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Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$, and $(B_t)_{t \ge 0}$ is a Brownian motion.

  1. Is the domain $\mathbb D \setminus [0, 1)$ regular?
  2. Is every simply connected domain regular?

Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$.

  1. Is the domain $\mathbb D \setminus [0, 1)$ regular?
  2. Is every simply connected domain regular?

Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$, and $(B_t)_{t \ge 0}$ is a Brownian motion.

  1. Is the domain $\mathbb D \setminus [0, 1)$ regular?
  2. Is every simply connected domain regular?
Source Link
Focus
  • 177
  • 5

Is every simply connected domain regular?

Recall that a domain $D \subseteq \mathbb C$ is called regular if for each point $x \in \partial D$, we have $\mathbf P_x\lbrack \tau_D = 0\rbrack = 1$, where $\tau_D = \inf\{t > 0 : B_t \notin D\}$.

  1. Is the domain $\mathbb D \setminus [0, 1)$ regular?
  2. Is every simply connected domain regular?