Subcritical Gaussian multiplicative chaos in the Wiener space: construction, moments and volume decay

We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos (Kahane in Ann Sci Math Quebec 9(2):105-150, 1985). Namely, if H-T (omega) is a random field defined w.r.t. space-time white noise (B) over dot and integrated w.r.t. Brownian paths in d >= 3, we consider the renormalized exponential mu(gamma),T, weighted w.r.t. the Wiener measure P-0(d omega). We construct the almost sure limit mu(gamma) = lim(T ->) (infinity) (mu gamma,T) in the entire weak disorder (subcritical) regime and call it subcritical GMC on the Wiener space. We show that mu(gamma) {omega: lim(T ->) (infinity) H-T (omega)/T(phi star phi) (0) not equal gamma} = 0 almost surely, meaning that mu(gamma) is supported almost surely only on gamma-thick paths, and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit mu(gamma) w.r.t. the mollification scheme phi in the sense of Shamov (J Funct Anal 270:3224-3261, 2016) - we show that the law of (B) over dot under the random rooted measure Q(mu gamma) (d(B) over dotd omega) = mu(gamma) (d omega, (B) over dot) P(d(B) over dot) is the same as the law of the distribution f (sic) (B) over dot (f) + gamma integral(infinity)(0) integral(Rd) f (s, y)phi(omega(s) - y)dsdy under P circle times P-0. We then determine the fractal properties of the measure around. -thick paths: -C-2 <= lim inf(epsilon down arrow 0 epsilon 2) log (mu) over cap (gamma) (vertical bar vertical bar omega vertical bar vertical bar < epsilon) <= lim sup(epsilon down arrow 0) sup(eta) epsilon(2) log <(mu)over cap>(gamma) (vertical bar vertical bar omega vertical bar vertical bar - eta vertical bar vertical bar < epsilon) <= -C-1 w.r.t a weighted norm vertical bar vertical bar center dot vertical bar vertical bar. Here C-1 > 0 and C-2 < infinity are the uniform upper (resp. pointwise lower) Holder exponents which are explicit in the entire weak disorder regime. Moreover, they converge to the scaling exponent of the Wiener measure as the disorder approaches zero. Finally, we establish negative and L-p (p > 1) moments for the total mass of mu(gamma) in the weak disorder regime.