1
vote
1answer
172 views

A stronger version of supramenability?

A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu_A$ on $G$ with $\mu_A(A)=1$. I'm interested in a seemingly stronger ...
1
vote
1answer
96 views

Nonstandard definition for the generator of a standard Ito diffusion

For a standard Brownian motion, the generator of the diffusion is $$ L = \frac12 \frac{d^2}{dx^2}. $$ Is there a nonstandard definition of this generator?
5
votes
0answers
134 views

Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence, a noise is black if it is not isomorphic to standard Gaussian white noise. Tsirelson showed the existence of black noise through the scaling limit ...
7
votes
2answers
318 views

Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons

Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a\lt b,$ and assigns an ...
19
votes
4answers
2k views

Nonstandard analysis in probability theory

I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books: Nelson (1987). Radically Elementary Probability Theory ...