# Reflected Brownian Motion

Let $Y(t)$ be a reflected Brownian motion, and $G(t)$ is the process which keeps count of number of times that $Y(t)$ has hit the X axis. How do I approach to find distribution of $G(t)$, or almost sure growth rate bound on $G(t)$.

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You will have $G(t) = \infty$ almost surely for $t>0$. But I believe what you are looking for is local time, see en.wikipedia.org/wiki/Local_time_%28mathematics%29 – Stephan Sturm Feb 28 '13 at 23:22
any suitable scaling for $G(t)$ would be fine....for example how does $\frac{G(n^{2}t)}{n}$ would behave as $n\to\infty$. – user24367 Feb 28 '13 at 23:32
Also your limit will be infinity a.s, as every term in the sequence will be infinity a.s.... If you believe that local time is not the right concept for your question, can you post more background on your problem? – Stephan Sturm Feb 28 '13 at 23:49
ok so my problem goes like this: I have a queuing system in heavy traffic with a state space collapse....$Y(t) = X(t) + G(t)$, where $Y(t)$ is reflected brownian motion and $X(t)$ is a Brownian motion and $G(t)$ causes reflection. What I want to show is that $Y(t)$ spends negligible time on the x-axis.... – user24367 Feb 28 '13 at 23:54
stated differently I want to show $Y(t)-G(t)$ and $Y(t)$ are almost same in certain sense....where $G(t)$ counts the number of times $Y(t)$ has hit the x-axis. – user24367 Feb 28 '13 at 23:59

As your reflected Brownian motion is nothing else then the absolute value of a Brownian motion you have $$\{t \in [0,T] : Y_t = 0\} = \{t \in [0,T] : \vert W_t \vert= 0\} = \{t \in [0,T] : W_t = 0\}$$ which is a Lebesgue null set (however, an uncountable one, as remarked in the comments). In point of view of distribution, $Y_t$ has the same law as the difference of the Maximum process of a Brownian motion and the Brownian itself, $\max_{0 \leq s \leq t} W_s - W_t$. For details see, e.g., Karatzas & Shreve, Brownian Motion and Stochastic Calculus, Chapter 3.6.C, p.210ff. More general, your problem is known as Skorohod problem (which should not be confused with the Skorohod embedding problem), and I believe there is a fair amount of queueing theoretic literature about.