Càdlàg Solutions of Cylindrical Stable Evolution Equations

Definitions and mathematical statement

Let $H$ denote the Hilbert space in the supplied record, and consider the linear stochastic evolution equation \[ \mathrm{d}X_t=AX_t\,\mathrm{d}t+\mathrm{d}Z_t, \] where $Z$ is described as a cylindrical stable process with coefficients $(\beta_k)_k$ and stable index $\alpha$. The supplied summability assumption is \[ \sum_k\frac{\beta_k^{\alpha}}{\gamma_k}<\infty. \] The record does not define $(\gamma_k)_k$, specify its relation to $A$, give the range of $\alpha$, or specify the signs and normalization of the coefficients. These omissions must be resolved before the displayed assumption determines a precise class of equations.

An $H$-valued path $x$ on a time interval $I$ is càdlàg in the norm topology if it is right-continuous at every point where $I$ extends to the right and has a left limit in $H$ at every point where $I$ extends to the left. Thus, at the applicable points, \[ \lim_{\substack{s\downarrow t\\s\in I}}\|x(s)-x(t)\|_H=0, \qquad \lim_{\substack{s\uparrow t\\s\in I}}x(s)\ \text{exists in }H. \] For a stochastic process, having càdlàg trajectories means that this path property holds almost surely. The source does not specify the time interval or whether it asks for the given process itself to have this property or for the existence of a càdlàg modification.

Source question, with unresolved qualifications. Under the displayed summability assumption, can conditions substantially weaker than \[ \sum_k\beta_k^{\alpha}<\infty \] ensure that the $H$-valued solution has càdlàg trajectories? The expression "substantially weaker" is not quantified in the record. Neither a proposed replacement condition nor the full hypotheses on the equation, initial data, and solution concept are supplied. Consequently, this is a research question requiring further specification, rather than a fully quantified conjecture; in particular, the record does not explicitly assert that the first summability condition alone suffices.

Short English statement

The question asks whether an $H$-valued solution of a linear evolution equation driven by cylindrical stable noise can have almost surely càdlàg paths under assumptions substantially weaker than $\sum_k\beta_k^{\alpha}<\infty$, while retaining $\sum_k\beta_k^{\alpha}/\gamma_k<\infty$. The replacement assumptions and several features of the equation are unspecified, so a precise theorem cannot yet be extracted.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001098$, exact source problem code catalogue entry, title: Càdlàg Solutions of Cylindrical Stable Evolution Equations. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2008.
• Supplied report citation: Infinite Dimensional Random Dynamical Systems and Their Applications (2009), Oberwolfach Reports, pp. 2815--2874, DOI 10.4171/owr/2008/50. Original report link: https://doi.org/10.4171/owr/2008/50. The supplied background describes the reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference, labeled "Peszat--Zabczyk path regularity": https://doi.org/10.1016/j.spa.2012.10.012.
• Supplied literature reference, labeled "Bodő--Cox--Jakubowski--Riedle": https://arxiv.org/abs/2605.13727.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes it as an unquantified conjecture concerning substantially weaker assumptions for Hilbert-valued càdlàg paths. It reports sharp later results in important linear and stable-noise settings, without establishment of the broad extracted claim. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is supplied provenance and has not been independently checked here.