Short answer: The countable product of probability measures is a well-defined object so the countability of periodic sequences is enough to conclude that the probability of drawing a periodic sequence ...

I know that Mathematical Reviews use the keywords to assign articles to reviewers since the MSC codes are sometimes too broad to be of much use. This does help find reviewers who can actually say ...

Semigroups of bounded $L^{2}$ operators are very important in probability. They in fact provide one of the main ways show the very close connection between a self-adjoint operator and a `nice' Markov ...

Alexander Teplyaev has been working on exactly this question this is a paper in which he and Michael Hinz show that the Navier-Stokes equation on a Sierpinski gasket is sensibly defined and that it ...

As Steve Huntsman points out in his comment this is just writing out the integral against the Hausdorff measure on the fractal domain in polar coordinates over the same fractal. Since the author does ...

If you intend to post your slides online after class then you run the risk of students not even taking notes/digesting the material on their own (I've had this feeling myself) or feeling that they don'...

From the wording of your question it is possible you are asking someone to write an entire historical overview for you. So instead what I did was spend a few minutes on Ye Olde Google and found this: ...

A quick, partial answer to your second question about the definition of fractals. If a fractal is generated by an iterated function system with a scaling ratio less than one then you do get a ...

The formula $$\mathbb{E} \int_{R+} \phi^{2}(t,\omega)dt$$ is a double integral a la Fubini-Tonelli. And if you did back there is probably a condition on the filtration saying that $\mathcal{F}_t \...

To build on what Andrey has mensioned about Brownian motion. The Laplacian is one of the points of connection between stochastic processes and analysis. The Laplacian appears as the infinitesimal ...

From the realm of probability there are determinental and permanental processes. Terry Tao has a nice post about determinental processes here. For instance "Examples of processes known to be ...

Hatano gives on page 19 of this article "Notes on Hausdorff Dimensions of Cartesian Product Sets" HIROSHIMA MATH. J.1 (1971), 17-25 as class of generalized symmetric Cantor sets whose products have ...

This is a partial answer but shows the kind of subtlety that makes the continuity of Brownian motion non trivial. If you try and take the first three axioms of Brownian motion and try to prove that ...

I think a fairly good demonstration is Persi Diaconis' machine to toss a perfectly standard US quarter to a single predetermined side with something like 99% accuracy. I have heard it said that he ...

I was listening to Lawler give a series of talks this summer and his sense of scaling limit is the same as is often used in defining Brownian motion as a scaling limit of simple random walks. In which ...

This may be buried in one of the references above, but for those don't wish to go through them all... The zeta function can arise as the trace of Hamiltonians governing physical systems. For example ...

David Applebaums' "Lévy processes and stochastic calculus" It looks like integrating with respect to an $\alpha$-stable process is addressed directly on page 212 with the Ito formula to follow ...

I'll admit I'm not number theorist but here is my take on why the field has to embrace the completions of $\mathbb{Q}$. It is a simple reason: $e$. Other branches of math have run across interesting ...

Some quick Googling brought me to this paper. The idea is to take the coefficients in the summation to be suitable independent random variables according to a suggestion of Mandelbrot. I can't ...

Spectral decimation is an inductive process where the eigenvalues of a natural Laplacian on "nice" fractals is computed inductively. The idea is that by using a sequence of finite graphs to ...

The canonical reference for this material (at least with the people I hang out with) is Ken Falconer's Fractal Geometry Mathematical Foundations and Applications. For most sets that are fully self-...

Suppose you had such a process that is not trivial. Suppose you have $W_s \neq W_0$. For $t >s$ we have assumed that $W_s$ and $W_t$ are independent, have mean zero, and the same distribution. Now ...

I really should avoid answering questions late at night. My original answer is muddled enough to not work. But here is what it should have been: Let $X_0 = \{0,1\}$ and $X_i = X_{i-1} \times X_0$. ...

Because the iterated function system that defines the Sierpinski gasket is a contraction mapping in the metric space of non-empty compact subsets of $\mathbb{R}^{2}$ with the Sierpinski gasket as its ...

I took a quick look around and Wolfram Mathworld has a great animation of an osculating circle here. Also the entry on osculating curves here has a nice definition. The intuition is that when the you ...

I'd like to add the book for one of the two number courses I've taken which was an entire senior seminar (last year of undergraduate) on transcendental numbers. Making Transcendence Transparent by ...

It does not because $x$ is just some element of your state space. You could conceivably choose $x$ as your sample point for any or all of the $\theta$'s. So what it would make sense to differentiate ...

As far as the QM goes, I don't know enough to really say, but books like Quantum Mechanics for Mathematicians would give a mathematical overview of the large pieces of QM. However, infinite ...

I think you may want to look at this MSE question. Amusingly enough, this question was to prepare a lecture for when the asker was covering my class last fall. This kind of argument is mentioned in ...

The Sierpinski gasket is not a good example for this because of the bound you saw on the degree of the graph. I'd venture to say that this is because the gasket falls into a class of fractals called ...