The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels

By Helmut Abels, M. Kassmann. 2006

67 citations recorded by the source corpus.

We consider the linear integro-differential operator $L$ defined by \[ Lu(x) =\int_\Rn (u(x+y) - u(x) - 1_{[1,2]}(\alpha) 1_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x)) k(x,y) \sd y . \] Here the kernel $k(x,y)$ behaves like $|y|^{-d-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is H\"older-continuous in the first variable, precise definitions are given below. The aim of this work is twofold. On one hand, we study the unique solvability of the Cauchy problem corresponding to $L$. On the other hand, we study the martingale problem for $L$. The analytic results obtained for the deterministic parabolic equation guarantee that the martingale problem is well-posed. Our strategy follows the classical path of Stroock-Varadhan. The assumptions allow for cases that have not been dealt with so far.

View Paper: https://www.semanticscholar.org/paper/423dc30de8798dc2e64355e1f812c3c8a90b1b3f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

On the Cauchy Problem for Integro-differential Operators in Hölder Classes and the Uniqueness of the Martingale Problem (2011, methodology): “Applying the Ito formula to the solution of (1), we prove the weak uniqueness of the solution to the martingale problem, generalizing so the uniqueness results in [1] and [13].”
On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem (2011, background): “In [1], the existence and uniqueness of a solution to (1) in Hölder spaces was proved analytically for m Hölder continuous in x, smooth in y and such that K ≥ m ≥ η > 0 (4) without assumption of homogeneity in y.”
Ergodicity of stochastic differential equations with jumps and singular coefficients (2017, background): “Then for any T > 0, p > d + 1 and α ∈ [1, 2], there is a constant c = c(T, p, d, α) > 0 such that for any stopping time τ and f ∈ L(T ),”
Regularity results for nonlocal parabolic equations (2013, background): “We assume that a : [0,∞)× Rd × Rd → [1, 2] satisfies a(t, x, y) = a(t, y, x) for all t, x, y and (μ(x, ·))x∈Rd is a family of measures satisfying the following assumptions.”
Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints (2012, background): “For instance, Harnack inequalities, heat kernel estimates, and Hölder continuity for L are the subjects of [13, 14]; the Dirichlet fractional Laplacian and Cauchy martingale problems for L are studied in [24] and [1], respectively.”
Schauder estimates for a class of non-local elliptic equations (2011, background): “We also refer the reader to [1], where a similar well-posedness result was proved under the condition that the principal part of the kernel is smooth in y and Hölder in x.”