A direct construction of the Wiener measure on C[0, ∞)

2022/04/21 by Pakshirajan, R. P., Sreehari, M.
#60J65 60G15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2204.09896

Abstract

Our construction of the Wiener measure on C=C[0, ∞) consists in first defining a set function φ on the class of all compact sets based on certain n-dimensional normal distributions, n = 1, 2,… using the structural relation at (\refE1.2) below. This structural relation, discovered by the first author, is recorded in his book (2013) on page 130. We then define a measure μ on the Borel σ-field of subsets of C which is the Wiener measure. This is done via a similar construction of the Wiener measure on C_a=C[0, a) where a > 0 is an arbitrary real number. The traditional way is to first construct the Brownian Motion process (BMP) and then, by proving it is a measurable mapping into (C, \mathscrC_∞), call the measure induced by the BMP on C the Wiener measure. In the present paper, we define the Wiener measure directly.

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