Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.

Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event

$$E_{\varepsilon} := \{|B_s| \geq \lambda s - \varepsilon\} \text{ for all } 0 \leq s \leq T\}.$$

Question: Can we describe the limiting distribution of $W$ conditional on $E_\varepsilon$ as $\varepsilon \to 0$? Does it converge, in law or some other weak sense to some limiting distribution?

Note that $|B_s| \sim \sqrt s$ with high probability, so this linear growth is in fact a rare event.

Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.

Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event

$$ E_T := \{|B_s| \geq \lambda s\ \text{ for all } 0 \leq s \leq T\}.$$

Question: Can we describe the distribution of $W$ on $C[0, T]$ conditional on $E_T$?

Note that $|B_s| \sim \sqrt s$ with high probability, so this linear growth is in fact a rare event on large time frames.

Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.

Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event

$$E_{\varepsilon} := \{|B_s| \geq \lambda s - \varepsilon\} \text{ for all } 0 \leq s \leq T\}.$$

Question: Can we describe the limiting distribution of $W$ conditional on $E_\varepsilon$ as $\varepsilon \to 0$? Does it converge, in law or some other weak sense to some limiting distribution?

Note that $|B_s| \sim \sqrt s$ with high probability, so this linear growth is fact a rare event.

Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.

Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event

$$E_{\varepsilon} := \{|B_s| \geq \lambda s - \varepsilon\} \text{ for all } 0 \leq s \leq T\}.$$

Question: Can we describe the limiting distribution of $W$ conditional on $E_\varepsilon$ as $\varepsilon \to 0$? Does it converge, in law or some other weak sense to some limiting distribution?

Note that $|B_s| \sim \sqrt s$ with high probability, so this linear growth is in fact a rare event.

Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.

Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event

$$E_{\varepsilon} := \{|B_s| \geq \lambda s - \varepsilon\} \text{ for all } 0 \leq s \leq T\}.$$

Question: Can we describe the limiting distribution of $W$ conditional on $E_\varepsilon$ as $\varepsilon \to 0$? Does it converge, in law or some other weak sense to some limiting distribution?

Let $W$ be a standard $d$-dimensional Brownian motion with $W_0 = 0$ almost surely.

Fix a constant $\lambda > 0$ and timeframe $T > 0$, and consider the event

$$E_{\varepsilon} := \{|B_s| \geq \lambda s - \varepsilon\} \text{ for all } 0 \leq s \leq T\}.$$

Question: Can we describe the limiting distribution of $W$ conditional on $E_\varepsilon$ as $\varepsilon \to 0$? Does it converge, in law or some other weak sense to some limiting distribution?

Note that $|B_s| \sim \sqrt s$ with high probability, so this linear growth is fact a rare event.