分形上的刘维尔量子引力研究
Towards Liouville Quantum Gravity on fractals
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中文总结 AI 辅助
受谢尔宾斯基地毯上分数高斯场构造启发，基于Kahane高斯乘性混沌方法构造分形上类似刘维尔量子引力的随机测度族，证明其反射不变性与标度自相似性，并构造对应刘维尔布朗运动类似物。
中文摘要 AI 辅助
受谢尔宾斯基地毯上分数高斯场最新构造的启发，我们研究协方差函数呈对数行为的高斯场，引入与这些场相关的参数化随机测度族，可视为刘维尔量子引力的类似物，其构造基于Kahane的高斯乘性混沌方法。我们证明了场和测度的反射不变性与标度自相似性，还构造了对应的刘维尔布朗运动类似物。
英文摘要
Motivated by the recent construction of fractional Gaussian fields on the Sierpinski gasket, we study those Gaussian fields whose covariance function exhibits a logarithmic behavior. We introduce a parametric family of random measures associated with these fields, which can be regarded as the analogue to Liouville quantum gravity. The construction is based on Kahane's approach via Gaussian multiplicate chaos. We establish reflection invariance and scaling self-similarity for both the fields and the measures, and in addition construct the corresponding analogue to Liouville Brownian motion.
