Local structure at the maximum and sharp persistence asymptotics of rough fractional Brownian motion

We consider a fractional Brownian motion $B$ with Hurst index $0 < H < 1/2$, and its maximiser $\tau$ on $[0,1]$.

We show that the rescaled process $a^H(B_{\tau+\cdot/a}-B_\tau)$ converges in $C_{\mathrm{loc}}(\mathbb R)$ to a limiting tangent law that is $H$-self-similar, supported on nonpositive paths pinned at zero, and rerooting-rescaling invariant: rerooting the limit process at its maximum on any fixed compact interval separated from zero and rescaling again asymptotically reproduces the same law.

We also show that the tangent law has a natural interpretation as "fractional Brownian motion conditioned to be nonpositive on the entire line".

As an application, we consider persistence probabilities for fractional Brownian motion.

A tilted variant of $B$ yields a different tangent law with a finite left horizon and an infinite right horizon and we show that \[ \mathbb P(B_t \leq 1\text{ for all }0 \leq t \leq T) \sim \frac{H\mathbb E[M]}{\Gamma(1/H)D_H}T^{-(1-H)}, \] where $D_H \in (0,\infty)$ has an explicit representation in terms of the tilted tangent law.