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?

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?