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comment Progressive measurability and functional composition
Continuity alone can't be sufficient; since $f$ is allowed to depend on $\omega$, you need some condition to ensure that it doesn't look into the future. For example if $Y(t)$ is any process that isn't progressively measurable, you could simply set $f(x,t,\omega)=Y(t,\omega)$. Then $f$ is certainly continuous in $x$ (even constant), and no matter what $X$ is, we have $f(X(t), t) = Y(t)$ which is not progressively measurable.
Jan
27
comment Reference : Special case of Banach-valued function integration by parts
What exactly is the definition of $\partial_t u$ here?
Jan
26
comment Can all unit-distance graphs have their vertices at algebraic integers?
What are you assuming about the graph $G$? Suppose I take the graph whose vertices are all the complex numbers, with an edge between two vertices at unit distance: $E = \{(z,w) : |z-w| = 1\}$. That seems to be unit distance by your definition (let $f$ be the identity function), but since it has uncountably many vertices we can't embed it in the algebraic numbers.
Jan
26
comment Convergence in distribution of stochastic equation solutions
Please add a link to the Math.SE post, and likewise, edit that post to add a link to this one.
Jan
23
comment Are Carnot groups (as Carnot Caratheodory metric spaces) doubling?
Are you using Haar measure, or some other measure?
Jan
23
comment Book on Convergence Concepts in Probability without Measure Theory
This site is focused only on research level questions, so textbook recommendations for undergrads are off topic. math.stackexchange.com would be a better place. But the rigorous definitions of the various modes of convergence of random variables are all in terms of measure theory, so I don't think you can really study this subject properly without understanding measure theory. I agree with @NoahS that it wouldn't be a waste of your time at all (provided you have the necessary background now); you'll just find it much easier in three years.
Jan
22
answered convergence in distribution and convergence of moments
Jan
21
comment Equivalence of Gaussian measures
@AbdelmalekAbdesselam: Yes, you're right. I got a square root backward in my head somewhere. Fixed.
Jan
21
revised Equivalence of Gaussian measures
trace class covariance
Jan
21
comment Do constrained random walks converge weakly to the Wiener measure on the space of constrained paths (that corresponds to the heat equation)?
Your requirement (2) seems wrong; you are essentially conditioning on your process being inside $U$ at time $t$, but it seems what you want is to condition on it staying inside $U$ all the way up to time 1. I have to say that unfortunately, I think answering this question properly requires rewriting a book on Brownian motion.
Jan
21
awarded  Popular Question
Jan
20
comment Equivalence of Gaussian measures
@MartinHairer: Oops, thank you. Fixed it.
Jan
20
revised Equivalence of Gaussian measures
1 implies 2
Jan
20
revised Equivalence of Gaussian measures
tags
Jan
20
answered Equivalence of Gaussian measures
Jan
20
comment Equivalence of Gaussian measures
No, I don't think this is true. For instance, $N(0,C)$ and $N(0, 2C)$ are typically mutually singular.
Jan
19
comment Problem on convergence in probability measres
@podu: Sorry about that. For probability measures it is true. See my edit.
Jan
19
revised Problem on convergence in probability measres
added 482 characters in body
Jan
19
revised Problem on convergence in probability measres
added 74 characters in body