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I'd like to know whether the following approach to defining a Brownian sheet has been done, and where I can find the details:

  1. Define a Weiner process $A(t)$ on the (infinite dimensional) vector space C([0,1]) with the sup norm.
  2. Let $W(s,0)$ be a usual Brownian motion.
  3. Let $W(s,t)$ be the value of the path-space Weiner process $A(t)$, started at $W(s,0)$.

The difficult parts of this approach would be making the definition in step 1 and in showing that our resulting $W(s,t)$ was in fact a Brownian sheet. I'd hope to be able to recursively define Brownian "sheets" with more parameters this way, and to be able to define Brownian sheets into more general spaces this way too.

Is this approach familiar to anyone? Does anyone see any major problems with it? In a best case scenario, does anyone know any references describing this?

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