9 problems

Variation-constant conjecture. For $H\neq 1/2$, the $(1/H)$-th variation along deterministic partitions differs from that along uniform Lebesgue partitions, more precisely,

Let $b$ be a finite signed measure on $\mathbb{R}$, let $H\in(0,1/2)$, and consider the stochastic differential equation … where $B^H$ is fractional Brownian motion with Hurst para…

Let $H>1/2$, and consider fractional Gaussian noise with Hurst parameter $H$. For any element of this process, project it onto any finite number of its subsequent elements, and den…

Let $B^H$ be a fractional Brownian motion with Hurst index $H\in(0,1)$, and let $\varphi(W)$ be the Hunt process on $\operatorname{gr}(B^H)$ induced by a Wiener process through the…

The dyadic local-time conjecture. Almost surely, $\widetilde{L}^{\pi_n}_t(x)$ converges uniformly for $(t,x)\in[0,T]\times\mathbb{R}$ to

Let $X=(X_x)_{x\in\mathbb R}$ be a self-similar process satisfying the growth condition needed to define the inviscid Burgers solution, and let $\mathcal L$ be the closure of the s…

Khokhlov-Molchan conjecture. One has

Let $B^H$ be fractional Brownian motion with Hurst parameter $H\in(0,1)$, let $A_t^H=\int_0^tB_s^H\,\mathrm{d}s$, and define $T_1^H=\inf\{t\ge0:A_t^H>1\}$. The fractional Brownian…

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