I am not a mathematican, please be patient if I ask something in a not appropriate way!
Let we suppose a Brownian motion with inital value of W(0)=0, and we look its possible realizations on time interval [0,T).
MY QUESTION: What is the probability density function (with closed form) of local time related to a given time (t) and value (x) (let we note it as L(x,t)) ?
The intuitive meaning of L(x,t): the probability density that the realization is "staying" at point x for time lasting t.
I marked "staying" because Brownian motion continuous but non differentiating function, it does not stay but cross the points. There is no probability of a given point but there is a probability density of crossings the given point on interval [0,T). Which can be computed as integral of Brownian motion with dirac delta (if I am not wrong).
I simulated the process by Matlab and based on the results it seems for me, that pdf consist of two parts.
If we look L(t,x) as a slice at a given x (I mean we slice the 2 dimensional "joint" pdf at a given x) we will get the "time profil" of L(x,t). First part is a profound density for t=0 (meaning the possibility that it never cross x on interval [0,t)). second part is similar to a half gaussian pdf (decreasing slope).
I have read articles related local time, but I did not find closed form representation of pdf of L(x,t) but I am sure that there should be, because this is relative simple problem in SDE problem. Maybe the first article of Levy in 1939 can have the formula (but I do not have read the article and I do not read French).
Thanks in advance. Tomi