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comment Duality argument to get $L^\infty-L^2$ inequality
In the last step, all we really need is that the semigroup $e^{-Ht}$ is a bounded self-adjoint operator, which follows from the fact that $H$ is self-adjoint. We don't need to use the fact that a heat kernel exists, which is quite a bit more difficult to prove.
1d
comment Example of a compact geodesic space, which is not doubling
I'm not sure of your definitions, but I think the Hilbert cube $[0,1]^{\mathbb{N}}$ may be a counterexample.
1d
comment Weighted global Holder property for Brownian motion paths
If $s=1$ and $t$ is large, the quantity in your last line is comparable to $|W_t|/t^{\alpha}$, which as you know has infinite supremum. So it can't hold in the form that you've stated.
1d
answered Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)
Feb
6
revised writing an integer as particular summation
edited tags
Feb
6
comment writing an integer as particular summation
Have you tried computing it for the first few $n$ and searching OEIS?
Feb
5
revised A question about Borel sets on the unit interval
added 31 characters in body
Feb
5
answered A question about Borel sets on the unit interval
Feb
5
comment A question about Borel sets on the unit interval
In your first paragraph, didn't you mean to say "every Baire subset $A$"? (Or equivalently "Borel")?
Feb
4
revised Hard maths on viXra?
removed closure complaint
Feb
4
comment Hard maths on viXra?
@StanleyYaoXiao: Certainly viXra has lots of garbage. But to play devil's advocate, since it was founded with the explicit intention of eschewing moderation or filtering, calling it "not trustworthy" kind of misses the point; it was never meant for anyone to trust it. It's a place where anyone can post math-ish PDFs, and readers can make of them what they will. They're upfront about this, and have never claimed to be anything else.
Feb
3
comment Brownian motion - probability of hitting an open subset of the sphere
Do note carefully that what I stated is not what you stated in the question; as @ChristianRemling notes, what you stated is false.
Feb
3
comment Brownian motion - probability of hitting an open subset of the sphere
What is true, of course, is: the probability that when the particle first hits the sphere, it does so at a point of $A$, is equal to the normalized surface measure of $A$. Because Brownian motion is invariant under rotations, its distribution at the hitting time of the sphere is a rotationally invariant measure on the sphere, which can only be surface measure. This holds for any measurable $A$, it need not be open.
Feb
2
comment How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?
Subellipticity is mostly useful as a substitute for elliptic regularity, and those definitions don't imply anything of the kind. Even if we strengthen them by adding a bracket-generating type condition, so that they do imply subellipticity in the sense of Folland, the proof of this implication is not trivial.
Feb
2
comment How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?
That talk shows two (inequivalent) definitions of "subelliptic", both of which are satisfied by the Heisenberg Laplacian, which is very easy to check. If you're using one of those definitions, then I don't understand what your question is. But those are strange definitions, since they don't lead directly to any useful analytic properties; and they're satisfied by truly degenerate operators like $\partial_x^2 + \partial_y^2$ on $\mathbb{R}^3$, or the 0 operator.
Feb
2
comment probability distribution
This site is not the place for your homework questions. Please go somewhere else.
Feb
2
comment How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?
So you mean this definition (as in Folland 1972): there exists $\epsilon > 0$ such that for each bounded open set $V$ there exists a constant $C$ such that for all $f \in C^\infty_c(V)$ we have $$\|f\|_{W^{\epsilon,2}}^2 \le C(|\langle \Delta f, f \rangle| + \|f\|_{L^2}^2)$$ If not, please state the definition you are using.
Feb
2
comment How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?
Also, for subellipticity, do you just want the "qualitative" statement "if $\Delta u \in C^\infty$ then $u \in C^\infty$", or do you want the "quantitative" subelliptic estimates on Sobolev norms?
Feb
2
comment How to prove that the Laplace oparator of the Heisenberg group is sub-elliptic?
Non-ellipticity is easy, just check that the principal symbol is degenerate (compute determinant or whatever you like). For subellipticity, are you happy with the "big gun" of Hormander's theorem, or do you want a direct argument?
Feb
2
comment Is a specific sequentially closed subset of $M([0,1])$ closed?
Readers <10K should note that "the above solution" refers to a now-deleted answer by Christian Remling; not to the answer by me, which I believe addresses the issue raised here.