Timeline for Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Feb 3, 2012 at 2:56 | comment | added | jzadeh | @AlexanderChervov. Thanks for your ideas but I am still left with the feeling that the RHS of * can be usesd to a come up with a measure that concentrates on something different than Holder continuous paths with modulus 1/2. Furthermore * is an expression for the probability density of Iterating Brownian motion and the density is not Gaussian and its transition probabilities do not satisfy Kolmogorov-Chapman so one is led to believe the induced measure is not a so called "Gaussian Measure". I wonder if * can give some way to study the induced measure of the IBM process itself. | |
| Jan 31, 2012 at 20:05 | comment | added | Alexander Chervov | Still, I am not clear. Still I would suggest consider the function which is f(i/n) = z_i and define measure (more precisely density of measure) of this function is integrand. I think playing with this we can do what you ask. | |
| Jan 31, 2012 at 18:57 | comment | added | jzadeh | @AlexanderChervov Thanks for your comment. I see your point and so to make things a little more clear how about this: Using equation * can we construct a measure on the space of continuous functions? Equation * has generalizations given by considering iterating fractional Brownian motion and so I am curious to see what type of (if any) measures on function spaces are induced by considering iterating certain classes of random walks an infinite number of times. | |
| Jan 31, 2012 at 12:18 | comment | added | Alexander Chervov | I am not sure I understand yours question. "Does * imply" - the word "imply" can be treated quite widely :) I guess you may want to define function f(y) such that f(y_i)~z_i or ~(z_i-z_{i-1}) for y_i something like i/n and then prove that when n-> inf "full measure set" of functions are continuous. This would imply positive answer. I guess this can be done and what you get will be Wiener measure... | |
| Jan 31, 2012 at 2:05 | history | edited | jzadeh | CC BY-SA 3.0 |
edited title
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| Jan 31, 2012 at 1:36 | history | asked | jzadeh | CC BY-SA 3.0 |