What exactly is the relation between the Wiener process and Wiener measure?

Question

The Wiener measure is (in the classical sense) a Gaussian measure on the Banach space $C[0,1]:=\{f:[0,1] \to \mathbb{R} \mid f\text{ is continuous and } f(0)=1\}$.

The Wiener process is a stochastic process whose definition can be found in any textbook. In any text, the stochastic integrals or stochastic differential equations use the notation $dW$ frequently, which should denote the Wiener measure, I suppose.

However, I cannot figure out the exact relation between the Wiener 'process' and 'measure'. Wikipedia says that the Wiener process induces the Wiener measure but what exactly does that mean?

I am afraid this question might not belong to MO but I ask here. Could anyone please clarify and help me understand?

Answer 1

The Wiener measure $w$ is the distribution of the Wiener process/random function $W$ on $C[0,1]$; that is, $$P(W\in A)=w(A)$$ for all Borel sets $A\subseteq C[0,1]$. Here "Borel sets" can be replaced by "open sets" or "closed sets".

Equivalently, $$Ef(W)=\int_{C[0,1]}f\,dw$$ for all (say) nonnegative Borel-measurable functions $f\colon C[0,1]\to\mathbb R$. Here "nonnegative Borel-measurable" can be replaced by (say) "bounded continuous".

Answer 2

We have to differentiate between the probability measure $w$ as a measure on the space $(C[0,1],\mathcal{B}(C[0,1]))$, the Wiener measure and the concept of an Ito integral $\int_{[0,t]} X_s dW_s$, $0 \leq t \leq 1$, where $W$ is the usual Wiener process. The Wiener measure is only a special probability measure. Of much more interest is of course the Ito integral. This is definitely not a special case of the integral in the answer of Iosif Pinelis. It requires different concepts. There are many good introductions to the Ito integral, f.i. Karatzas/Shreve (1988), Brownian Motion and Stochastic Calculus, to mention only one.