Tokunaga and Horton self-similarity conjecture for fractional Brownian motions

Fractional Brownian motion is described by a centered Gaussian process $B^H_t$, $t\in[0,1]$, with Hurst index $0<H<1$. Its level set tree is denoted by $\operatorname{tree}(B^H)$, and $\operatorname{shape}(\operatorname{tree}(B^H))$ is its combinatorial shape. For $i,k\ge 1$, let $T_{i(i+k)}(B^H)$ and $T_k$ denote the corresponding Tokunaga coefficients, and let $\eta_r(B^H)$ denote the Horton ratios.

Fractional Brownian motion self-similarity conjecture. The tree $\operatorname{shape}(\operatorname{tree}(B^H))$ is Tokunaga self-similar with

$$T_{i(i+k)}(B^H)=T_k=c^{k-1},\qquad c=2H+1,$$

for $i,k\ge 1$. Equivalently, the Horton self-similarity parameters should satisfy

$$\eta_r(B^H)=2+H+\sqrt{H^2+2},\qquad r\ge 1.$$

This conjecture is motivated by the established Horton and Tokunaga self-similarity of the level set tree of regular Brownian motion and by numerical simulations. The appropriate sense in which the limiting self-similarity relations should hold remains to be determined.