# White Noise Space and Local Time

This question follows from the answer I gave to the question "Wiener Meets Sobolev" in the MathStackExchange Forum.

I was wondering in the context of White Noise Space if the Local Time at x of a pre-Brownian motion is a notion that can be properly defined.

Best regards

PS:

I posted this very same question on MathSE but didn't received any answer, so I allow myself to post it here (as advised by a MathSE Forum member).

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Hi I have found some articles about Chaos decomposition of Local Times but it is still unclear to me how I can precisley related those kind of decomposition to the White Noise Space.(e.g. Nualart Vives "Chaos Expansions and Local Times") Regards –  The Bridge Dec 16 '11 at 11:16

I'm not sure what a pre-Brownian motion is-- my guess is that it is another name for white noise. If you want a notion of local time of white noise, my guess is that you would take some sort of formal derivative of local time for Brownian motion and then perhaps move the derivative over using some notion of integration by parts. This is vague, but then again, the question was a bit vague.

For both Brownian motion and fractional Brownian motion one space in which you can do these sorts of things is the Hida distribution space. This space is an extension of the typical L2 space in which the Wiener chaos lives (along the lines of the Nualart Vives paper you are citing). In short, typical L2 random variables have chaos decompositions, but this notion can be extended to random distributions (called Hida distributions) which is the dual of an appropriate test function space. The best reference I can think of for this is the SPDE book by Holden, Oksendal, Uboe, and Zhang. In particular, one can take a formal derivative of Brownian motion and show that is lives in this space (for example see the paper of elliot and van der Hoek in 2003).

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@ Paul Jung : Hi thank you very much for your interest. I have heard about Hida's Distribution spaces but couldn't take time to investigate them properly to see if they were the right approach to this issue. To answer your remark, yes the White Noise (slightly abusively applied to indicators functions) is the pre-Brownian motion that I mentioned in my first post. Anyway I would be ultimately interested in a representation of pre-local time that doesn't use Chaoses (if it exists) but rather use an explicit sub-family of test functions. Sorry I can't be more precise in my statement. Best regards –  The Bridge Jun 13 '12 at 6:27
hmmm... i don't know of anything. I would try discretizing by using a random walk where each step is a small Gaussian, and using an approximation of the delta distribution. A nice approximation to the delta function is again a Gaussian with very small variance. –  Paul Jung Jun 13 '12 at 16:38

Hi this post is unfortunately not to answer the question which is still open on my side but rather to try to extend it to a more general case.

Here it is,

We are given $S$ a continuous Semi-martingale. As it is (uniquely) decomposable into a continuous local martingale process $M$ and a continuous Finite Variation process $F$ (theorem 1), it is a special semi-martingale, and its decomposition into a continuous local martingale process and a compensator process (i.e. a continuous FV process) is given by $M$ and $F$.

I am interested in the way I can define explicitely such processes ($S,M,F$) through the White Noise Space Analysis.

Both references or direct answers are ok

Best regards

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