ would be the critical value for the above phenomena. We confirm that this intuition is indeed correct. Furthermore, we uncover a novel phase for $\alpha<1$ in which the second moment explodes sub-exponentially, distinct from the behavior observed in Euclidean space.