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Return of 2D Brownian motion to the origin

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$(X_{z+t}-X_{z})_{t\geq 0}$ satisfies "Strong Markov Property" where $X$ is càdlag process and $z$ discrete stopping time.

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Given a stopping time $\tau,$ there is a decreasing discrete stopping times sequence which converges to $\tau$

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Proof of Strong Markov Property in Liggett

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$X$ adapted right-continuous process and stopping times $\tau_{n}\rightarrow\tau$ implies $X_{\tau_{n}}\rightarrow X_{\tau}$

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List Question

convergence towards infinity of jumping times of Levy processes

Prove that $\tau$ is a stopping time

Levy measure of borel sets away from $0$

Return of 2D Brownian motion to the origin

Convergence of stopping times and limit of a right continuous process

$(X_{z+t}-X_{z})_{t\geq 0}$ satisfies "Strong Markov Property" where $X$ is càdlag process and $z$ discrete stopping time.

Sum of probabilities equal expectations

Stopped process is $F_T$-measurable

Stopped $L^2$ - martingale is again $L^2$ integrable

Interpretation of the resolvent of Brownian motion

Existence of probability density of a functional of Brownian motion

$T_{O}=\inf\{t\geq 0:X_{t}\in O\}$ is stopping time with $O$ open set.

Given a stopping time $\tau,$ there is a decreasing discrete stopping times sequence which converges to $\tau$

Proof of Strong Markov Property in Liggett

$X$ adapted right-continuous process and stopping times $\tau_{n}\rightarrow\tau$ implies $X_{\tau_{n}}\rightarrow X_{\tau}$

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