43 problems

Vanishing conjecture.

Perturbation invariance conjecture. If $w$ is $(\rho,\gamma)$-irregular, then $w^\varphi$ is also $(\rho,\gamma)$-irregular.

Let $(t,x)\in\mathbb{R}_+\times\mathbb{R}^3$. For each $N$, let $\mathbf{u}_N$ denote the corresponding approximating solution, and let $U_{t,x}$ be a neighborhood of $(t,x)$. Loca…

Let $E$ be a Banach space with finite cotype. Assume that $-A$ is injective and sectorial of angle less than $\frac{\pi}{2}$ on $E$ and admits a bounded $H^\infty$-calculus. Let…

Matrix-valued Feynman–Kac representation conjecture. Under suitable regularity conditions, the solution of the system is

Drift-elimination conjecture. In a gauge-theoretical setting, and by further applying stochastic analysis, one could do away with the drift in any dimension other than $1$.

Let $\boldsymbol{\boldsymbol{\boldsymbol{A}}}$ be a local rough path lift (local RPL) defined on a Borel set $B earrowtext{?}$ in the sample space $a9$. **Null-domain conjecture fo…

A local rough path lift (local RPL) is the object studied in this section, with the preceding results established under a square-integrability hypothesis. Square-integrability-free…

Moment-bound conjecture. The same bound holds for all $2\leq p<\infty$.

The $G$-framework is built from a family of volatility-controlled paths, with the driving path represented by $B_t^\theta=\int_0^t\theta_s\,dW_s$ for controls…

Consider stochastic parabolic equations in spatial dimension $d$, driven by colored noise whose spatial correlation is described by a Riesz kernel with parameter $\alpha$, and let…

Let $W$ be the Brownian motion and let $G$ be the Green kernel appearing in the definition of the renormalised Amperean area. Write $I_W$ for the recentred self-intersection local…

The paper considers multidimensional perturbed pantograph equations, with property … and … holds under conditions … should be true. The conjecture proposes a multidimensional exten…

Manifold semimartingale compatibility conjecture. An analogous result to the Euclidean Brownian-motion case should hold for manifolds and general driving semimartingales: under an…

Let $s^+(t)$ and $s^-(t)$ denote the upper and lower boundaries of the spread, let $\u03bb$ be the parameter appearing in the Stefan problem, let $b1$ be the drift parameter, and l…

Let $E$ denote the deterministic Lévy area correction appearing when approximating a stochastic system by smooth processes, so that the limiting stochastic integral has the form ……

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…

Small-friction derivative estimate. It is conjectured that

The conjectured first-order heat-kernel approximation. If $c$ is three times continuously differentiable and…

Covariance-setting conjecture. For noises in this class, the covariance setting will prove to be a useful tool.

Stochastic multiparameter extension conjecture. Using the stochastic reconstruction theorem in place of the deterministic multi-parameter sewing lemma should allow the results of t…

Lê's stochastic reconstruction conjecture. A result analogous to the stochastic sewing lemma should hold in the context of Hairer's reconstruction theorem.

Invariant-property conjecture. For any pair $(\gamma,\rho)$, there exists a property $\mathcal{P}$ such that:

Let $w$ be a fractional Brownian motion in $\mathbb{R}^d$, let $K$ be any Schwartz function satisfying … and let $p$ be as in the conjecture. Catellier's conjecture. The estimate ……

Normalization conjecture. The factor $t^{-\frac16}$ is the correct normalization, and the limit should satisfy an equation involving a weighted local time of the solution.