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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 |
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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
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Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space?
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Jan
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asked | Does the existence of an asymtpotic density imply the existence of a measure on infinite dimensional (path) space? |
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Apr
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accepted | Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)? |
Apr
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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
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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 |
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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. |
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What structure is needed to define a Gaussian distribution on a given space?
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answered | What structure is needed to define a Gaussian distribution on a given space? |
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Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
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Mar
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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. |
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Question about a Limit of Gaussian Integrals and how it relates to Path Integration (if at all)?
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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 ** |