ABSTRACT

As mentioned at the end of §1.1, the white noise https://www.w3.org/1998/Math/MathML"> B ˙ ( t ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203733813/a40443c5-6a81-47d1-8410-417833651c5b/content/eq125.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is a generalized function on an infinite dimensional space. First we shall specify such a space. A common choice is the Schwartz space https://www.w3.org/1998/Math/MathML"> S ( ℝ ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203733813/a40443c5-6a81-47d1-8410-417833651c5b/content/eq126.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of real-valued rapidly decreasing functions on https://www.w3.org/1998/Math/MathML"> ℝ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203733813/a40443c5-6a81-47d1-8410-417833651c5b/content/eq127.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . A function ξ on https://www.w3.org/1998/Math/MathML"> ℝ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203733813/a40443c5-6a81-47d1-8410-417833651c5b/content/eq128.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> is called rapidly decreasing if it is smooth and for all nonnegative integers n and k, https://www.w3.org/1998/Math/MathML"> | x n ξ ( k ) ( x ) |   →   0             as   | x |   →   ∞ . https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203733813/a40443c5-6a81-47d1-8410-417833651c5b/content/eq129.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>