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I am trying to understand how the first passage time density of Brownian motion with drift is modified by the presence of waiting times that are distributed as a power law

In other words, what is the density function for first passage time of a Brownian motion with drift when the movement "pauses" for time intervals that follow a power law distribution

I believe the density function would be the convolution of inverse Gaussian and power law distributions. Is there an expression for this? Any help would be much appreciated

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