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Doctoral Student in Probability jzadeh@math.purdue.edu http://math.purdue.edu/~jzadeh

Sep
24
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Feb
3
comment Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
@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
comment Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
@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
revised Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
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Jan
31
asked Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
Apr
17
awarded  Nice Question
Apr
5
accepted Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
Apr
5
comment Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
Is is clear that $T$ has a fixed point because of your comments that $K$ is Hilbert-Schmidt?
Apr
5
comment Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
Thanks for your help. I am am not sure I completely understand the iteration process in terms of $T$. Dont you want to show $\lim_{n\rightarrow \infty } T^{n} \phi = \phi$?
Mar
27
comment What structure is needed to define a Gaussian distribution on a given space?
sorry is it appropriate to remove this as answer then? One last thing I can add is the construction of "brownian motion" in the free potability setting where you are working over von neumann algebras of operators as in section 1.1 of iecn.u-nancy.fr/~nourdin/4th-moment-Wigner-KNPS.pdf. Just in case you were unaware of an example of a type of Brownian motion taking values in a non commutative space.
Mar
27
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Mar
27
revised What structure is needed to define a Gaussian distribution on a given space?
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Mar
27
answered What structure is needed to define a Gaussian distribution on a given space?
Mar
26
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Mar
25
revised Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
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Mar
25
comment Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
Concerning 2) and and the comment following 3) I have been moving back and forth between analyzing the iterated density of $X_n$ and analyzing the behavior of its moment generating function and these comments really apply to the mgf.
Mar
25
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Mar
25
revised Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
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Mar
25
comment Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
The density of $X_n(t)$ is given by the iterated integral in **
Mar
25
comment Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
Yes this is true if you look at this from a probabilistic perspective you can argue by self similarity. So set $X_n(t)=B_n(B_{n-1}(...(B1(t))...))$ where $B_i$ is two-sided Brownian motion. Then the following equalities hold in distribution: $X_n(t)=t^{\frac{1}{2^n}} X_n(1)$ taking limits on both sides we see that the random variable $\lim_{n\rightarrow \infty} X_n(t)$ depends only on $X_n(1)$ (i.e. is time invariant). Other authors have made this more rigorous (there is a proof that the asymptotic density is time invariant based on method of moments)