On the martingale problem associated with nondegenerate Lévy operators

The existence and uniqueness in Sobolev spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order α ∈ (0, 2) is investigated. The principal part of the operator has kernel m(t, x, y)/|y| d+α with a bounded nondegenerate m, Hölder in x and measurable in y. The lower order part has bounded and measurable coefficients. The result is applied to prove the existence and uniqueness of the corresponding martingale problem.MSC classes: 45K05, 60J75, 35B65

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“…We say that a probability measure P on (D, D) is a solution to the (s, x, L)martingale problem (see [16], [11]…”

Section: Martingale Problemmentioning

confidence: 99%

“…We denote S(s, x, L) the set of all solutions to the problem (s, x, L)-martingale problem. A modification of Theorem 5 in [11] is the following statement.…”

Section: Martingale Problemmentioning

confidence: 99%

On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem

Mikulevicius

Pragarauskas

2014

Journal of Differential Equations

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“…It follows from [8] and [10] that, under the assumptions above, the martingale problem for L is well-posed. That is, there is a conservative strong Markov process X = (X t , P x , x ∈ R Theorem 5.1 Under the assumptions above, the Harnack inequality holds for X.…”

Section: Non-symmetric Markov Processes With No Diffusion Componentmentioning

confidence: 99%

Harnack inequality for some classes of Markov processes

Song

Vondraček

2004

Mathematische Zeitschrift

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“…See, e.g., [4,5,19,20,22,23,26,28] and the references therein. In particular, Komatsu [20] and Mikulevicious-Pragarauskas [22] considered martingale problem for a class of non-local operators that is directly related to L b . In fact, the uniqueness of the martingale problem for (L b , S(R d )) stated in Theorem 1.3 above is a direct consequence of [20,Theorem 3], while it follows from [22,Theorem 5] that for any bounded b satisfying (1.2) and (1.19), there is a unique solution to the martingale problem (L b , C ∞ c (R d )).…”

Section: Introductionmentioning

confidence: 99%

Perturbation by non-local operators

Chen¹,

Wang²

2018

Ann. Inst. H. Poincaré Probab. Statist.

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Suppose that d ≥ 1 and 0 < β < α < 2. We establish the existence and uniqueness of the fundamental solution q b (t, x, y) to a class of (typically nonsymmetric) non-local operatorsis a strictly positive continuous function and it uniquely determines a conservative Feller process X b , which has strong Feller property. The Feller process X b is the unique solution to the martingale problem of (L b , S(R d )), where S(R d ) denotes the space of tempered functions on R d . Furthermore, sharp two-sided estimates on q b (t, x, y) are derived. In stark contrast with the gradient perturbations, these estimates exhibit different behaviors for different types of b(x, z). The model considered in this paper contains the following as a special case. Let Y and Z be (rotationally) symmetric α-stable process and symmetric β-stable processes on R d , respectively, that are independent to each other. Solution to stochastic differential equations dX t = dY t + c(X t− )dZ t has infinitesimal generator L b with b(x, z) = |c(x)| β .

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On the martingale problem associated with nondegenerate Lévy operators

Cited by 31 publications

References 8 publications

On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem

On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem

Harnack inequality for some classes of Markov processes

Perturbation by non-local operators

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