User mark peletier - MathOverflow

Reference for proof that $C_b^* = rba$

2011-07-18T11:34:51Z

The following theorem seems to have folk status:

The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, regular, finitely additive Borel set functions.

This fact is often mentioned (for instance in the answer to http://mathoverflow.net/questions/44183/dual-of-bounded-uniformly-continuous-functions) but I'm having great difficulty actually finding a reference. Often Dunford & Schwartz is mentioned as a reference; D&S defines $rba$, but doesn't prove the connection to the dual of $C_b$. Hildebrandt 1934 proves a characterization in terms of limits of Stieltjes integrals, but that is still some steps away from the characterization above. I haven't been able to find anything coming closer than this.

Does anyone know of a real proof of this statement? Am I maybe overlooking a very simple proof?

Answer by Mark Peletier for Probabilistic Solution of the Porous Medium Equation

2011-07-18T13:28:53Z

The processes described by Andre work by having the interaction act at the level of the mobility of the particles.

There are other ways, too. One is in the work of Philipowski (see also Figalli & Philipowski). Here the idea is to take interactions of potential type, i.e. for instance

$ dX^i = -\sum_{j\not=i} \nabla W_\epsilon(X^i-X^j)\, dt + \delta \, dB^i. $

The parameter $\epsilon$ is the spatial range of $W$, and in the limit $\epsilon\to0$ the interaction becomes purely local, and leads to a nonlinear diffusion term. If one also lets $\delta\to0$, then the purely Brownian contribution also vanishes. Only the nonlinear diffusion is then left.

Answer by Mark Peletier for Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

2011-07-18T13:18:34Z

The equation

$ u_t = u_{xx} + u_{yy} $

is a gradient flow, or gradient descent, in the following sense. You should think of the equation as being placed in the space $L^2$. The Fréchèt derivative of the functional $E$ is the linear mapping

$ \displaystyle E'(u): v \mapsto -2\iint \nabla u\cdot \nabla v = -2\iint v(u_{xx}+u_{yy}), $

where for simplicity I'm assuming that the boundary conditions don't give rise to boundary terms in the partial integration. The second version, after partial integration, is relevant because it's written in the form of an $L^2$ inner product, allowing us to write the Fréchèt derivative as

$ E'(u)\cdot v = (-2(u_{xx}+u_{yy}),v)_{L^2} =: (\mathrm{grad}\ E(u),v)_{L^2} $

The $L^2$-gradient flow of $E$ is then the equation

$ u_t = -\mathrm{grad}\ E(u) = 2(u_{xx}+u_{yy}). $

Comment by Mark Peletier

2011-08-14T18:14:26Z

Thanks @Theo and @Gerard - I obviously overlooked Theorem IV.6.2, possibly because of the different notation. Thanks for pointing that out!

Comment by Mark Peletier

2011-07-22T12:47:46Z

Of course, in higher dimensions the gradient can not be just any vector-valued $L^\infty$-function $f = (f_i)$, since it satisfies the distributional identity $ \dfrac{\partial f_i}{\partial x_j} = \dfrac{\partial f_j}{\partial x_i}. $ So you don't get the whole of $L^\infty$, only those functions that satisfy this identity.

Comment by Mark Peletier

2011-07-22T12:30:49Z

Tapio, why do you want the entropy of your finitely additive translation-invariant measure to be finite? In the limit as $n\to\infty$ the Shannon entropy of the uniform measure on $[n]$ converges to infinity. Wouldn't you therefore expect that such a `uniform' measure on $\mathbb N$ should have infinite entropy?