A Remark on Walsh’s Brownian Motions

Let Z_t, 0 ≤ t ≤ 1, be a Walsh’s Brownian motion, with natural past filtration https://www.w3.org/1998/Math/MathML">F≤thttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2558.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and future filtration https://www.w3.org/1998/Math/MathML">F≥1−thttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2559.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>. We say that Z_t is past-and-future equivalent to a Brownian motion https://www.w3.org/1998/Math/MathML">B˜thttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2560.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>, if https://www.w3.org/1998/Math/MathML">F≤t≡F˜≤thttps://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2561.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and https://www.w3.org/1998/Math/MathML">F≥1−t≡F˜≥1−t,0≤t≤1https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2562.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>, where https://www.w3.org/1998/Math/MathML">F˜(⋅)https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2563.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> are the natural filtrations of https://www.w3.org/1998/Math/MathML">B˜https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332838/58f4c041-1317-4a7c-b51a-e794ca13209e/content/eq2564.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>. The object of this paper is to show that, for certain choices of Z_t, although both Z_t and Z_1−t have the Brownian (previsible) representation property, Z_t is not past-and-future equivalent to any Brownian motion.