  We construct and study properties of an infinite dimensional analog of
Kahane's theory of Gaussian multiplicative chaos \cite{K85}. Namely, we
consider a random field defined with respect to space-time white noise
integrated w.r.t. Brownian paths in $d\geq 3$ and construct the infinite volume
limit of the normalized exponential of this field, weighted w.r.t. the Wiener
measure, in the entire weak disorder (subcritical) regime. Moreover, we
characterize this infinite volume measure, which we call the {\it subcritical
GMC on the Wiener space}, w.r.t. the mollification scheme in the sense of
Shamov \cite{S14} and determine its support by identifying its {\it thick
points}. This in turn implies that almost surely, the subcritical GMC on the
Wiener space is singular w.r.t. the Wiener measure. We also prove, in the
subcritical regime, existence of negative and positive ($L^p$ for $p>1$)
moments of the total mass of the limiting GMC, and deduce its H\"older
exponents (small ball probabilities) explicitly. While the uniform H\"older
exponent (the upper bound) and the pointwise scaling exponent (the lower bound)
differ for a fixed disorder, we show that, as the disorder goes to zero, the
two exponents agree, coinciding with the scaling exponent of the Wiener
measure.
