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 HT(ω ) H T ( ω ) is a random field defined w.r.t. space-time white noise Ḃ B ˙ and integrated w.r.t. Brownian paths in d≥ 3 d ≥ 3 , we consider the renormalized exponential μ γ ,T μ γ , T , weighted w.r.t. the Wiener measure P₀(dω ) P 0 ( d ω ) . We construct the almost sure limit μ _γ = lim T→ ∞ μ γ ,T μ γ = lim T → ∞ μ γ , T in the entire weak disorder (subcritical) regime and call it subcritical GMC on the Wiener space . We show that aligned μ _γ \ω : lim T→ ∞ HT(ω )/T(φ φ )(0) ≠ γ \=0 almost surely, aligned μ γ { ω : lim T → ∞ H T ( ω ) T ( ϕ ⋆ ϕ ) ( 0 ) ≠ γ } = 0 almost surely, meaning that μ _γ μ γ is supported almost surely only on γ γ - thick paths , and consequently, the normalized version is singular w.r.t. the Wiener measure. We then characterize uniquely the limit μ _γ μ γ w.r.t. the mollification scheme φ ϕ in the sense of Shamov (J Funct Anal 270:3224–3261, 2016) – we show that the law of Ḃ B ˙ under the random rooted measure Qμ _γ(dḂdω )= μ _γ (dω ,Ḃ)P(dḂ) Q μ γ ( d B ˙ d ω ) = μ γ ( d ω , B ˙ ) P ( d B ˙ ) is the same as the law of the distribution f↦ Ḃ(f)+ γ ∫ ₀^∞ ∫ Rᵈ f(s,y) φ (ω ₛ-y) ds dy f ↦ B ˙ ( f ) + γ ∫ 0 ∞ ∫ R d f ( s , y ) ϕ ( ω s - y ) d s d y under P ⊗ P₀ P ⊗ P 0 . We then determine the fractal properties of the measure around γ γ -thick paths: $$-C_2 ≤ ε ↓ 0 ε ^2 log {{μ }}_γ ( ω < ε ) ≤ ε ↓ 0 _η ε ^2 log {{μ }}_γ ( ω -η < ε ) ≤ -C_1- C 2 ≤ lim inf ε ↓ 0 ε 2 log μ ^ γ ( ‖ ω ‖ < ε ) ≤ lim sup ε ↓ 0 sup η ε 2 log μ ^ γ ( ‖ ω - η ‖ < ε ) ≤ - C 1 w.r.t a weighted norm · $$ ‖ · ‖ . Here $$C_1>0$$ C 1 > 0 and $$C_2<∞ $$ C 2 < ∞ are the uniform upper (resp. pointwise lower) Hölder 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 Lᵖ L p ( $$p>1$$ p > 1 ) moments for the total mass of μ _γ μ γ in the weak disorder regime.