Nice to connect. I have a question related to your mathoverlflow response (https://mathoverflow.net/questions/59739/gaussian-processes-sample-paths-and-associated-hilbert-space).

Let X be a generic topological space and k : X x X -> R be a reproducing kernel indexed on X. Let H_k be the RKHS of functions f : X -> R associated to k. Let be the inner product of H_k. Consider a Gaussian process GP(0,k) supported on X.

In the (machine learning) GP community people often refer to Driscoll’s theorem which states that if the RKHS H_k is infinite dimanensional then any sample f ~ GP does not belong to H_k with probability 1. Suppose that H_k has a countable orthonormal basis e = {e_1, e_2, …}.

Is it possible for you expand your argument (based on cylindrical measures) to justify that for any sample f ~ GP and any basis element e_n, the quantity is well defined, despite the fact that f is not in H_k a.s.?

In various papers in the GP community people use arguments that are specific to the particular choice of kernel k. It would be very nice to have a more general justification that doesn’t depend on the particular choice of kernel k.

Many thanks,

Cris

]]>Can you please elaborate why in the proof of theorem 10 in the direction A is adapted and the equality means that is adapted.

]]>George, indeed

> “when I refer … only applies to the classical situation, ”

— but since you DID NOT objects to the J.P.’s

> “… the first coin measured …”

I think my comment was in fact addressed to both of you!

@ J.P.:

I cannot imagine any situation (with quantum bosonic coins) where what you wrote can make any sense. (It is not surprising that people prefer to use creation and annihilation operators to avoid such linguistic “black holes”.)

Could you explain what you meant in more palatable terms?

]]>Aha! NOW it is more clear! Thanks!

Note that the paragraph we discuss mentions “the first condition” twice — in different meanings! So when you say “the second condition”, IMO it is more or less mandatory to clarify as in “the second condition of the following lemma”…

]]>You wrote:

> “See Localization.”

This was indeed a very keen remark! Well, almost!

I think you wanted to (essentially) refer to the argument in the proof of Lemma 13 (https://almostsuremath.com/2009/12/23/localization/#scn_local_lem13), correct?

it was indeed what I have been missing. On the other hand, Google can find NO OTHER PLACE in your notes where you refer to this lemma, or provide the link from your reply. Given that a lot of people are going to read your notes non-linearly, IMO giving more cross-references would be VERY helpful!

> “this does help.”

— but only when one is fluent with the notions and/or when one is reading your notes IN ORDER… Given that your notes are SO GREAT even when reads them “in random order” (as in “after googling”), a few extra crosslinks would make the reading yet MORE GREAT ;–) !!! Thanks!

]]>I revisited the 1st paragraph of the end-notes in “Further Properties of the Stochastic Integral”, and NOW I can see that it INDEED (essentially) says that your notion of X-integrability “is the same as the usual one”. However, I want to stress that it is not 100%-explicitly stated, and on my first reading this (in the context of discussing the differences!) lead to my confusion (indicated above).

A lot of thanks, and I appreciate your patience with me!

]]>Sorry to be responding a million years later. Absolutely I was talking about non-commuting measurements. The comment was made while I was starting to put some ideas together for this: https://arxiv.org/abs/2104.02817

]]>That text refers to “the following lemma” which, in context, is Lemma 6. The link does explain the second condition of Lemma 6. Maybe it was not clear which lemma was being referred to?

]]>Here’s a rough argument: if is smooth and goes to zero at t=0 and for large t then . Integrating, taking expectations and removing the martingale term, which has expectation zero, then . As the density of M_t is given by the second derivative of C(t,x) wrt x, then . Using some integration by parts, . As can be any arbitrary smooth function of compact support, this gives the result.

This is just a general formula I have in my mind and have used frequently, so put it in this post without explaining. I have a rigorous proof in Lemma 3.3 here.