64 problems

Persistence exponent conjecture. One should have

Non-Gaussianity conjecture. The term $I_i(t)$ is not Gaussian.

Consider a particle system in $\mathbb{R}^{d}$ whose particles move according to a symmetric $\alpha$-stable Lévy process and undergo critical finite-variance branching at rate…

The preceding construction defines processes $X^{\mu^H}$ and establishes convergence to fractional Brownian motion $B_H$. Extension conjecture. This result should remain true in th…

Consider fractional Brownian motions with Hurst parameter $H\in (1/2,1)$ as intermediate processes between smooth and Brownian-semimartingale vortex filaments, and assume that the…

Let $B^H=(B_t^H)_{t\in[0,1]}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$, and let … be its graph. Fraser–Sahlsten conjecture. Almost surely, … Fraser and Sahl…

Let $Y=Y_{\alpha,b,H}$ be the fractional Wiener–Weierstrass process, let $K=\min\{1,-\log_b(\alpha)\}$, and let $\rho(s,t)=|s-t|^H$. Strong local nondeterminism conjecture. If…

Complex fractional Ornstein–Uhlenbeck estimation conjecture. The strong consistency and the asymptotic normality of the estimator $\hat{\gamma}_T$ hold for the complex-valued fract…

Let $H\in(0,1)$ be the Hurst parameter, and consider the numerical method for an autonomous stochastic differential equation driven by fractional Brownian motion, with the stochast…

Compensated Riemann–Stieltjes/Skorohod-sum conjecture. The two limiting sums should coincide:

Let $H\in(0,1)$ be the Hurst parameter of fractional Brownian motion, and let $\gamma$ denote the Hölder regularity exponent of the coefficient $\sigma$. Pathwise uniqueness failur…

Let $H\in[1,\infty)\setminus\mathbb{N}$ be the Hurst parameter of fractional Brownian motion, and let $\gamma$ denote the Hölder regularity exponent of the coefficient $\sigma$. Th…

Let $B^H$ be fractional Brownian motion with Hurst parameter $H$, and let $W^{\sigma^2,T}$ be a branching Brownian motion with time-inhomogeneous variance profile $\sigma^2(s/T)$,…

Let $B_t^H$ be a self-repellent fractional Brownian motion with Hurst parameter $H$ in dimension $d$, and let $R_T$ denote its radius of gyration, … Suppose that, up to a possible…

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,

Variation conjecture. The $(1/H)$-th variation of the fake fractional process along $\pi$ will be linear with slope $C_{1/H}$.

Spohn's conjecture. The equilibrium fluctuations of the current should converge in the limit to a $1/4$ fractional Brownian motion.

Let $p\in[1,\infty)$, let $b\in L_p(\mathbb{R},\mathbb{R})$, and consider the stochastic differential equation with drift $b$ driven by fractional Brownian motion with Hurst parame…

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 $H\in(1/4,1/2)$, and consider elliptic, non-commutative, $C_{b}^{\infty}$ vector fields in the rough regime. Generic lack-of-Gaussian-tail conjecture. The lack of Gaussian tail…

Let $B^H$ be a fractional Brownian motion with Hurst index $H$, viewed as a process on the flat torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$, and let…

Local averaging conjecture. The Lebesgue integral should satisfy the local approximation

Let $H\in(0,1)$ and let $B^H$ be fractional Brownian motion. Define the integrated fractional Brownian motion by … Its persistence probability has polynomial asymptotics with persi…

Consider the time non-homogeneous linear test equation driven by fractional Brownian motion … where $H$ is the Hurst parameter, $\kappa$ is the time-inhomogeneity parameter, and…