Let $W$ be a standard Brownian motion. Define the upper left Dini derivative $D^-W$ by


$$D^-W\_t := \limsup\_{h \to 0^-} \frac{W\_{t+h} - W\_t}{h}.$$


Fix $a > 0$, and define the stopping time $\tau$ by


$$\tau := \inf \{t > 0 \, | \, W\_t \geq a\}$$


**Question:** Is it true that $D^- W\_\tau = +\infty$, almost surely?



Upper left Dini derivative of Brownian motion at a hitting time
The derivative is indeed infinite, basically because Brownian motion does not have points that are too regular in a sense.


A *slow point* for a realisation of Brownian motion is a time $t$ such that
$$ \sup\_{h\to0^+}\frac{|W\_{t+h}-W\_t|}{h^{1/2}}<\infty. $$
For the purposes of this answer, let me call a *very slow point* a time $t$ such that the above supremum is (strictly) less than 1.



> 
> **Theorem** (Davis) (Greenwood & Perkins)**.**
> 
> 
> Almost surely, the set of very slow points is empty.
> 
> 
> 


It follows by time reflection that at all points, either the upper or lower left Dini derivative of Brownian motion is infinite. Since the lower left Dini derivative is zero at $\tau$, then the upper one must be infinite.




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*A conditioned limit theorem for random walk and Brownian local time on square root boundaries,* P. Greenwood and E. A. Perkins (1983). [Link.](https://doi.org/10.1214/aop/1176993594)


*On Brownian slow points,* B. Davis (1983). [Link.](https://doi.org/10.1007/BF00532967)


