Sharp Density Estimates for General Gaussian Rough Paths

Definitions and mathematical statement

A Gaussian process is a family of random variables whose every finite subfamily has a joint Gaussian distribution; vector-valued versions are defined by requiring all finite collections of scalar coordinates to be jointly Gaussian. For terminology, a standard real fractional Brownian motion with Hurst parameter $H\in(0,1)$ is a centered Gaussian process $(B_t^H)_{t\geq0}$ with covariance \[ {\mathbb{E}}[B_s^H B_t^H]=\frac12\left(s^{2H}+t^{2H}-|t-s|^{2H}\right),\qquad s,t\geq0. \] This definition fixes terminology only: the supplied record specifies neither a Hurst-parameter range for the intended estimates nor the dimension or normalization of the driving process. For an ${\mathbb{R}}^d$-valued random variable $Y$, a Lebesgue density is a measurable function $p:{\mathbb{R}}^d\to[0,\infty)$ satisfying \[ \mathbb P(Y\in A)=\int_A p(y)\,{\,\mathrm{d}} y \quad\text{for every Borel set }A\subseteq{\mathbb{R}}^d. \] The record does not identify the random variable whose density is to be estimated.

Source question. Can the sharp density-estimate results be extended from fractional Brownian motion to more general Gaussian processes?

Unresolved formulation. The title places the question in the setting of Gaussian rough paths, but the supplied statement does not specify the fractional-Brownian-motion estimates to be extended, their mathematical form, or their hypotheses. It also does not identify whether the density concerns the driving process, a solution of a rough differential equation, or another functional. No equation, coefficient assumptions, nondegeneracy condition, covariance regularity, rough-path lift, parameter range, time regime, or meaning of sharpness is supplied. Consequently, neither a quantified class of admissible Gaussian processes nor a precise extension theorem can be stated from this record alone. These missing qualifications remain part of the draft and must not be replaced by an assertion covering every Gaussian process.

Short English statement

The question asks whether sharp density estimates known for fractional Brownian motion extend to a broader class of Gaussian processes, in the rough-path setting indicated by the title. The baseline estimates, density target, and admissible class are unspecified, so a precise theorem formulation requires further source information. The supplied status is partially solved; this is a dated curation label, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, supplied record with permanent ID $30004576$, title Sharp Density Estimates for General Gaussian Rough Paths, source problem code catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied report citation: New Directions in Rough Path Theory (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/40, pp. 1955--2019. https://doi.org/10.4171/owr/2020/40. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page is supplied.
• Supplied literature-source label: "Baudoin, Hairer and others, Gaussian rough differential equations." https://annals.math.princeton.edu/2010/171-3/p15. The label and attribution are reproduced as supplied; their bibliographic accuracy and relevance to the intended estimates have not been independently verified.
• Supplied literature-source label: "Varadhan, density estimates for fractional Brownian motion." https://doi.org/10.1016/j.spa.2014.09.012. The label and attribution are reproduced as supplied; their bibliographic accuracy and relevance to the intended estimates have not been independently verified.
• Supplied literature-source label: "OWR 2020/40 report." https://doi.org/10.4171/owr/2020/40.
• Supplied literature assessment, checked at the exact recorded date 2026-08-22: the status is partially solved. The assessment states that general Gaussian rough-path density theorems cover processes beyond fractional Brownian motion under structural hypotheses, while the precise sharp estimates and unrestricted extension asked for remain unresolved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.