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By Kiyosi Ito

A scientific, self-contained remedy of the speculation of stochastic differential equations in limitless dimensional areas. incorporated is a dialogue of Schwartz areas of distributions in terms of chance thought and endless dimensional stochastic research, in addition to the random variables and stochastic procedures that take values in limitless dimensional areas.

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Also we define the HS p-norm IXI P SO for X = X a> eL p by where {en} is an ONE in d3>, p); this norm is well defined independently of the choice of the ONE. Similarly we define IYI^ for Y_= Y^eLJS). XeLp yields a unique linear random functional X on (S>p, p) whose p-norm equals ||X||P. We denote X by the same notation X if there is no possibility of confusion. We can choose an ONB on da, p) to define the HS p-norm of X. If the HS p-norm of X is finite, then X yields an jSp-variable by taking a p-regular version Z n X(e n )e n , denoted by X again, if there is no possibility of confusion, where {en} is the dual norm of {«„} and similarly for YeL u .

Suppose that E is a vector space. If then Y = {Yf,feE} is called a linear random functional on E, where it should be noted that the exceptional P-null set for (1) may depend on a, b, f and g. (1) means that Y: E —> L0(O) is linear. Suppose that E is a topological space with topology T. If then Y: E —» L0(fl) is called r-continuous. Suppose that ET = (E, T) is multi-Hilbertian. Even if Y = {Yf, /eE} is a r-continuous linear random functional on E, the sample functional Y(o>) = (Yf(co), /eE) is not always an E ^-valued random variable.

1. 3>^ is a separable Hilbert space with dual norm q'. By modifying B, on a P-null w-set for each t, we can assume that B, e3>'q for every o>eO. Observing that B, - Bs is centered Gaussian and that we obtain 42 CHAPTER 2 setting E = 3>,X = (<-s)~1/2(B,-B5) and a =2 in the next theorem. Using Kolmogorov's continuous version theorem extended to processes with values in a complete separable metric space, we can find a q'-continuous version of {B,}, denoted by {B,} again. Since q

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