### Abstract

It is shown that if φ(f) ∝_{Rd
} φ(y) f(y) dy is a Markoff random field and X_{α} are multiplicative functionals of φ (with E(X_{α}) = 1) which converge locally in L_{1}, then there exists a locally Markoff random field φ_{*} such that E(exp(iφ_{*}(f))) = lim_{α} E(X_{α} exp(iφ(φ))). We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take X_{α} proportional to exp(-λ∝_{R2
} : P(φ(y)) : g_{α}(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of g_{α} → 1 and small λ, {X_{α}} does converge locally in L_{1} and that the corresponding φ_{*} is stationary.

Original language | English (US) |
---|---|

Pages (from-to) | 44-61 |

Number of pages | 18 |

Journal | Journal of Functional Analysis |

Volume | 14 |

Issue number | 1 |

DOIs | |

State | Published - 1973 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

**The construction of stationary two-dimensional Markoff fields with an application to quantum field theory.** / Newman, Charles M.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - The construction of stationary two-dimensional Markoff fields with an application to quantum field theory

AU - Newman, Charles M.

PY - 1973

Y1 - 1973

N2 - It is shown that if φ(f) ∝Rd φ(y) f(y) dy is a Markoff random field and Xα are multiplicative functionals of φ (with E(Xα) = 1) which converge locally in L1, then there exists a locally Markoff random field φ* such that E(exp(iφ*(f))) = limα E(Xα exp(iφ(φ))). We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take Xα proportional to exp(-λ∝R2 : P(φ(y)) : gα(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of gα → 1 and small λ, {Xα} does converge locally in L1 and that the corresponding φ* is stationary.

AB - It is shown that if φ(f) ∝Rd φ(y) f(y) dy is a Markoff random field and Xα are multiplicative functionals of φ (with E(Xα) = 1) which converge locally in L1, then there exists a locally Markoff random field φ* such that E(exp(iφ*(f))) = limα E(Xα exp(iφ(φ))). We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take Xα proportional to exp(-λ∝R2 : P(φ(y)) : gα(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of gα → 1 and small λ, {Xα} does converge locally in L1 and that the corresponding φ* is stationary.

UR - http://www.scopus.com/inward/record.url?scp=50849149137&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=50849149137&partnerID=8YFLogxK

U2 - 10.1016/0022-1236(73)90029-3

DO - 10.1016/0022-1236(73)90029-3

M3 - Article

VL - 14

SP - 44

EP - 61

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -