QuoteOriginally posted by: caperoverQuoteOriginally posted by: AlanQuoteOriginally posted by: caperoverHi, I read some papers on Laplace-Beltrami equation but cannot put everyting together. I guess some connections are missing. Can someone help?What is the difference and relationship between heat equation in 2D Euclidean space and on a Plane? Thanks,CRI'll take a stab at part of it.You have to distinguish various things: the coordinates vs. the metricFor the "heat equation in 2D Euclidean space", you could have1. The heat equation in 2D Euclidean space in the usual Cartesian coordinates. 2. The heat equation in 2D Euclidean space in polar coordinates.Both eqns describe the same "physics", but they look (outwardly) different and the Laplace-Beltrami operator gives you the right Laplacian.A visual plot of the evolution of the solution of either equation, say starting from a Dirac mass at the origin, would look exactly the same.For the "heat equation in the plane" , you could have3. The heat equation in a plane with a Euclidean metric.4. The heat equation in a plane with a non-Euclidean metric.Again 3 is describing the same physics as 1. and 2., but 4 is fundamentally different.4. describes the evolution of Brownian motion where there is intrinsic curvature. A standard example is the Poincare (half)-plane with the metric ds^2 = (dx^2 + dy^2)/y^2, on (-infty < x < infty, y > 0)which has constant negative curvature. A visual plot of the evolution of the solution of 4., starting from a Dirac mass at (x=0,y=1), looks very different from the Euclidean case. In fact, this heat kernel solution for 4., due to McKean, is known (lesniewski.us/papers/working/ProbDistrForSABR.pdf) --suggest you plot it to see my point.Thanks, Alan. I got you out of water. I will try to re-phrase my questions in your definitions.First, does the PDE solution have anything to do with the embedded curves or surfaces, for example, ||x|| = 1 for Poincare half-plane?I understand the relationship between 1 and 2, just change of variable or transformation of cooridiate, and as you said they describe exactly the same physics, but I don't quite understand 3. What does it exactly mean by "in the plane" or on the curve in a 2D case? Suppose we have the solution for a 2D PDE df/dt=0.5y^2( d^2f/dx^2 + d^2f/dy^2) in the Poincare half-plane with embedded curves ||x|| =1. Is there any relationship between the solution and the curves ||x|| = 1? How to visualize this? A Brownian with constant negative convexity, starting from a Dirac mass at (x=0,y=1), how will it evolve?Basically, how do we visualize the evolution of a PDE on a Riemannian manifold? or, intuitively how metric impacts the PDE coefficients?Second, is there any connection between 1 and 4?In the case of SABR with rho = 0, by changing variable the heat equation in the Euclidean space becomes the heat equation on Poincare half-plane. is there some more general connection?Third, why do we resort to PDE on non-Euclidean geometry? does it have anything to do with local cooridates?Some asymptotic expansions and aymptotic solutions have been derived for Laplace-Beltrami equations. Is that the only reason? In physics, sometimes it is convenient to solve a system in local coordiates, but in finance I did not see people mentioning that except for asymptotic solutions.Thanks a lot!Briefly, I will just say that visualization comes from experience; also you can treat 2D problems with a metricas equivalent to a curved surface in Euclidean 3-space and picture motions (Brownian or otherwise) on that.For example, you can treat (part of the) Poincare half-plane as equivalent to the surface shown: (Note: left-right is y; the strip (-pi < x < pi) is the "theta" of the figure, with the ends of the strip identified. It is a partial embedding) Re: "why do we resort to PDE on non-Euclidean geometry?"Any parabolic evolution equation associated to dX = b(X) dt + a(X) dB, (X = n-vector; a(X) = n x n matrix)can be interpretted in two different, but ultimately equivalent, ways:1. Standard interpretation: the "particle" is diffusing in our familiar everyday (Euclidean) world, butthere is a spatially dependent diffusion (matrix) coefficient. Unless a(X) = constant (and ignore the drift),this is definitely _not_ standard Brownian motion. B(t) of the sde is a standard BM for Euclidean n-space.2. Alternative interpretation: the particle is diffusing on a Riemannian manifold (R^n,g) with metric g(x) = a^(-1)(x).Its motion is a "standard Brownian motion for _that_ manifold", perturbed by a (different) drift. For example, the beta=0 SABR process (with free motion across F=0) is a standard BM on the figure above, at least until the time the particle moves halfway round the circular direction. Sometimes point of view 2 is very useful; for example, for the small-time smile problem in finance.
Last edited by Alan
on September 19th, 2011, 10:00 pm, edited 1 time in total.