Density Estimates and Short-Time Asymptotics for Rough Equations

Definitions and mathematical statement

A real standard fractional Brownian motion with Hurst parameter $H\in(0,1)$ is a centered Gaussian process $(B_t^H)_{t\geq0}$ with covariance \[ {\mathbb{E}}[B_s^H B_t^H]=\frac12\left(s^{2H}+t^{2H}-|t-s|^{2H}\right). \] This definition fixes terminology only: the source does not specify the admissible Hurst parameters, the dimension or covariance structure of the driving process, or the interpretation of the stochastic equation.

For an equation in the intended class, write $X_t^x\in{\mathbb{R}}^n$ for its solution started at $x$, whenever such a solution is defined. If its time-$t$ law admits a Lebesgue density, denote that density by $p_t(x,y)$, so that \[ \mathbb P(X_t^x\in A)=\int_A p_t(x,y)\,{\,\mathrm{d}} y \] for every Borel set $A\subseteq{\mathbb{R}}^n$. This notation refers to the law from the specified initial condition and does not assume a Markov transition semigroup.

A uniform density estimate means a bound valid throughout a specified set of time, initial-point and terminal-point parameters, with bounding constants independent of those parameters on that set. A short-time asymptotic expansion describes the behavior of the density as $t\downarrow0$, with an explicitly specified normalization, expansion scale, remainder and parameter regime. The source supplies none of these specifications, and it does not define the intended meaning of sharpness.

Research question supplied by the record. How do sharp uniform density estimates for stochastic differential equations driven by fractional Brownian motion relate to existing short-time asymptotic expansions for elliptic and hypoelliptic stochastic differential equations?

More precisely, the requested comparison concerns the information that the estimates and expansions provide about the same densities in regimes where their hypotheses overlap. The source does not state a particular implication, equivalence, comparison formula or remainder estimate to be proved. It also does not specify the coefficient assumptions, the ellipticity or hypoellipticity conditions, or whether the expansions being compared use the same driving process. Consequently, this is a research question requiring further specification, rather than a fully quantified conjecture. These missing qualifications cannot be supplied from the record alone.

Short English statement

The problem asks how uniform bounds for densities of equations driven by fractional Brownian motion connect with short-time density expansions in elliptic and hypoelliptic settings. The supplied record does not specify the equations, hypotheses, uniformity regimes or precise relationship sought, so a unique theorem statement cannot yet be extracted. Its partially solved label is supplied dated curation, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30004574$, title "Density Estimates and Short-Time Asymptotics for Rough Equations," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• Supplied report citation: New Directions in Rough Path Theory (2021), Oberwolfach Reports, pp. 1955--2019, DOI 10.4171/owr/2020/40. https://doi.org/10.4171/owr/2020/40. The record associates the problem with proposed year 2020; this is retained separately from the citation's year 2021.
• Geng, Ouyang and Tindel, "density estimates" (label supplied by the registry). https://arxiv.org/abs/2007.16178.
• Geng, Ouyang and Tindel, "hypoelliptic estimates" (label supplied by the registry). https://arxiv.org/abs/2008.01229.
• Inahama and Naganuma, "short-time asymptotics" (label supplied by the registry). https://arxiv.org/abs/1902.05219.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and reports that sharp short-time density estimates and separate full asymptotic expansions are available for elliptic and hypoelliptic rough differential equations, while a general equivalence or complete relation between the theories is not established. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent verification.