Re: Spectrum, Markov, Large Deviations, and PI
From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 11/17/04
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Date: Wed, 17 Nov 2004 02:58:03 +0000 (UTC)
On 16 Nov 04 10:42:35 -0500 (EST), Osher Doctorow wrote:
>Sub-Markovian semigroups are defined and explained in "Towards an
>L^p potential theory for sub-Markovian semigroups: kernels and
>capacities," by Niels Jacob and Rene L Schilling of U. Wales and
>U. Sussex respectively, 2002 or 2003 preprint, www.maths.sussex.ac.uk
>/Staff/RLS/preprints/kern.pdf. That paper involves considerable
>functional/operator analysis which is almost presented from the
>beginning and is in itself interesting relative to PI because of
>the kernel representations for L^p sub-Markovian semigroups of form:
>1) T_t u(x) = I[u(y)pt(x,y)]dy
The paper of Jacob and Schilling referred to above emphasizes the
close relationship of its topics with the theory of Dirichlet forms
and especially the study of (r,p)-capacities throughout, and since
Probable Influence (PI) has close connections with volume via
Lebesgue measure, let's discuss this a bit.
The connection of Probability-Statistics with volume and Lebesgue
measure or measures in general is often ignored by applied people
unless it is in one of the computer-intensive fields like fractals
and chaos where it may be pursued out of conformity and recently
"publish-ability" in computer-intensive journals. This is not very
different from ignoring the A and B in ABC so to speak. Probability
may not look like a volume or area or length in rushing through
elementary or even first upper division probability or statistics
courses, but mathematicians have a tendency to introduce the deepest
factors last in college course sequences despite the common exper-
ience of pushing more and more graduate material into undergraduate
courses (the latter is mostly to prevent the gap between research
and applications from becoming impossible to narrow so to speak, in
my opinion).
What would delight applied people, if they were aware of it, is that
concrete reality does help to save the day because probability is
often applied to real things or events or processes in real space-
time that near the surface of the earth looks Euclidean or Euclidean-
like. In such places, objects like stones have volumes, as do
objects like pools of water or glasses of water, and these are
essentially Lebesgue measure. Volume, area, and length in Euclidean
and Euclidean-like space are examples of Lebesgue measure (respect-
ively 3-dimensional, 2-dimensional, one-dimensional. A point is
0-dimensional, as are discrete points even if infinitely "countable".
When probabilities are applied to bounded objects in Euclidean or
Euclidean-like spaces, we have for continuous or connected objects
like stones, pools, pieces of paper, and even sticks or pencils:
1) Probability = a (some) finite bounded measure
In fact, if you consider only objects with maximum measure 1 on some
scale (which may be as long as the known universe so to speak),
you have a type of probability which is proportional to the volume
or area or length of the object (volume if the object is 3-dimen-
sional in a 3-dimensional space, etc.), except that it becomes 0
for lower dimensional objects like 0 or 1 or 2 dimensional in a
3-dimensional space. This doesn't restrict analysis of lower
dimensional objects much because they can be studied without re-
garding them as embedded in higher dimensional space, and things
only become 0 if the lower dimensional objects are studied as
objects that are contained in 3 dimensional space for example.
So Lebesgue measure for bounded objects on Euclidean or Euclidean-
like space, and hence volumes, areas, lengths of bounded objects can
be studied as probabilities. Likewise, all of calculus and differ-
ential equations and real and complex analysis and even functional
analysis when restricted to these types of spce can be studied as
probabilities.
Not all probabilities are Lebesgue measures. For example, when
you assign a probability of 1/6 to a fair die landing on a particular
face with a particular number of dots, like 6 dots, when thrown in
the air so that no preference is given to throwing it in a particular
direction except "up", then you haven't wildly deviated from
Lebesgue measure, but what about a "loaded" die that has probability
of 3/4 of landing on the 6-dot face? That's not Lebesgue measure
even though the other faces have 1/4 total probability.
Capacity-dimension in fractals makes contact with the above because
in Euclidean spaces, capacity dimension on closed bounded (that is
to say, compact) sets equals Hausdorff dimension, and the latter
is relatively easy to give examples of. For any real number r, if
you decrease the size of an object in D Euclidean dimensions by
1/r in each direction of space (see www.vanderbilt.edu/AnS/psychology
/cogsci/chaos/workshop/Fractals.html for more details), the length
or area or volume depending respectively on whether the object and
space are together regarded as 1, 2, or 3 dimensional (D = 1, 2, or
3), decreases to M = r^D times the original length, area, or volume
respectively, and solving for D gives D = log(M)/log(r) so for
fractals which correspond to "fractional dimensions in between
integer dimensions" in ordinary language, D can be a non-integer and
is the Hausdorff dimension. The a-capacity itself is
defined as:
2) [inf[I[I[//x - y//^(a-n) dm(x)dm(y)]^(-1/a) (if bounded)
for I integrals over the compact subset, say A, of a Euclidean
space and 0 < a < n where n is the ordinary dimension of the
Euclidean space and //.// is the Euclidean norm and m is a positive
Borel measure that is 1 on A and inf is over all such measures m.
The a-capacity is indexed by a subscript a, and the capacity-
dimension is u if the a-capacity for all a > u is 0 and u > 0.
When these conditions do not hold, the capacity-dimension is set
equal to 0.
Osher Doctorow
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