Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

Chen, Zhenqing; Fukushima, Masatoshi

doi:10.1515/9781400840564

Cited by 120 publications

(239 citation statements)

References 0 publications

Supporting

Mentioning

237

Contrasting

Order By: Relevance

“…By [22], Theorem 1.6.3, this implies that A is recurrent. Since, by [10], Theorem 2.1.10, recurrence implies conservativeness, in conclusion we get (here the hypothesis m(X ) < ∞ is essential)…”

Section: 1 One Hasmentioning

confidence: 55%

“…Here we briefly recall the one-to-one correspondence between regular Dirichlet forms and Markov processes. We refer to [19], [20], [22] and [10] for more details and proofs. Let the Dirichlet form F A be regular on X and let e t A , t ≥ 0, be the semi-group on L 2 (X ) generated by the corresponding Markovian operator A.…”

Section: Dirichlet Forms and Hunt Processesmentioning

confidence: 99%

“…Some analytical properties of the Dirichlet form F A and the corresponding resolvents R A λ and semigroups e t A correlate with paths behavior of Z A . Here we recall the main results following [22] and [10] to which we refer for more details and proofs. For simplicity from now on in this subsection we suppose that m(X ) < ∞.…”

Section: Dirichlet Forms and Paths Behaviormentioning

confidence: 99%

“…In particular Silverstein found a characterization, again in terms of Dirichlet spaces on the boundary, of the Markovian resolvents R λ ≥ R 0 λ dominating a given one R 0 λ . For recent developments of boundary theory of Dirichlet forms and symmetric Markov processes, we refer to the book [10] by Chen and Fukushima. Here our approach is different from the ones described above: we build on the theory of self-adjoint extensions as initiated by Kreȋn [K], Visȋk [52], Birman [5] and Grubb [26]. In particular Grubb characterized all self-adjoint extensions of a given symmetric elliptic differential operator on a domain with a smooth boundary in terms of (non-local) boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Markovian extensions of symmetric second order elliptic differential operators

Posilicano

2014

Math. Nachr.

View full text Add to dashboard Cite

Key words Self-adjoint extensions, Markovian operators, elliptic differential operators MSC (2010) Primary, 47B25, Secondary, 47B38, 35J25Let ⊂ R n be bounded with a smooth boundary and let S be the symmetric operator in L 2 ( ) given by the minimal realization of a second order elliptic differential operator. We give a complete classification of the Markovian self-adjoint extensions of S by providing an explicit one-to-one correspondence between such extensions and the class of Dirichlet forms in L 2 ( ) which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of an additive decomposition of the bilinear forms associated to the extensions, the second one uses the additive decomposition of the resolvents provided by Kreȋn's formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell-type boundary conditions. Some properties of the extensions, and of the corresponding Dirichlet forms, semigroups and heat kernels, like locality, regularity, irreducibility, recurrence, transience, ultracontractivity and Gaussian bounds are also discussed.

show abstract

Section: 1 One Hasmentioning

confidence: 55%

Section: Dirichlet Forms and Hunt Processesmentioning

confidence: 99%

Section: Dirichlet Forms and Paths Behaviormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Markovian extensions of symmetric second order elliptic differential operators

Posilicano

2014

Math. Nachr.

View full text Add to dashboard Cite

show abstract

“…Then the following statements are equivalent due to Theorem 3.5.6 in [2] and a similar proof of Lemma 2.1:…”

Section: (B) and µ(A N ∩ B) ↑ µ(B) Hence There Exists Some N Such Thmentioning

confidence: 83%

The uniqueness of symmetrizing measure of Markov processes

Ying

Zhao

2010

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

Kopfer

Sturm

2018

Comm Pure Appl Math

View full text Add to dashboard Cite

We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness, and regularity. Of particular interest are properties that characterize the underlying space as a super-Ricci flow as previously introduced by the second author [51]. Our main result yields the equivalence of F dynamic convexity of the Boltzmann entropy on the (time-dependent) L 2 -Wasserstein space, F monotonicity of L 2 -Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time), F gradient estimates for the heat flow acting on functions (forward in time), and F a Bochner inequality involving the time-derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI flow for the (time-dependent) energy in L 2 -Hilbert space and the dual heat flow on probability measures as the unique backward EVI flow for the (timedependent) Boltzmann entropy in L 2 -Wasserstein space.

show abstract

Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

Cited by 120 publications

References 0 publications

Markovian extensions of symmetric second order elliptic differential operators

Markovian extensions of symmetric second order elliptic differential operators

The uniqueness of symmetrizing measure of Markov processes

Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

Contact Info

Product

Resources

About