A Family of Log-Correlated Gaussian Processes
Yizao Wang · Journal of Theoretical Probability · 2025
Abstract A family of log-correlated Gaussian processes indexed by metric spaces is introduced, when the metric is conditionally negative definite. These processes arise as the limit of bi-fractional Brownian motions indexed by ( H , K ) scaled by $$K^{-1/2}$$ K - 1 / 2 as $$K\downarrow 0$$ K ↓ 0 with $$H\in (0,1/2]$$ H ∈ ( 0 , 1 / 2 ] fixed. When the metric is in addition a measure definite kernel, stochastic-integral representations of the generalized processes when evaluated at a test function are provided. The introduced processes are also shown to be the scaling limits of certain aggregated models.