Persistence Exponents of Multiply Integrated Brownian Motion

Definitions and mathematical statement

Let $(B_t)_{t\geq0}$ be standard real Brownian motion with $B_0=0$. Define the iterated integrals by \[ A_t^{(1)}=\int_0^t B_s\,{\,\mathrm{d}} s, \qquad A_t^{(d)}=\int_0^t A_s^{(d-1)}\,{\,\mathrm{d}} s \quad(d\geq2, \ t\geq0). \] Here $d$ denotes the number of integrations in the supplied reconstruction, rather than a specified spatial dimension. In particular, $A_0^{(d)}=0$ for every $d\geq1$.

The supplied problem asks for the persistence exponent $\theta_d$ governing the long-time probability that $A^{(d)}$ stays positive, for every integer $d\geq2$. For a specified positive persistence probability $p_d(T)$, a logarithmic power-law exponent would be defined by \[ \theta_d=-\lim_{T\to\infty}\frac{\log p_d(T)}{\log T}, \] provided this limit exists. This formula describes a possible precise exponent convention; the supplied record does not specify the persistence event defining $p_d(T)$, its time interval, conditioning, or boundary convention.

Consequently, the requested computation remains qualified by an unresolved formulation issue. Strict positivity on an interval containing time zero is impossible because $A_0^{(d)}=0$. A suitable observation interval, conditioning, or alternative initial or boundary convention must be identified from the source before a nondegenerate persistence problem can be stated. None is selected here, and existence of the exponent is not asserted.

Short English statement

The problem asks for the persistence exponents of repeatedly integrated Brownian motion for every integration order $d\geq2$. The supplied record leaves the positivity event and boundary conventions unspecified, so the precise probability whose decay is to be measured requires clarification. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30002931$, source code catalogue entry, title Persistence Exponents of Multiply Integrated Brownian Motion. Project registry paths are data/unsolvedmath/problems.json and catalog/problems.json; their contents were not independently inspected.
• Supplied report citation: Interplay of Analysis and Probability in Applied Mathematics (2016), Oberwolfach Reports, pp. 1989--2064, DOI 10.4171/owr/2015/35. https://doi.org/10.4171/owr/2015/35. The record assigns proposed year 2015 and supplies no individual proposer.
• Supplied literature reference: Newman--Stein, blocking and persistence. https://arxiv.org/abs/cond-mat/9904208.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that only the one-dimensional exponent is known exactly, while higher-dimensional exponents remain open despite numerical and scaling results. This is supplied dated curation, not an independently verified assessment; its dimensional terminology requires reconciliation with the integration-order parameter above.
• The registry labels the statement corrected_verified and reports that the official report was checked to restore the recursive definition and identify the exponent. These are supplied provenance claims; no source visit or independent verification is claimed here.