Symmetric Stable Laws and Stable-Like Jump-Diffusions

Kolokoltsov, Vassili N.

doi:10.1112/s0024611500012314

Cited by 183 publications

(219 citation statements)

References 18 publications

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“…In case B t (x) = 0 and A t (x) = A(x) being time independent this result is obtained in [32] (the existence of the Green function in case ψ = 0 being previously obtained in [31]). It remains to observe that the proof from [32] generalizes straightforwardly to the present non-homogeneous situation.…”

Section: Exists a Unique (Classical) Solution G T In C ∞ (X) To The Cmentioning

confidence: 52%

“…The well posedness of the Cauchy problem for equation (37) under the conditions of the Theorem and with bounded ψ was shown first seemingly in [43]. In [32], [33] it was shown that in this case there exists a Green function G(t, x, x 0 ; [u]) of equation (37), which is continuously differentiable in t, x, x 0 for t > 0 and satisfies the Chapman-Kolmogorov equation. In [31] the same result was obtained for Γ k being a finite sum of fractional powers of the Laplace operator (stable laws).…”

mentioning

confidence: 99%

“…for ψ = 0. On the other hand, the series representation for G(t, x, x 0 ; [u]) obtained in [32] for non-vanishing ψ shows that the integral operator Γ can be treated as a small perturbation that does not affect the validity of (38).…”

mentioning

confidence: 99%

“…In [31] the same result was obtained for Γ k being a finite sum of fractional powers of the Laplace operator (stable laws). Both proofs from [32] and [31] can be easily generalized to give the existence of a differentiable Green function under the above conditions on ψ . Moreover, as the same holds for the dual equation and the dual equation preserves constants it follows that the solution of (37) preserves the L 1 -norm.…”

mentioning

confidence: 99%

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Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Kolokoltsov

2006

J Stat Phys

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Semigroups of positivity preserving linear operators on measures of a measurable space X describe the evolutions of probability distributions of Markov processes on X. Their dual semigroups of positivity preserving linear operators on the space of measurable bounded functions B(X) on X describe the evolutions of averages over the trajectories of these Markov processes. In this paper we introduce and study the general class of semigroups of non-linear positivity preserving transformations on measures that is non-linear Markov or Feller semigroups. An explicit structure of generators of such groups is given in case when X is the Euclidean space R d (or more generally, a manifold) showing how these semigroups arise from the general kinetic equations of statistical mechanics and evolutionary biology that describe the dynamic law of large numbers for Markov models of interacting particles. Well posedness results for these equations are given together with applications to interacting particles: dynamic law of large numbers and central limit theorem, the latter being new already for the standard coagulation-fragmentation models.

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Section: Exists a Unique (Classical) Solution G T In C ∞ (X) To The Cmentioning

confidence: 52%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Kolokoltsov

2006

J Stat Phys

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“…and to evaluate the inverse Fourier transform with the aid of Polya integrals; see for instance [11].…”

Section: Proposition 23mentioning

confidence: 99%

Gradient estimates and symmetrization for Fisher–KPP front propagation with fractional diffusion

Roquejoffre

Tarfulea

2017

Journal de Mathématiques Pures et Appliquées

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In this paper, we study gradient decay estimates for solutions to the multidimensional Fisher-KPP equation with fractional diffusion. It is known that this equation exhibits exponentially advancing level sets with strong qualitative upper and lower bounds on the solution. However, little has been shown concerning the gradient of the solution. We prove that, under mild conditions on the initial data, the first and second derivatives of the solution obey a comparative exponential decay in time. We then use this estimate to prove a symmetrization result, which shows that the reaction front flattens and quantifiably circularizes, losing its initial structure.

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Transition probabilities of Lévy‐type processes: Parametrix construction

Kühn

2018

Mathematische Nachrichten

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We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol q(x,ξ) with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.

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Symmetric Stable Laws and Stable-Like Jump-Diffusions

Cited by 183 publications

References 18 publications

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

Gradient estimates and symmetrization for Fisher–KPP front propagation with fractional diffusion

Transition probabilities of Lévy‐type processes: Parametrix construction

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