Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators

Eberle, Andreas

doi:10.1007/bfb0103045

Cited by 170 publications

(266 citation statements)

References 0 publications

Supporting

Mentioning

260

Contrasting

Unclassified

Order By: Relevance

“…Proof It follows from Corollary 6.2 and a result of Eberle (page 115, [12]) that Markov uniqueness is equivalent to ID 2,1 = 0 W 2,1 ∞ . Proposition 6.8 then shows that (1) and (2) are equivalent.…”

Section: Proposition 68 Suppose Conditionmentioning

confidence: 92%

“…Following Eberle [12], consider the space of bounded functions in 0 W 2,1 closed under the weak Sobolev norm, which we shall denote by 0 W 2,1 ∞ .…”

Section: Theorem 65mentioning

confidence: 99%

“…The latter concept relates to the uniqueness of a Markov process on the path space which would play the role of a Brownian motion (or Ornstein-Uhlenbeck process), see e.g. Eberle [12] and §6 below. Note that Aida has shown that such operators on certain finite co-dimensional submanifolds of Wiener space [3] are essentially self adjoint, and similarly for paths and loops on Lie groups [4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Itô maps and analysis on path spaces

Elworthy

2007

Math. Z.

View full text Add to dashboard Cite

We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross-Sobolev spaces of differentiable functions and proving their intertwining with solution maps, I, of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator d and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given I, is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.

show abstract

Section: Proposition 68 Suppose Conditionmentioning

confidence: 92%

“…Following Eberle [12], consider the space of bounded functions in 0 W 2,1 closed under the weak Sobolev norm, which we shall denote by 0 W 2,1 ∞ .…”

Section: Theorem 65mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Itô maps and analysis on path spaces

Elworthy

2007

Math. Z.

View full text Add to dashboard Cite

show abstract

“…To this end, we use an approximative criterium of Eberle [17]. The result of this section will be useful to prove results in Sect.…”

Section: Extension Of P Tmentioning

confidence: 94%

“…Moreover, (36) implies that the Kolmogorov operator (K, F C ∞ b ) is dissipative on L p (μ β,ν ) for any 1 ≤ p < ∞ (see Lemma 1.8 in [17]). Hence it is closable in L p (μ β,ν ) (see [26]).…”

Section: Extension Of P Tmentioning

confidence: 99%

Invariant Measures of Gaussian Type for 2D Turbulence

Bessaih

Ferrario

2012

J Stat Phys

View full text Add to dashboard Cite

Gaussian measures μ β,ν are associated to some stochastic 2D models of turbulence. They are Gibbs measures constructed by means of an invariant quantity of the system depending on some parameter β (related to the 2D nature of the fluid) and the viscosity ν. We prove the existence and the uniqueness of the global flow for the stochastic viscous system; moreover the measure μ β,ν is invariant for this flow and is the unique invariant measure. Finally, we prove that the deterministic inviscid equation has a μ β,ν -stationary solution (for any ν > 0).

show abstract

Measurable Riemannian structures associated with strong local Dirichlet forms

Hino

2013

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators

Cited by 170 publications

References 0 publications

Itô maps and analysis on path spaces

Itô maps and analysis on path spaces

Invariant Measures of Gaussian Type for 2D Turbulence

Measurable Riemannian structures associated with strong local Dirichlet forms

Contact Info

Product

Resources

About