I heard a comment in a seminar about geometric analysis that it is very beneficial to regard Radon-Nikodym derivative of a real function as a stochastic process over a $\sigma$-algebra. When I asked for more details, the speaker told me it could be done by branching, so I was wondering

(1)How exactly this is constructed?

(2)How this insight can actually be used in the field of geometric analysis or other related field?


As the answer shown, it can be regarded as a stochastic process of generating $\sigma$-algebra, yet not a general algebra.


(1) (Modified from [1]pp.246-247,312-313) First we sample $Z$ from $Unif[0,1)$, since $\mathbb{R}$ is an Archimedes field, for a fixed $n$ we can find such a $k$ that $\frac{k}{2^n}\leq Z< \frac{k+1}{2^n}$. Define random variable $Y_n=\frac{k}{2^n}$, this is a random variable because $k$ depends on $Z$ and $Z$ is random.

Now by our construction $Y_n\leq Z<Y_n+\frac{1}{2^n}$, $Y_n,Z$ coincide in the first $n$ digits in a binary expansion. For any Lipschitz-continuous function $f$, we can use the limit of a sequence of bounded functions $f_n$ to approximate it. Therefore wlog, assume $f$ is bounded over $[0,1)$.

Consider the jump random variable defined by $X_n=2^n\{f(Y_n+\frac{1}{2^n})-f(Y_n)\}$. $Z\mid Y_0,\cdots Y_n$ is conditionally distributed as $Unif[Y_n,Y_n+\frac{1}{2^n})$. $$E(X_{n+1}\mid Y_0,\cdots Y_n)=2^{n+1}E[f(Y_{n+1}+\frac{1}{2^{n+1}})-f(Y_{n+1})\mid Y_0,\cdots Y_n]$$ $$=2^{n+1}\int_{[Y_n,Y_n+\frac{1}{2^n})}2^n[f(u+\frac{1}{2^{n+1}})-f(u)]du$$ $$=2^n\{f(Y_n+\frac{1}{2^{n}})-f(Y_n)\}=X_n$$ Thus $X_n$ is a martingale w.r.t. the sigma-algebra $\sigma\{Y_0,\cdots Y_n\}$, if we take $X_\infty=lim_{n\rightarrow\infty}X_n$, that is to say we investigate the stationary/invariant measure (corresponding to $X_\infty$)of the stochastic process we defined above, thn $X_\infty=g(Z)$ is the RN derivative of $f$.

Therefore RN derivative of any Lipschitz continuous function can be represented as the stationary measure of the (branching) martingale $X_n$ w.r.t to the $\sigma$-algebra generated by $Y_0,\cdots Y_n$ defined above.

(2) One of the benifits that I feel is that this gives an explicit way of constructing an extension of functional. RN Theorem can be proven by using Hahn-Banach Theorem[2], this insight might give you a constructive way of expressing the RN derivative in stochastic calculus. In statistics, the branching technique could be used to construct sample functions of many processes like Dirichlet process and Polya tree.

For geometric analysis, the only tangential reference that I know of is [3]. If you think RN derivative as a vector field. If you regard the RN derivate as a stochastic process I think it is also natural to regard it as a flow as [4].

From the perspective of operator algebra, this construction allows us to see conditional expectation as sort of RN derivate, and hence a differential operator.

[1]Karlin, S., and H. M. Taylor. "A first course in stochastic processes Academic." New York (1975).

[2]Radon-Nikodym Theorem and Conditional Expectation http://www.dam.brown.edu/people/huiwang/classes/Am264/Archive/cond_expe.pdf

[3]Stroock, Daniel W. An introduction to the analysis of paths on a Riemannian manifold. No. 74. American Mathematical Soc., 2005.

[4]Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2008.

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