Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2021-04-07 , DOI: 10.1016/j.jfa.2021.109029 Morris Brooks , Giacomo Di Gesù
We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisfies a Kramers-type formula in the limit of vanishing noise. We work with finite-dimensional lattice approximations and establish semiclassical estimates which are uniform in the dimension. Our key estimate shows that the constant separating the two exponentially small eigenvalues from the rest of the spectrum can be taken independently of the dimension.
中文翻译:
无限维双井模型的清晰隧道估算
我们考虑了半经典状态下四次双阱能量函数的随机量化,并为基态能量的指数小分裂导出了最优渐近性。我们的结果提供了一些零度的半经典维滕拉普拉斯算子的有限维已知的一些尖锐隧道估计的无穷大形式。从随机的角度来看,它证明了有限体积中的随机一维Allen-Cahn方程的谱隙在噪声消失的极限内满足Kramers型公式。我们使用有限维晶格近似,并建立维数均匀的半经典估计。我们的主要估算结果表明,可以将两个指数较小的特征值与其余频谱分开的常数可以与维数无关。