 Alan
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### Laplace-Beltrami equation

No: BM on a Riemannian manifold is not (generally) described by a spatially homogeneous pde. It is still given by dp/dt = Lap p, where Lap is the Laplace-Beltrami operator.Unless the manifold is (essentially) E^n, in no set of coordinates is this spatially homogeneous, AFAIK.There are various useful sets of coordinates, but none of them will get you to (d^2/dx^2 + d^2/dy^2 + ... )My point was about different ways to interpret and visualize the stochastic process underlying dp/dt = Lap p.To answer your second question directly, I believe the answer is "generally no", unless you can infer from theequation that the metric is Euclidean. For example, if I gave you dp/dt = Lap p for E^2, but in polar coordinates,and you didn't recognize it, you could still discover from the eqn. that a transformation to dp/dt = (d^2/dx^2 + d^2/dy^2) p was possible. How? From the pde, you can read off a metric;from the metric, you could compute a curvature and find it was 0. This would tell you. But not for the Poincare model; no such coordinate transformation will work,as the (Gaussian) curvature is negative and this is a coordinate system invariant.
Last edited by Alan on September 20th, 2011, 10:00 pm, edited 1 time in total. Cuchulainn
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### Laplace-Beltrami equation

QuoteSuppose I have a heat equation with spatially inhomogenous coefficients. Is there a way to map Euclidean space to a surface such that the heat equation on the surface has constant coefficients?In general, it might be possible but it might take forever. The new PDE (is it a PDE?) will be even more scary than the original. From a computational point of view there are few advantages and you rarely see this approach being taken as far as I can see.PDE with non-constant coefficients is not so tricky?Maybe some kind of Cole-Hopf transformation is useful, in this case reduce Burgers' equation to a linear heat equation. Eventually, numerics are needed to get numbers.
Last edited by Cuchulainn on September 20th, 2011, 10:00 pm, edited 1 time in total.
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### Laplace-Beltrami equation

Thank everyone. Now below is what I understand so far:What is the difference between heat equation in 2D Euclidean space and on a Plane with non-Euclidean metric? What is the relationship between the solution of the PDE and the surface?The difference is between Laplace operator and Laplace-Beltrami operator. A heat equation on a Plane with non-Euclidean is just like a heat equation in 2D Euclidean space in local coordinates, which varies smoothly on the surface though.PDE on a 2D plane with non-Euclidean metric evolves on the curved surface in E3 Is the solution of a Laplace-Beltrami equation defined on local coordinates?It is in global coordinates. In local coordinates it is a Laplace equation.Is Laplace-Beltrami operator corresponding to one type of change of variable?No, changing variable does not change curvatureThis helps me intuitively understand what is a PDE on Riemannian manifold. Now, here is a practical question:How does considering a PDE on Riemannian manifold help solve PDEs in Finance?Because of Varadhan's theorem, Mckean's formula, results by Minakshisundaram-Pleijel-De Witt-Gilkey, or differential geometry provides some tools to achieve asymptotic expansion of the Laplace-Beltrami operator?Thanks.
Last edited by caperover on September 20th, 2011, 10:00 pm, edited 1 time in total. Alan
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### Laplace-Beltrami equation

QuoteOriginally posted by: caperover Now, here is a practical question:How does considering a PDE on Riemannian manifold help solve PDEs in Finance?Thanks.Here is one very nice example.The simple stochastic volatility setupdS = sig(t) S dB(t)d sig(t) = sig(t) dW(t), where dB dW = rho dtis a very natural one in finance, even apart from the SABR-model work of Hagan et al.I discussed versions of this in my book as special cases of the GARCH diffusion process.I am fairly sure Engle and Bollerslev, when they developed GARCH, were not thinking about Riemannian geometry (even though Hagan et al were). Once you have GARCH, thinking about the diffusion limit is very natural and was taken up by Nelsonand various follow-ups. Again, these people were not thinking about Riemannian geometry.Now, it turns out that there is a closed-form solution for the transition density of this systemin terms of the one for BM on H^3.H^3 is the generalization of the Poincare half-plane model I discussedto the coordinate space R+^3 = {{x1,x2,x3}: x1 in R, x2 in R, x3 > 0}.The metric is ds^2 = (dx1^2 + dx2^2 + dx3^2)/x3^2Notice that this space has 3 spatial variables (x1,x2,x3), while the original problem has only two (S,sig)It is extremely unlikely, IMO, this solution would ever have been noticed without knowing the H^3 connection.You could play around with the PDE forever and never think to move to a *higher* dimensional space. So, here is a problem that arises naturally in finance, but has a solution found using the heat kernelassociated to a fundamental space in Riemannian geometry.
Last edited by Alan on September 21st, 2011, 10:00 pm, edited 1 time in total. croot
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### Laplace-Beltrami equation

QuoteIs the solution of a Laplace-Beltrami equation defined on local coordinates? It is in global coordinates. In local coordinates it is a Laplace equation. You are wrongly assuming that global coordinates exist. Another example, it might look like a Laplace equation around one point in the corresponding normal coordinates, but generally not in a neighbourhood of said point. QuoteBecause of Varadhan's theorem, Mckean's formula, results by Minakshisundaram-Pleijel-De Witt-Gilkey,Quite an impressive list of references/ideas, good luck with that. ppauper
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### Laplace-Beltrami equation ppauper
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### Laplace-Beltrami equation

and obviously, you're just solving in different geometries caperover
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### Laplace-Beltrami equation

QuoteOriginally posted by: crootQuoteIs the solution of a Laplace-Beltrami equation defined on local coordinates? It is in global coordinates. In local coordinates it is a Laplace equation. You are wrongly assuming that global coordinates exist. Another example, it might look like a Laplace equation around one point in the corresponding normal coordinates, but generally not in a neighbourhood of said point. Suppose you have a PDE, without thinking of Riemannian manifold at all. Isn't the solution, say f(x,y), defined in the global coordinate? caperover
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### Laplace-Beltrami equation

QuoteOriginally posted by: AlanQuoteOriginally posted by: caperover Now, here is a practical question:How does considering a PDE on Riemannian manifold help solve PDEs in Finance?Thanks.Here is one very nice example.The simple stochastic volatility setupdS = sig(t) S dB(t)d sig(t) = sig(t) dW(t), where dB dW = rho dtis a very natural one in finance, even apart from the SABR-model work of Hagan et al.I discussed versions of this in my book as special cases of the GARCH diffusion process.I am fairly sure Engle and Bollerslev, when they developed GARCH, were not thinking about Riemannian geometry (even though Hagan et al were). Once you have GARCH, thinking about the diffusion limit is very natural and was taken up by Nelsonand various follow-ups. Again, these people were not thinking about Riemannian geometry.Now, it turns out that there is a closed-form solution for the transition density of this systemin terms of the one for BM on H^3.H^3 is the generalization of the Poincare half-plane model I discussedto the coordinate space R+^3 = {{x1,x2,x3}: x1 in R, x2 in R, x3 > 0}.The metric is ds^2 = (dx1^2 + dx2^2 + dx3^2)/x3^2Notice that this space has 3 spatial variables (x1,x2,x3), while the original problem has only two (S,sig)It is extremely unlikely, IMO, this solution would ever have been noticed without knowing the H^3 connection.You could play around with the PDE forever and never think to move to a *higher* dimensional space. So, here is a problem that arises naturally in finance, but has a solution found using the heat kernelassociated to a fundamental space in Riemannian geometry.A nice example. Thanks a lot. Cuchulainn
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### Laplace-Beltrami equation

QuoteOriginally posted by: ppauperand obviously, you're just solving in different geometriesLike cartesian, cylindrical and spherical?
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### Laplace-Beltrami equation

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: ppauperand obviously, you're just solving in different geometriesLike cartesian, cylindrical and spherical?Laplace-Beltrami is on the surface of a sphere ! caperover
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### Laplace-Beltrami equation

QuoteOriginally posted by: ppauperQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: ppauperand obviously, you're just solving in different geometriesLike cartesian, cylindrical and spherical?Laplace-Beltrami is on the surface of a sphere !Laplace-Beltrami is a functional of metric. why does it have to be a sphere? frenchX
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### Laplace-Beltrami equation

Laplace Beltrami is a differential operator applyable to any Riemannian manifold. So I agree that it can be applyed on other topology than a sphere. DevonFangs
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### Laplace-Beltrami equation

And after great Alan shed light, here we come back to my very silly, first reply. caperover
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### Laplace-Beltrami equation

QuoteOriginally posted by: DevonFangsAnd after great Alan shed light, here we come back to my very silly, first reply.when you step into the same river for a second time, ...
Last edited by caperover on September 27th, 2011, 10:00 pm, edited 1 time in total.  