<blockquote>Quote<hr><i>Originally posted by: <b>MathDork</b></i>Actually, I found the answer, and the answer is yes if p = .5 and no if p is anything else. Check this out for a proof:http://www.stats.ox.ac.uk/~northrop/tea ... .pdfAnyone have an inuitive reason why a semetric random walk won't revist in 3 dimensions?<hr></blockquote>------------------------------------------------------------------------------------------------------------------------------------------------------Durrett (Stochastic calculus) has a nice discussion of this. Intuitively as many have mentioned, the space is getting larger, but why is d = 2 the borderline case?This is not intuitive, but ultimately it all gets traced to the spherically symmetric solutions of:Laplacian f(x) = 0, in d-dimensions. These solutions are, with |x| = RR ( d =1)log R (d = 2)R^(2-d) (d >= 3)Only the d = 1 and d = 2 solutions diverge as R -> infinity and that leads (see Durrett) to recurrence.If you let d be any real number > 0, then d = 2 is the dividing line and d = 2 + eps is completely different from d <= 2.

Thinking aloud.....there must be more to this issue of recurrent or transient motion in "Cartesian space." What happens to a random walk on the surface of a sphere? I.e. 3D space with walk constrained to a 2D surface. What if the random walk is in a 1D parametric space & I later use a known map to a 3D curve? So what other qualifications must be made with regards to recurrence in 2D & transience in 3D?

Forget my question ..... realized that the properties of recurrence & transience apply when "motion in each dimension is driven" by an independent BM.....

Processes on curved manifolds are interesting WB.

Thanks chiral, hope to read up more on the "fundamentals" one of these days (feel like I'm still at square one!!)A possibly unsatisfactory way of getting a sense of the distinction in 2d/3d might be to think of the projection of a 3d curve on the plane. 2d motion described by a curve in the (x,y) plane, and the 3d motion an extension of the 2d case by introducing the z-axis. Now corresponding to ANY self-intersecting 2d curve there surely are an infinite number of non-intersecting 3d curves for which the particular 2D curve is the projection onto the plane. So when we think of the "likelihood" of self-intersection in 2D versus 3D, it would appear that for every 2d self-intersecting curve there are many many more (infinitely?) non-intersecting 3D curves. Thus self-intersection is more likely in 2d. Okay, this is a lame attempt.....a wild shot...

No need to thank me, I just meant that when you go to >=3d, don't think curved yet. Think Euclidean first.

> Anyone have an inuitive reason why a semetric random walk won't revist in 3 dimensions?The way I look at it intuitively is that the probability of revisiting the origin is 1 if and only if the particle returns an infinite number of times to the origin. (I suspect there's an elementary proof if someone challenged me on this.)So to get the expected number of revisits, sum up the probabilities of revisiting after 2n moves for n = 1, 2, 3, ...In the 1-d case the prob of a particle at the origin after 2n moves.Prob(x(n) = 0) = C(2n, n) 1/2^n, By Stirling's approx, as n gets large, P(x(n)=0) is approx 1/sqrt(pi*n)So the expected number of returns to the origin is infinite because the sum of 1/n^(1/2) diverges as n->inf.The 2d case:Prob(x(n) = 0, y(n) = 0) = Prob(x(n)=0) Prob(y(n)=0) = (C(2n, n) 1/2^n)^2, But (C(2n, n) 1/2^n)^2 is proportional to 1/n as n gets large, so the expected number of returns to the origin is infinite in the 2d case because the harmonic series (sum of 1/n as n->Inf) diverges but BARELY so(*).Now in the 3d case we finally have a convergent sum, summing terms (C(2n, n) 1/2^n)^3, each term of order 1/n^(3/2) for n large, and a finite expected number of revisits implies prob of ever returning is < 1.(*) The harmonic series sum 1 + ½ + 1/3 + ¼ + ... finally reaches 100 after summing about 10^43 terms. For this and other interesting properties of Euler’s constant, harmonic series, the gamma function, Prime Number Theorem, Riemann's Hypothesis and interconnections thereof see Gamma: Exploring Euler's Constant

Very nice and thanks for the book recommendation.One correction: I believe you meant to say that:In the 1-d case the prob of a particle at the origin after 2n moves isProb(x(n) = 0) = C(2n, n) (1/2)^(2 n), and then all is well. p.s.: Here's what seems a tricky variation: calculate the probability that the3-d symmetric random walk will *ever* return to the origin.

Thanks Alan -- and yes indeed, 2n (not n) should appear in all the powers of 1/2.As for the prob that the 3d symmetric random walk ever returns to the origin:Instead of giving a # I'll say that if I were running a casino that offered odds to gamblers who wanted to bet on the 'return to origin' event, I'd give 3-1 odds and enjoy a very slim margin BTW, I've enjoyed your book, Option Valuation under Stochastic Volatility, very much.

Thank you so much -- your casino sounds like it's under 'expert' management.

Quotehttp://www.stats.ox.ac.uk/~northrop/teaching/b10/polya.pdfLink seems broken. Can anyone provide any updated link?Regards,

If the price movement is pure random, there will be not way to make profit at all.