ISSN:
1432-0916

Source:
Springer Online Journal Archives 1860-2000

Topics:
Mathematics
,
Physics

Notes:
Abstract Nelson's free Markoff field on ℝ l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ℝ1, mapping a class of distributions φ(x,t) on ℝ l ×ℝ1 to mean zero Gaussian random variables φ with covariance given by the inner product $$\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 $$ . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl〉0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ℝ l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.

Type of Medium:
Electronic Resource