Infinite Goldie dimensions

Hypoellipticity in Infinite Dimensions

We consider semilinear parabolic stochastic PDEs driven by additive noise. The question addressed in this note is that of the regularity of transition probabilities. If the equation satisfies a Hörmander ‘bracket condition’, then any finite-dimensional projection of the solution has a smooth density with respect to Lebesgue measure. One key ingredient in the argument is a bound on ‘Wiener polyn...

Harnack Inequalities in Infinite Dimensions

We consider the Harnack inequality for harmonic functions with respect to three types of infinite-dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori mod...

Capturing Functions in Infinite Dimensions

The following are notes on stochastic and parametric PDEs of the short course in Paris. 1 Lecture 4: Capturing Functions in Infinite Dimensions Finally, we want to give an example where the problem is to recover a function of infinitely many variables. We will first show how such problems occur in the context of stochastic partial differential equations. 1.1 Elliptic equations: general principl...

Spinor Types in Infinite Dimensions

The Cartan Dirac classification of spinors into types is generalized to infinite dimensions. The main conclusion is that, in the statistical interpretation where such spinors are functions on Z2 , any real or quaternionic structure involves switching zeroes and ones. There results a maze of equivalence classes of each type. Some examples are shown in L2(T) . The classification of spinors leads ...

Infinite Frameworks in Two Dimensions