Wiener process
Conformal invariance in two dimensions

Let W (t) {\displaystyle W(t)} be a two-dimensional Wiener process, regarded as a complex-valued process with W (0) = 0 ∈ C {\displaystyle W(0)=0\in \mathbb {C} }. Let D ⊂ C {\displaystyle D\subset \mathbb {C} } be an open set containing 0, and τ D {\displaystyle \tau _{D}} be associated Markov time: τ D = inf { t ≥ 0 | W (t) ∉ D }. {\displaystyle \tau _{D}=\inf\{t\geq 0|W(t)\not \in D\}.} If f: D → C {\displaystyle f:D\to \mathbb {C} } is a holomorphic function which is not constant, such that f (0) = 0 {\displaystyle f(0)=0}, then f (W t) {\displaystyle f(W_{t})} is a time-changed Wiener process in f (D) {\displaystyle f(D)} (Lawler 2005). More precisely, the process Y (t) {\displaystyle Y(t)} is Wiener in D {\displaystyle D} with the Markov time S (t) {\displaystyle S(t)} where Y (t) = f (W (σ (t))) {\displaystyle Y(t)=f(W(\sigma (t)))} S (t) = ∫ 0 t | f ′ (W (s)) | 2 d s {\displaystyle S(t)=\int _{0}^{t}|f'(W(s))|^{2}\,ds} σ (t) = S − 1 (t): t = ∫ 0 σ (t) | f ′ (W (s)) | 2 d s. {\displaystyle \sigma (t)=S^{-1}(t):\quad t=\int _{0}^{\sigma (t)}|f'(W(s))|^{2}\,ds.}
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