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Electronic Resource
Springer
Communications in mathematical physics 35 (1974), S. 215-233
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
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