Heat content for convolution semigroups

Cygan, Wojciech; Grzywny, Tomasz

doi:10.1016/j.jmaa.2016.09.051

Cited by 16 publications

(36 citation statements)

References 21 publications

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“…For the existence of the limit (18) for a fixed x ∈ R d the behaviour of f close to x ∈ R d is crucial. For instance, if X t ∶= t is a deterministic drift process, then the limit (18) exists if, and only if, f is differentiable at x. This means that we have to make an assumption on the local regularity of f at x, typically Hölder continuity or differentiability.…”

Section: Pointwise Limitsmentioning

confidence: 99%

On the domain of fractional Laplacians and related generators of Feller processes

Kühn

Schilling

2019

Journal of Functional Analysis

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In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo-and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy-and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder-Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type processfor functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed x ∈ R d if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of (X t ) t≥0 contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.At the end of Section 5 we discuss several examples, including stable-like dominated processes (Example 5.4), solutions of Lévy-driven SDEs (Example 5.6), stable-like processes and relativistic stable-like processes (Example 5.5). Basic definitions and notationWe consider the Euclidean space R d with the canonical scalar product x⋅y ∶= ∑ d j=1 x j y j and the Borel σ-algebra B(R d ) generated by the open balls B(x, r) ∶= {y ∈ R d ; y − x < r} 3

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Section: Pointwise Limitsmentioning

confidence: 99%

On the domain of fractional Laplacians and related generators of Feller processes

Kühn

Schilling

2019

Journal of Functional Analysis

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“…The heat content with respect to Lévy processes, especially Brownian motions, has been studied extensively, see, for instance, . The spectral heat content

Q_{D}^{(2)} false(t false)

with respect to Brownian motion has also been studied a lot (see ).…”

Section: Introductionmentioning

confidence: 99%

“…The heat content with respect to Lévy processes, especially Brownian motions, has been studied extensively, see, for instance, . The spectral heat content

Q_{D}^{(2)} false(t false)

with respect to Brownian motion has also been studied a lot (see ). In , a two‐term small time expansion for

Q_{D}^{(2)} false(t false)

was established for bounded

C^{1, 1}

domains and in a three‐term small time expansion for

Q_{D}^{(2)} false(t false)

was obtained for bounded domains with

C^{3}

boundary.…”

Section: Introductionmentioning

confidence: 99%

Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic α‐stable processes

Park

Song

2019

Bull. London Math. Soc.

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In this paper, we study the small time asymptotic behavior of the spectral heat content Q (α) D (t) of an arbitrary bounded C 1,1 domain D with respect to the subordinate killed Brownian motion in D via an (α/2)-stable subordinator. For all α ∈ (0, 2), we establish a two-term small time expansion for Q(α) D (t) in all dimensions. When α ∈ (1, 2) and d 2, we establish a three-term small time expansion for Q(α) D (t).

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“…The asymptotic behaviors of the heat content and the spectral heat content have been studied intensively in the case of Brownian motion, see and –. Recently significant progress has also been made in studying the heat content and the spectral heat content with respect to Lévy processes with discontinuous sample paths, see . The asymptotic behaviors of the heat content and the spectral heat content with respect to symmetric stable processes were studied in .…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the exact asymptotic behavior of the spectral heat content of bounded open intervals with respect to symmetric stable processes in

double-struckR

was established in . The asymptotic behavior of the heat content with respect to general Lévy processes was studied in (see also for a generalization). In , an asymptotic expansion of the heat content with respect to some isotropic compound Poisson processes with compactly supported jumping kernels was established.…”

Section: Introductionmentioning

confidence: 99%

Spectral heat content for Lévy processes

Grzywny

Park

Song

2018

Mathematische Nachrichten

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In this paper we study the spectral heat content for various Lévy processes. We establish the small time asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in Rd, d≥1. We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in double-struckR with respect to symmetric Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally, we establish that the small time asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.

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Heat content for convolution semigroups

Cited by 16 publications

References 21 publications

On the domain of fractional Laplacians and related generators of Feller processes

On the domain of fractional Laplacians and related generators of Feller processes

Small time asymptotics of spectral heat contents for subordinate killed Brownian motions related to isotropic α‐stable processes

Spectral heat content for Lévy processes

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