32 problems

Vanishing conjecture.

Let $\tau$ be a labeled rooted tree, let $|\tau|$ denote its degree, let $\tau!$ denote its tree factorial, and let $q_\gamma(\tau)$ be the quantity defined in the paper for a posi…

Fix an interval $T\subset\mathbb{R}$, and let $R\in\Omega\mathcal{C}(T)$ be a process. Let $p:\mathbb{R}^+\to\mathbb{R}^+$ be an increasing function with $p(0)=0$, and let $\psi$ b…

Time-dependent Riccati feedback conjecture. The optimal control retains the same feedback form as in the time-independent case, with $\bm{\chi}^{\mathbb{X}}$ solving the Riccati eq…

For a sparse multivariate Hawkes process, let $A_T^{ij}$ denote the cross-area between coordinates $i$ and $j$ over a time horizon $T$, and let the associated excitation graph enco…

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…

Chen–Hermite Gaussianity conjecture. Then $(X_t)_{t\in[0,T]}$ must be a Gaussian process. This would establish joint Gaussianity from Gaussian single-increment laws and the Chen–He…

Chen–Hermite balancing conjecture. The Chen–Hermite balancing condition should be strong enough to imply that the increments are jointly Gaussian; equivalently, every Chen–Hermite…

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…

A signature tensor is the tensor obtained by evaluating the signature construction at an arbitrary level, generalizing the signature matrices of membranes. All-signature-tensor con…

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 $\mathfrak{L}$ be the index set, let $T$ be the pre-Lie algebra constructed from multi-indices over $\mathfrak{L}$, and let $D$ be its derivation-type pre-Lie product. Write…

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

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…

The construction uses vector fields $csigma$ in a Sobolev space $H^k(\mathbb{R}^d;\mathbb{R}^d)$ with $k$ large enough for Sobolev embedding to apply and yield a rough driver. Regu…

Non-zero area anomaly conjecture. Roughly speaking, any reasonably asymmetric RWRE from this ballistic class should have a non-zero area anomaly.

Pure rough path signature-tail asymptotics conjecture. For every pure $m$-rough path, the tail asymptotics equals its local $m$-variation, namely

Geometricity conjecture. Upon taking $\alpha$ slightly smaller, the algebraic weak geometricity constraints should imply that ${{\mathbf B}}$ is geometric.

Improved spatial regularity conjecture. The pressure $\pi$ has better spatial regularity than membership in $\bH_{\perp}^{-3}$ suggests. The preceding lemma and system formulation…

The paper considers Yang–Mills theory on $\mathbb{R}^{2}$ in axial gauge, where quantum gauge fields are approximated by classical smooth gauge fields. Smooth approximation conject…

Let $\gamma$ be a tree-reduced continuous path with bounded variation, and let $g$ be its signature, with $g_n$ denoting the $n$-th term and $\|\cdot\|_{\mathrm{proj}}$ the project…

Davie's conjecture. For any SDE, it should be possible to prove

Let $d4b7$ be a variation function, and let the Loève–Young inequality refer to an inequality controlling the integral of a product in terms of the $d4b7$-variations of the integra…

A weakly geometric rough path is a rough path taking values in the corresponding weakly geometric rough path space, with its signature defined by the associated multiplicative func…