Persistence exponent for rectangular Euclidean domains

Determine whether, for the multidimensional fractional Brownian field indexed by a domain K = [0,1]^k × [-1,1]^{d-k}, the persistence probability satisfies P(sup_{t∈T^K} B_H(t) < 1) = T^{-(d-kH)+o(1)} as T → ∞.

Background

The paper reviews known persistence exponents for multidimensional fractional Brownian fields when the origin lies in the interior of a bounded domain and when it lies on a smooth boundary. These results suggest that the exponent may depend on how many coordinate directions place the origin on an edge or boundary of the indexing domain.

For the rectangular domains K = [0,1]^k × [-1,1]^{d-k}, the proposed exponent d − kH interpolates between the previously known cases. The statement is attributed to a conjecture of Molchan and is identified by the authors as unresolved.