On hitting times of conditioned diffusions

I have a question of a conditioned diffusion processes. This question is somewhat related to an argument which appears in this paper: P.

Let $ D=\{z=(x,y) \in \mathbb{R}^2 \mid |y|<1\}$ and $ K=\{(x,y) \in D \mid x<1\}$ . We denote $ X=(X_t,P_z)$ by the absorbing Brownian motion on $ D$ conditioned to hit $ K$ . We let $ T_{K}=\inf\{t \ge 0 \mid X_t \in K\}$ .

My question

We set $ S=\inf\{t \ge 0 \mid \text{ the second coordinate of }X_{t}=-1/2\}$ .

Can we prove the following?: \begin{align*} (1) \quad \lim_{x \to +\infty}\inf_{z=(x,y) \in D \ \text{ with }-1/2<y<1}P_{z}(T_{K}<S)=0. \end{align*}

A claim similar to (1) should hold for more general conditioned diffusions on $ D$ . Can we prove (1) with some universal argument? I would like to know whether similar claims to (1) can be proved for more general conditioned diffusions.

Is there a pattern to inform the user to enter the full search string before hitting the search button?

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Any suggestions of patterns used to call out that a search fields requires the legit value before the user goes about searching would be greatly appreciated. Thanks in advance.

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Probability of Random walk with 2 absorbing walls hitting one wall during N steps

Assume a i.i.d. one dimensional random walk S with symmetrical probabilities of 1/2 for a unit plus or minus step. Start is at S = 0 and absorbing walls are at S = – B and S = + A. What is the probability Pa that the particle will be absorbed at S = A before reaching S = – B or escaping unabsorbed during or by N steps. Assume N > 2A + B and assume if necessary A > B.

What is the probability Pu that the particle hits neither wall in N steps?

The solution for N -> infinity or until particle is absorbed is Pa = b / a + b. But for finite N this is solved tediously by counting outcomes for specific cases. Can anyone find a closed solution for Pa in terms of A, B and N?

This is called Gambler’s Ruin.

Hitting probability of a transient diffusion process

I have a question about properties of transient diffusion process.

In the case of $ d$ -dimensional Brownian motion $ B=(B_t,P_x)$ ($ d \ge 3$ ), we can prove that \begin{align} (1)&\quad 0<P_{x}(\sigma_{K}<\infty)\quad \text{ for any }x \in \mathbb{R}^d,\ (2)&\quad \lim_{|x| \to \infty,\ x \in \mathbb{R}^d}P_{x}(\sigma_{K}<\infty)=0. \end{align} Here, $ K$ is a compact subset of $ \mathbb{R}^d$ with positive Lebesgue measure and $ \sigma_{K}=\inf\{t \ge0 \mid X_t \in K\}$ . $ |\cdot|$ is the Eucledean metric. Note that (2) follows from the heat kernel estimate of the Brownian motion and the strong Markov property.

My question

We consider the next diffusion $ X=(X_t,P_x)$ on $ \mathbb{R}^d$ : \begin{equation*} X_t=x+\int_{0}^{t}a(X_s)\,dB_s+\int_{0}^{t}b(X_s)\,ds, \end{equation*} where $ a$ and $ b$ are bounded continuous functions on $ \mathbb{R}^d$ and $ B$ is the $ d$ -dim Brownian motion starting from the origin. We assume that $ X$ is transient.

Is there conditions on $ a$ and $ b$ such that $ \inf_{x \in \mathbb{R}^d}P_{x}(T_{K}<\infty)>0?$

Here, $ T_K=\inf\{t>0 \mid X_t \in K\}$ .

Does hitting a creature with a magical creature counts as magical damage?

Half-Orc Barbarian Conan has been Enlarged, making him Large. During a fight against some Couatls, he managed to grappled one and beat it to death.

Now, having already something (the body of the dead Couatl) in his hands and being a tad affected by its current rage, Conan decides to strike a second Couatl with the first one. Laughs all around the table as the DM rules that he can indeed wield the corpse as an improvised weapon (bludgeoning), given the situation.

A Couatl is immune to non-magical bludgeoning, among other things. But given the fact that the first Couatl is a magical creature and has the Magic Weapons feature, does the damage counts as magical damage?

Magic Weapons: The couatl’s weapon attacks are magical.

If yes, would any “magical creatures” work for this purpose or only ones with the Magic Weapons feature?

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