Nate Eldredge
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 18h awarded Custodian 19h comment Are rays in Carnot groups straight? @YCor: Yeah, what you say is true, but I don't think it resolves the question. For instance, in the Heisenberg group, I think it's simply true that all rays lie in $\exp(V)$. You can draw a length-minimizing geodesic from the identity to a point outside of $\exp(V)$ (its projection in the $xy$-plane looks like an arc) but if you try to extend it indefinitely, it ceases to be globally length-minimizing (specifically, after the arc has completed a full circle and started wrapping around again). So the extension won't be a ray. 1d revised Are compactly supported continuous functions dense in the Continuous functions of Sobolev space? typo fix 2d answered Are compactly supported continuous functions dense in the Continuous functions of Sobolev space? Apr 27 comment Convergence of an rcll process along a random subsequence The question is clear now, I just don't have any suggestions as to the answer. Apr 26 comment Probability theory without deductive closure Should this be tagged math-philosophy? Apr 26 comment Convergence of an rcll process along a random subsequence I don't understand your use of the word "simultaneous" in 1. The usual interpretation of "convergence as $s \to \infty$" would be: for every $\epsilon$ we have $\lim_{s \to \infty} P(d(X_s, c)>\epsilon) = 0$. That is, for every $\delta$ and every $\epsilon$ there exists $s_0$ such that for all $s > s_0$ we have $P(d(X_s, c) > \epsilon) < \delta$. But you get that for free from the assumption that $X_{s_n} \to c$ i.p. for all nonrandom sequences $s_n$. If that is not what you want, please state even more precisely where the quantifiers should go. Apr 25 comment Strong maximum principle for heat equation. Positivity of solution One way to get this (possibly overkill) is from the parabolic Harnack inequality. Apr 25 comment Convergence of an rcll process along a random subsequence Can you be very precise in defining what you have and what you want? For instance, in the first paragraph, do you have $X_{s_n} \to c$ i.p. for deterministic sequences of $s_n$ only, or random sequences? What precise statement are you looking for in #1? Note that the statement "$X_s \to c$ almost surely" does not get you #2 either: convergence along random sequences is much much stronger, and might need something like $L^\infty$ convergence. Apr 25 answered The pointwise Lipschitz-ness of a function on a dense set, implies its pointwise Lipschitz-ness everywhere? Apr 24 awarded Nice Answer Apr 22 comment The pointwise Lipschitz-ness of a function on a dense set, implies its pointwise Lipschitz-ness everywhere? What about taking the Cantor staircase function? Apr 20 comment Why the sub-Laplacian $\Delta_{sub}$ on the Heisenberg group $H^3$ is sub-elliptic but not elliptic? Didn't you already ask this question? mathoverflow.net/questions/229958/… Apr 19 comment Existence of probability distributions/measures/spaces and mathematical expectations in some functional spaces Also, surely you must know that, for example, a continuous stochastic process induces a probability measure (sometimes called its distribution or law) on the space of continuous paths? That is what Liviu is talking about. Apr 19 comment Existence of probability distributions/measures/spaces and mathematical expectations in some functional spaces In some sense asking about $\mathbb{R}^\mathbb{R}$ is particularly problematic because its topology and measurable structure are really nasty. Most interesting examples of probability measures on function spaces live on Polish vector spaces. Apr 19 comment Existence of probability distributions/measures/spaces and mathematical expectations in some functional spaces I don't think I understand your question. What exactly do you mean by a "probability distribution" and what exactly would it mean for it to be "uniform"? One common interpretation of "uniform" in vector spaces is "translation invariant", but there is no translation invariant probability measure on any vector space, not even on $\mathbb{R}$. Maybe you just want a translation invariant measure, i.e. a Lebesgue measure, but it's well known that there is no Lebesgue measure on infinite-dimensional spaces. Apr 19 revised Measurability of integrals with respect to different measures uncountable ordinal Apr 19 answered Measurability of integrals with respect to different measures Apr 19 comment The borel $\sigma-$algebra of the set of probability measures Perhaps it should be made clear that the algebra you mention needs to contain all the open sets. Certainly the indicator of an open set (or at least an open ball) can be written as a pointwise limit of continuous functions, but we are taking advantage of the metric here. Apr 19 comment Total variation, Wasserstein, and Prokhorov metrics on countably infinite discrete spaces Well, that will certainly give you a different topology than the other two, provided there exist any measures with infinite $p$th moments, which happens iff your space $X$ is unbounded. So I guess all three topologies will agree iff $X$ is bounded and has the discrete topology.