## Abstract

Let μ = (μ^{1},... ,μ^{d}) be such that each μ^{i} is a signed measure on R^{d} belonging to the Kato class K_{d,1}. The existence and uniqueness of a continuous Markov process X on R^{d}, called a Brownian motion with drift μ, was recently established by Bass and Chen. In this paper we study the potential theory of X. We show that X has a continuous density q^{μ} and that there exist positive constants c_{i}, i = 1, ⋯ , 9, such that c_{1}e^{-c2t}t^{-d/2}e^{-c3|x-y|2/2t} ≤ q^{μ} (t,x,y) ≤ c_{4}e^{c5t}t^{-d/2} e^{-c6|x-y|2/2t} and |∇_{x}q^{μ}(t,x,y)| ≤ c_{7}e^{c8t} t^{-d+1/2} e^{-c9|x-y|2/2t} for all (t,x,y) ∈ (0, ∞) × R^{d} × R^{d}. We further show that, for any bounded C^{1,1} domain D, the density q ^{μ,D} of X^{D}, the process obtained by killing X upon exiting from D, has the following estimates: for any T > 0, there exist positive constants C_{i}, i = 1, ⋯,5, such that C_{1}(1 Λ ρ(x)/√t)(1 Λ ρ(y)/√t)t^{-d/2}e ^{-C2|x-y|2/t} ≤ q^{μ,D} (t,x,y) ≤ C_{3} (1 Λ ρ(x)/√t)(1 Λ ρ(y)/√t)t^{-d/2}e ^{-C4|x-y|2/t} and |∇ _{x}q^{μ,D} (t,x,y)| ≤ C_{5}(1Λ ρ(y)/√t)t^{-d+1/2}e ^{-C4|x-y|2/t} for all (t,x,y) ∈ (0, T] × D × D, where ρ(x) is the distance between x and ∂D. Using the above estimates, we then prove the parabolic Harnack principle for X and show that the boundary Harnack principle holds for the nonnegative harmonic functions of X. We also identify the Martin boundary of X^{D}.

Original language | English (US) |
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Pages (from-to) | 635-688 |

Number of pages | 54 |

Journal | Illinois Journal of Mathematics |

Volume | 50 |

Issue number | 3 |

DOIs | |

State | Published - 2006 |

## ASJC Scopus subject areas

- General Mathematics