## Abstract

Suppose that X_{t} is the standard Brownian motion in R^{d}, d ≥ 3, that ρ ∈ H^{1}(R^{d}) is a bounded continuous function such that |∇ρ|^{2} belongs to the Kato class and μ is a measure belonging to the Kato class. Let A^{[ρ]}_{t} be defined as A^{[ρ]}_{t} = ρ(X_{t}) - ρ(X_{0}) - ∫^{t}_{0} ∇ρ(X_{s}) · dX_{s}, and let A^{μ}_{t} be the continuous additive functional with μ as its Revuz measure. Define A_{t} as the sum of the two additive functionals above. Then the semigroup defined as T_{t}f(x) = E^{x}(e^{At}f(X_{t})) is called a generalized Schrödinger semigroup. In this paper we identify the quadratic form corresponding to (T_{t}) as (E ̃, H^{1}(R^{d})) with E ̃(u, v) = 1 2 ∫ ∇u(x) · ∇v(x) dx + ∫R^{d} ∇(uv)(x) · ∇ρ(x) dx - ∫_{Rd}u(x) v(x) μ(dx).

Original language | English (US) |
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Pages (from-to) | 358-378 |

Number of pages | 21 |

Journal | Journal of Functional Analysis |

Volume | 125 |

Issue number | 2 |

DOIs | |

State | Published - Nov 1 1994 |

Externally published | Yes |

## ASJC Scopus subject areas

- Analysis