I am trying to formulate the measure of event

$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,

where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise disjoint compact non-empty sets. I am looking for $\mu_{W}(E)$. Any solutions?

One guess is:

$\mu_{W}(E)=\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{B} \int_{A} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}+ $

$+\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{A} \int_{B} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}-$

$-\int_{0}^{\infty}\int_{C} \int_{\mathbb{R}^{d}/C} p(x_{1},x_{2},t_{1}) dx_{1}dx_{2}dt_{1}$

The first term is hitting A and then B, the second is the converse and the last term is hitting C.

The inner integral is the startpoint of the Brownian motion. For the first two terms ,the outer integral has $t_{1}$ as the startpoint, to denote the transition from A to B.

thanks Ilya. Still I would appreciate if someone can give a precise answer.

I am trying to formulate the measure of event

$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,

where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise disjoint compact non-empty sets.

Q1: I am looking for $\mu_{W}(E)$. Any solutions?

One guess is:

$\mu_{W}(E)=\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{B} \int_{A} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}+ $

$+\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{A} \int_{B} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}-$

$-\int_{0}^{\infty}\int_{C} \int_{\mathbb{R}^{d}/C} p(x_{1},x_{2},t_{1}) dx_{1}dx_{2}dt_{1}$

The first term is hitting A and then B, the second is the converse and the last term is hitting C.

The inner integral is the startpoint of the Brownian motion. For the first two terms ,the outer integral has $t_{1}$ as the startpoint, to denote the transition from A to B.

Q2:Expressing hitting set A but not C, in terms of their individual hitting probabilities $P(H(A))$ and $P(H(C))$.

I guess $P_{\mathbb{R}^{d}/\bar{C}}(H(A))-P_{\partial A}(H(C))$.

thanks Ilya. Still I would appreciate if someone can give a precise answer.

I am trying to formulate the measure of event

$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,

where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise disjoint compact non-empty sets. I am looking for $\mu_{W}(E)$. Any ideas?

One guess is:

$\mu_{W}(E)=\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{B} \int_{A} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}+ $

$+\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{A} \int_{B} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}-$

$-\int_{0}^{\infty}\int_{C} \int_{\mathbb{R}^{d}/C} p(x_{1},x_{2},t_{1}) dx_{1}dx_{2}dt_{1}$

The first term is hitting A and then B, the second is the converse and the last term is hitting C.

The inner integral is the startpoint of the Brownian motion. For the first two terms ,the outer integral has $t_{1}$ as the startpoint, to denote the transition from A to B.

thanks Ilya. Still I would appreciate if someone can give a precise answer.

I am trying to formulate the measure of event

$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,

where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise disjoint compact non-empty sets. I am looking for $\mu_{W}(E)$. Any solutions?

One guess is:

$\mu_{W}(E)=\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{B} \int_{A} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}+ $

$+\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{A} \int_{B} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}-$

$-\int_{0}^{\infty}\int_{C} \int_{\mathbb{R}^{d}/C} p(x_{1},x_{2},t_{1}) dx_{1}dx_{2}dt_{1}$

The first term is hitting A and then B, the second is the converse and the last term is hitting C.

The inner integral is the startpoint of the Brownian motion. For the first two terms ,the outer integral has $t_{1}$ as the startpoint, to denote the transition from A to B.

thanks Ilya. Still I would appreciate if someone can give a precise answer.

I am trying to formulate the measure of event

$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,

where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise disjoint compact non-empty sets. I am looking for $\mu_{W}(E)$. Any ideas?

One guess is:

$\mu_{W}(E)=\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{B} \int_{A} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}+ $

$+\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{A} \int_{B} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}-$

$-\int_{0}^{\infty}\int_{C} \int_{\mathbb{R}^{d}/C} p(x_{1},x_{2},t_{1}) dx_{1}dx_{2}dt_{1}$

The first term is hitting A and then B, the second is the converse and the last term is hitting C.

The inner integral is the startpoint of the Brownian motion. For the first two terms ,the outer integral has $t_{1}$ as the startpoint, to denote the transition from A to B.

I am trying to formulate the measure of event

$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,

where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise disjoint compact non-empty sets. I am looking for $\mu_{W}(E)$. Any ideas?

One guess is:

$\mu_{W}(E)=\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{B} \int_{A} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}+ $

$+\int_{t_{1}}^{\infty}\int_{0}^{\infty} \int_{A} \int_{B} \int_{\mathbb{R}^{d}/(A\cup B)}p(x_{1},x_{2},t_{1})p(x_{2},x_{3},t_{2}) dx_{1}dx_{2}dx_{3}dt_{1}dt_{2}-$

$-\int_{0}^{\infty}\int_{C} \int_{\mathbb{R}^{d}/C} p(x_{1},x_{2},t_{1}) dx_{1}dx_{2}dt_{1}$

The first term is hitting A and then B, the second is the converse and the last term is hitting C.

The inner integral is the startpoint of the Brownian motion. For the first two terms ,the outer integral has $t_{1}$ as the startpoint, to denote the transition from A to B.

thanks Ilya. Still I would appreciate if someone can give a precise answer.