Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

Kim, Panki; Song, Renming

doi:10.1007/s00208-007-0110-6

Cited by 10 publications

(41 citation statements)

References 26 publications

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“…As for assumption (A) (i), see e.g. Theorem 2.4 in [19] and Section 4 in [20]. In particular, we deduce from Section 4 in [20] in conjunction with Jentzsch's theorem that assumption (A) is satisfied for Schrödinger-type semigroups with drift and potential in a Kato class (as defined in the next section) and with the zero Dirichlet boundary condition in a bounded C 1,1 -domain E ⊆ R d , where the measure m can be chosen to be the Lebesgue measure on E.…”

Section: Remarkmentioning

confidence: 98%

Uniform Conditional Ergodicity and Intrinsic Ultracontractivity

Knobloch

Partzsch

2009

Potential Anal

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We study two properties of semigroups of sub-Markov kernels, namely uniform conditional ergodicity and intrinsic ultracontractivity. In this paper we investigate the relationship between these two properties and we provide sufficient criteria as well as characterisations of them. In particular, our considerations show that, under suitable assumptions, the second property implies the first one. We also introduce a property called compact domination and show how this property and the parabolic boundary Harnack principle are related to the aforementioned properties. Furthermore, we apply these results in some special cases.

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Section: Remarkmentioning

confidence: 98%

Uniform Conditional Ergodicity and Intrinsic Ultracontractivity

Knobloch

Partzsch

2009

Potential Anal

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“…In [12], we also showed that there exist positive constants c 1 and c 2 depending on D via its diameter such that for any…”

Section: Green Function Estimates and 3g Theoremmentioning

confidence: 93%

“…In particular, for every y ∈ D and ε > 0, G D (·, y) is regular harmonic in D \ B(y, ε) with respect to X (see Theorem 2.9(1) in [12]). We recall here the scale invariant Harnack inequality from [11].…”

Section: Green Function Estimates and 3g Theoremmentioning

confidence: 96%

Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

Kim

Song

2007

Journal of Mathematical Analysis and Applications

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In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains.Informally the Schrödinger-type operators we consider are of the form L + μ · ∇ + ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to K d,1 and ν is a signed measure belonging to K d,2 . In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C 1,1 -domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains.

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“…We should point out that the above mentioned result of [2] hold in a more general setting and is actually valid for the case when the drift b is a Radon measure that satisfies (3), although in this case the notion of a solution to (2) has to be defined in a more general sense. Later, analytical and probabilistic properties of the solution (X t ) t≥0 to (2) with a drift |b| ∈ K d−1 were investigated by Kim and Song, see [9,10,11,12]; among many other things, they obtained two-sided Gaussian estimates for the heat kernel of (X t ) t≥0 .…”

Section: Introductionmentioning

confidence: 99%

Brownian Motion with Singular Time-Dependent Drift

Jin

2016

J Theor Probab

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Abstract. In this paper we study weak solutions for the following type of stochastic differential equationA solution X = (Xt) t≥s for the above SDE is called a Brownian motion with time-dependent drift b starting from (s, x). Under the assumption that |b| belongs to the forward-Kato class F K α d−1 for some α ∈ (0, 1/2), we prove that the above SDE has a unique weak solution for every starting point (s, x) ∈ [0, ∞) × R d .

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Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains

Cited by 10 publications

References 26 publications

Uniform Conditional Ergodicity and Intrinsic Ultracontractivity

Uniform Conditional Ergodicity and Intrinsic Ultracontractivity

Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

Brownian Motion with Singular Time-Dependent Drift

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