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?



On the range of Holder continuity of Brownian motion
**Claim:** Let $0<\alpha<1/2$ and $0<C<\infty$. 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 $$q \in G^-:=\{q \in {\mathbb Q}: \psi\_q^-(q)<W\_q\} \,,$$ 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.


