Space and time inversions of stochastic processes and Kelvin transform

Alili, Larbi; Chaumont, Loïc; Graczyk, Piotr; Żak, Tomasz

doi:10.1002/mana.201700152

Cited by 5 publications

(2 citation statements)

References 39 publications

(151 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then the transformed process T (X δ t ) is symmetric with respect to 1/(|x| 2d−2δ )dx and the Dirichlet form generated by T (X δ t ) is written as τt has the same law as that of T (X δ t ). In other words, X δ t has the inversion property (cf, [2]). By the time-change, the recurrence and transience are invariant, and thus the transience of X δ implies that of X δ .…”

Section: Symmetric α-Stable Process: Recurrence and Transiencementioning

confidence: 99%

Criticality of Schrödinger Forms and Recurrence of Dirichlet Forms

Takeda¹,

Uemura²

2021

Preprint

View full text Add to dashboard Cite

Introducing the notion of extended Schrödinger spaces, we define the criticality and subcriticality of Schrödinger forms in the same manner as the recurrence and transience of Dirichlet forms, and give a sufficient condition for the subcriticality of Schrödinger forms in terms the bottom of spectrum. We define a subclass of Hardy potentials and prove that Schrödinger forms with potentials in this subclass are always critical, which leads us to optimal Hardy type inequality. We show that this definition of criticality and subcriticality is equivalent to that there exists an excessive function with respect to Schrödinger semigroup and its generating Dirichlet form through h-transform is recurrent and transient respectively. As an application, we can show the recurrence and transience of a family of Dirichlet forms by showing the criticality and subcriticaly of Schrödinger forms and show the other way around through h-transform, We give a such example with fractional Schrödinger operators with Hardy potential.

show abstract

Section: Symmetric α-Stable Process: Recurrence and Transiencementioning

confidence: 99%

Criticality of Schrödinger Forms and Recurrence of Dirichlet Forms

Takeda¹,

Uemura²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…For example, the convex minorant of a Cauchy meander has infinitely many faces in any neighborhood of the origin since the set of its slopes is a.s. dense in R. Hence, if a generalisation of the description in [PR12] to other Lévy meanders existed, it could work only if the sole accumulation point is the right end of the interval. Moreover, the scaling and time inversion properties of Brownian motion, not exhibited by other Lévy processes [GY05,ACGZ19], are central in the description of [PR12].…”

mentioning

confidence: 99%

$\varepsilon$-strong simulation of the convex minorants of stable processes and meanders

Cázares,

Mijatović,

Bravo

2019

Preprint

View full text Add to dashboard Cite

Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisaiton to construct ε-strong simulation (εSS) algorithms for the convex minorant of stable meanders, the finite dimensional distributions of stable meanders and the convex minorants of weakly stable processes. We prove that the running times of our εSS algorithms have finite exponential moments. We implement the algorithms in Julia 1.0 (available on GitHub) and present numerical examples supporting our convergence results.

show abstract

Criticality of Schrödinger forms and recurrence of Dirichlet forms

Takeda¹,

Uemura²

2023

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Introducing the notion of extended Schrödinger spaces, we define the criticality and subcriticality of Schrödinger forms in the manner similar to the recurrence and transience of Dirichlet forms. We show that a Schrödinger form is critical (resp. subcritical) if and only if there exists an excessive function of the associated Schrödinger semigroup and the Dirichlet form defined by h h -transform of the excessive function is recurrent (resp. transient). We give an analytical condition for the subcriticality of Schrödinger forms in terms of the bottom of spectrum. We introduce a subclass K H {\mathcal {K}}_H of the local Kato class and show a Schrödinger form with potential in K H {\mathcal {K}}_H is critical. Critical Schrödinger forms lead us to critical Hardy-type inequalities. As an example, we treat fractional Schrödinger operators with potential in K H {\mathcal {K}}_H and reconsider the classical Hardy inequality by our approach.

show abstract

Space and time inversions of stochastic processes and Kelvin transform

Cited by 5 publications

References 39 publications

Criticality of Schrödinger Forms and Recurrence of Dirichlet Forms

Criticality of Schrödinger Forms and Recurrence of Dirichlet Forms

$\varepsilon$-strong simulation of the convex minorants of stable processes and meanders

Criticality of Schrödinger forms and recurrence of Dirichlet forms

Contact Info

Product

Resources

About