# Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?

This question is related to the following question Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?

A couple of authors have observed that composing a random walk an infinite number of times gives an asymptotic time invariant density. The original reference is "Fractional diffusion equations and processes with randomly varying time" Enzo Orsingher, Luisa Beghin http://arxiv.org/abs/1102.4729. Roughly speaking I am curious if this notion of iterating a random walk infinitely often and that fact that this iteration converges to some fixed density imply the existence of an infinite dimensional measure.

The line (3.14) of Orsingher and Beghins paper reads for $t > 0$ and $x \in \mathbb{R}$ $$(*) \qquad\lim_{n \rightarrow \infty} 2^{n} \int_{0}^{\infty} \ldots \int_{0}^{\infty} \frac{e^{\frac{-x^2}{2z_1}}}{\sqrt{2 \pi z_1}} \frac{e^{\frac{-{z_1}^2}{2z_2}}}{\sqrt{2 \pi z_2}} \ldots \frac{e^{\frac{-{z_n}^2}{2t}}}{\sqrt{2 \pi t}} \mathrm{d}z_1 \ldots \mathrm{d}z_n = e^{-2 |x|}$$

Since (*) is very similar to normalizations carried out in computing the propagator in quantum mechanics or just the formulations of path integrals in general I was curious about how rigorous we could make the following statements. Also the way I have seen these type of constructions carried out is either via the the standard definition of Wiener measure on finite dimensional "cylinder sets" or some application of Bochner-Milnos combined with a normalization of Gaussian measure on $\mathbb{R}^n$. So I am wondering if this is something contained within the construction of wiener measure or other infinite dimensional measures on Banach spaces.

1) Does (*) imply the existence of a measure on the space of continuous functions with finite support (paths)?

2) If such a measure does exist is it equivalent to Wiener measure?

• 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... – Alexander Chervov Jan 31 '12 at 12:18
• @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. – jzadeh Jan 31 '12 at 18:57
• 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. – Alexander Chervov Jan 31 '12 at 20:05
• @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. – jzadeh Feb 3 '12 at 2:56