On the range of Holder continuity of Brownian motion

It is known that Brownian motion is almost surely locally Holder continuous, on a range that is random, i.e. depends on the particular path. This question explores the maximal range on which Brownian motion is Holder continuous.

Let $$W$$ be a standard Brownian motion, and let $$C > 0$$ and $$0 < \alpha < \frac{1}{2}$$ be constants.

Define the parametrized family of random variables $$H_{C, \alpha}$$ by

$$H_{C, \alpha} := \sup \big \{T > 0 \, \big | \, |W_t - W_s| \leq C |t - s|^\alpha \text{ for all } t, s \in [0, T] \big \}$$

Question: What is known about the probability distribution of $$H_{C, \alpha}$$? Does it admit a density, or an expression in terms of known distributions?

• A trivial remark: this is equivalent to asking what is the distribution of the optimal Hölder constant $C_\alpha$ for the Brownian motion $W_t$, $t \in [0,1]$. No idea whether this has been studied, though. Commented Dec 10, 2021 at 8:28

Claim: Let $$0<\alpha<1/2$$ and $$0. The random variable $$H_{C,\alpha}$$ has an absolutely continuous distribution.
Proof: Let $$\{{\cal F}_t\}_{t \ge 0}$$ be the standard Brownian Filtration. Given a non-negative rational $$q$$, define the $${\cal F}_q$$-measurable random boundary functions $$\psi_q^+:[q,\infty) \to (-\infty,\infty)$$ and $$\psi_q^-:[q,\infty) \to (-\infty,\infty)$$ by $$\psi_q^+(t)=\inf_{s \le q} \; [W_s+C(t-s)^\alpha]$$ and $$\psi_q^-(t)=\sup_{s \le q} \; [W_s-C(t-s)^\alpha]\,.$$ Observe that both these functions are a.s. continuous, and $$\psi_q^+$$ is nondecreasing. For $$q \in G^+=\{q \in {\mathbb Q}: \psi_q^+(q)>W_q\} \,,$$ consider the stopping time $$H_{C, \alpha}^+(q) := \inf \big \{t \ge q \,: W_t = \psi_q^+(t)\} \,.$$ Similarly, for $$q \in G^-:=\{q \in {\mathbb Q}: \psi_q^-(q) consider the stopping time $$H_{C, \alpha}^-(q) := \inf \big \{t \ge q \,: W_t = \psi_q^-(t)\} \,.$$ Since $$W$$ is locally $$\beta$$-Holder continuous for $$\beta \in (\alpha,1/2)$$, we infer that with probability 1, we have $$H_{C,\alpha} \in \{H_{C, \alpha}^+(q) : q \in G^+\} \cup \{H_{C, \alpha}^-(q) : q \in G^-\} \,,$$ so by symmetry, it suffices to prove that for fixed $$q \in G^+$$, the stopping time $$H_{C, \alpha}^+(q)$$ has an absolutely continuous distribution. We deduce this from the next Lemma using the Markov property of $$W$$ at time $$q$$. QED
Lemma: Let $$\psi:[0,\infty) \to [r,\infty)$$ be a nondecreasing function, with $$\psi(0)=r>0$$. Denote $$H:=\inf \{t \ge 0 : W_t=\psi(t)\},$$ where the infimum of the empty set is $$\infty$$. Then for all $$t,\epsilon>0$$ and some absolute constant $$A$$, we have
$$P\Bigl(H \in [t,t+\epsilon]\Bigr) \le Ar^{-2}\epsilon \,.$$
Proof: For $$b>0$$, let $$\tau_b:=\inf \{t \ge 0 : W_t=b\}.$$ Then $$\tau_1$$ has a Levy distribution, with density bounded above by some absolute constant $$A>0$$. See e.g. https://en.wikipedia.org/wiki/First-hitting-time_model Since $$\tau_b$$ has the same law as $$b^2 \tau_1$$, the density of $$\tau_b$$ is bounded above by $$Ab^{-2}$$. Take $$b=\psi(t) \ge r$$. Then $$P\Bigl(H \in [t,t+\epsilon]\Bigr) \le P\Bigl(\tau_b \in [t,t+\epsilon]\Bigr) \le Ab^{-2}\epsilon \,,$$ and the lemma follows.