Timeline for On a reflecting Brownian motion and its boundary local time

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Sep 3, 2016 at 20:22	comment	added	sharpe		Thank you for your answer. I think it may be difficult to show $P_x(L_t \ge \alpha) \ge P_{1/2}(L_t \ge \alpha)$.
Sep 3, 2016 at 20:12	vote	accept	sharpe
Sep 3, 2016 at 18:55	comment	added	Nawaf Bou-Rabee		Motivated by your question, I added a bit more detail to my answer.
Sep 3, 2016 at 18:52	history	edited	Nawaf Bou-Rabee	CC BY-SA 3.0	added 551 characters in body
Sep 3, 2016 at 8:57	comment	added	sharpe		Thank you for your kind answer. But I am interested in boundary local time. In $1$D case, boundary local time $L_t$ may be characterized by $E_{x}[L_t]=\lim_{\epsilon \to 0}u_{\epsilon}(t,x)/\epsilon$, $u_{\epsilon}(t,x)=\int_{0}^{t}1_{[0,\epsilon]}(X_s)\,ds+\int_{0}^{t}1_{[1-\epsilon,1]}(X_s)$. Futhermore, $E_{0}[L_t]=E_{1}[L_t]$ should hold and $E_{x}[L_t] \ge E_{1/2}[L_t]$ for every $x \in[0,1]$. In this case, can we show $P_{x}(L_t \ge \alpha ) \ge P_{1/2}(L_t \ge \alpha)$?
Sep 3, 2016 at 7:35	vote	accept	sharpe
Sep 3, 2016 at 7:41
Sep 2, 2016 at 22:06	history	edited	Nawaf Bou-Rabee	CC BY-SA 3.0	added 206 characters in body
Sep 2, 2016 at 22:00	history	edited	Nawaf Bou-Rabee	CC BY-SA 3.0	added 206 characters in body
Sep 2, 2016 at 19:39	history	answered	Nawaf Bou-Rabee	CC BY-SA 3.0