Let X=(X_t)_t≥ 0 and Y=(Y_t)_t≥ 0 be independent Markov processes taking values in a common metric measure space (E,d,mu). Define the collision-time and collision-point sets by T_X,Y:=t≥ 0:X_t=Y_t and C_X,Y:=X_t:t∈ T_X,Y. Determine exact Hausdorff gauge functions h_T and h_C for which the Hausdorff measures H^h_T(T_X,Y) and H^h_C(C_X,Y) are nontrivial, and characterize these measures, in particular in terms of the temporal and spatial marginals of the collision measure. The question is posed for broad classes of independent Markov processes and remains open in full generality.

Xiao’s open question on Hausdorff measures of collision sets

Let $X=(X_t)_{t\ge 0}$ and $Y=(Y_t)_{t\ge 0}$ be independent Markov processes taking values in a common metric measure space $(E,d,\mu)$ . Define the collision-time and collision-point sets by $T_{X,Y}:=\{t\ge 0:X_t=Y_t\}$ and $C_{X,Y}:=\{X_t:t\in T_{X,Y}\}$ . Determine exact Hausdorff gauge functions $h_T$ and $h_C$ for which the Hausdorff measures $\mathcal{H}^{h_T}(T_{X,Y})$ and $\mathcal{H}^{h_C}(C_{X,Y})$ are nontrivial, and characterize these measures, in particular in terms of the temporal and spatial marginals of the collision measure. The question is posed for broad classes of independent Markov processes and remains open in full generality.

References

Primary source

arXiv

Additional references

Exact Hausdorff measures and fine geometry of collisions of independent Markov processes — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A new paper gives conditional answers for important families of processes, but the general question remains open.

Xiao’s question asks for the exact Hausdorff measures of collision-time and collision-point sets for broad classes of independent Markov processes, and for their relationship with collision-measure marginals. The available source identifies Xiao as the proposer but gives no proposal date.

August 2026 conditional advance

On August 26, 2026, the arXiv paper Exact Hausdorff measures and fine geometry of collisions of independent Markov processes claimed local-dimension, support, logarithmic-limsup, and exact Hausdorff-measure results in a natural regime, including stable processes, diffusions on nested fractals, and stable-like jump processes. The conclusions require Ahlfors-regularity and heat-kernel hypotheses, so they constitute substantial claimed progress rather than a full solution; the claim is unverified.

Current status (as of August 2026): A conditional Hausdorff-measure advance is claimed for several important process classes, while Xiao’s question in full generality remains open and the claim is unverified.

Sources

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