From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Table 2

List of notations.

𝑎,𝑏,𝑐,𝐾,𝑁,𝜅𝑁,𝑝𝑘	Given parameters
𝜃𝑗,𝜑𝑗	Roots of ±1
𝜉𝑘,𝑆𝑛,𝑋𝑡,𝑋^𝜀𝑡	Pseudorandom variables, pseudoprocesses
𝒢𝑠,𝒢𝑋	Infinitesimal generators
𝜎^−𝑎,𝜎^+𝑏,𝜎𝑎𝑏,𝑆^−𝑎,𝑆^+𝑏,𝑆𝑎𝑏	Overshooting times and locations related to pseudorandom walk
𝜏^−𝑎,𝜏^+𝑏,𝜏𝑎𝑏,𝑋^−𝑎,𝑋^+𝑏,𝑋𝑎𝑏	Overshooting times and locations related to pseudo-Brownian motion
𝜏𝑎^𝜀−,𝜏𝑏^𝜀+,𝜏^𝜀𝑎𝑏,𝑋𝑎^𝜀−,𝑋𝑏^𝜀+,𝑋^𝜀𝑎𝑏
Δ,Δ^+,Δ^−	Discrete Laplacian, finite difference operators
𝐴𝑎1,…,𝑎𝑁,𝐴𝑘𝑎1,…,𝑎𝑁𝛼1,…,𝛼𝑁	Vandermonde-like determinants
𝑈𝑢1,…,𝑢2𝑁,𝑈ℓ𝑢1,…,𝑢2𝑁
𝑈𝑘ℓ𝑢1,…,𝑢𝑘−1,𝑢𝑘+1,…,𝑢2𝑁
𝑈𝑝𝑞𝑟𝑢1,…,𝑢𝑝+𝑟,𝑈𝑝𝑞𝑟,ℓ𝑢1,…,𝑢𝑝+𝑟𝛼1,…,𝛼𝑝+𝑟
𝑈𝑝𝑞𝑟,𝑘ℓ𝑢1,…,𝑢𝑘−1,𝑢𝑘+1,…,𝑢𝑝+𝑟
𝑉𝑣1,…,𝑣𝑁,𝑉ℓ𝑣1,…,𝑣𝑁
𝑉𝑘ℓ𝑣1,…,𝑣𝑘−1,𝑣𝑘+1,…,𝑣𝑁
𝑈(𝑧),𝑈𝑘ℓ(𝑧),𝑉(𝑧),𝑉𝑘(𝑧),𝐷(𝑧),𝐷𝑘(𝑧,𝜁)
𝐃(𝜆),𝐃^+𝑘(𝜆,𝜇),𝐃^−𝑘(𝜆,𝜇)
𝒮𝑝𝑞𝑟,𝒮𝑝𝑞𝑟,𝑘ℓ	Other determinants
𝐺𝑘(𝑧),𝐺(𝑧,𝜁),𝐻^−𝑎,ℓ(𝑧),𝐻^+𝑏,ℓ(𝑧),𝐻𝑎𝑏,ℓ(𝑧)	Generating functions
𝐿𝑘(𝑧,𝜁),𝐿𝑘(𝑧,𝜁),𝑃(𝑧,𝜁),𝑃𝑘(𝑧,𝜁)	Polynomials
𝑃^+𝑏,𝑗(𝑥),𝑃𝑎𝑏,𝑗(𝑥),𝐾𝑚(𝑥),𝐾𝑚(𝑥)
𝐼^+𝑎𝑏,𝑗,𝐼^−𝑎𝑏,𝑗,𝐈^+𝑎𝑏,𝑗,𝐈^−𝑎𝑏,𝑗	Integrals and sums
𝒟^+,𝒟^−,ℰ	Sets
𝑀1,𝑀∞	Bounds
𝑎𝑗(𝑧),𝑏𝑗(𝑧),𝑢𝑗(𝑧),𝑣𝑗(𝑧),𝑤(𝑧)	Miscellaneous
𝛼𝑗(𝑧),𝛼𝑗(𝑧),𝜖𝑗,𝜀𝑗(𝑧),u𝑗(𝑧),v𝑗(𝑧)
𝐴𝑗(𝑧),𝐵𝑗(𝑧),𝑀𝑘(𝑧),𝑅𝑗(𝑧),𝑎𝜀,𝑏𝜀
𝑠ℓ(𝑧),𝑝𝑘(𝑧),𝑠^+𝑘,ℓ(𝑧),𝑝^+𝑘(𝑧),𝑠^−𝑘,ℓ(𝑧),𝑝^−𝑘(𝑧)	Sums and products
𝑠𝑘,𝑚𝑢1,…,𝑢𝑘−1,𝑢𝑘+1,…,𝑢𝑝+𝑟
𝐀,𝐁,𝐂𝑖^(𝑘),𝐋,𝐔,𝐔^′,𝐔^′′	Matrices
𝑉^→𝑚	Vectors