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Approximate factorizations for non-symmetric jump processes

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Pith's one-line read Under $C^{1,\mathrm{Dini}}$ boundary regularity, the paper proves sharp two-sided heat kernel estimates for three classes of jump processes, with explicit boundary-decay exponents for killed and censored $\alpha$-stable processes.

desk verdict Genuinely new heat kernel estimates for stable processes in Dini-type domains, but the proof of the key factorization theorem is omitted and pointed to previous work, so the central mechanism is not fully verifiable in this text. read the letter →

arxiv 2504.14763 v2 pith:RXYPD3MO submitted 2025-04-20 math.PR math.AP

Soobin Cho , Renming Song This is my paper · ORCID
classification math.PRmath.AP MSC 60J4560J5060J7647G2035K08
keywords heatkernelGreenfunctionregionalfractionalLaplacianstable-likeprocessesapproximatefactorizationC^{1Dini}opensetscensoredstableprocessnon-symmetric
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves sharp two-sided heat kernel estimates for non-local jump processes in rough open sets, with explicit boundary-decay factors. The tool is an approximate factorization: on short time scales the Dirichlet heat kernel is, up to universal constants, the product of two boundary-decay factors and the free-space heat kernel. The paper extends this factorization to a broader class of Feynman–Kac processes by removing two earlier assumptions, and shows that the heat-kernel factorization is equivalent to a Green-function factorization. The payoff is explicit estimates for killed $\alpha$-stable processes in $C^{1,\mathrm{Dini}}$ domains for all $\alpha\in(0,2)$, for censored $\alpha$-stable processes in the same domains for $\alpha\in(1,2)$, and for non-symmetric stable processes with direction-dependent boundary exponents.

What carries the argument

The load-bearing identity is the approximate factorization $p(t,x,y)\asymp P_x(\zeta>t)\hat P_y(\hat\zeta>t)\bar q(t,x,y)$ for the Dirichlet heat kernel, with the analogous Green-function factorization $G(x,y)\asymp P_x(\zeta>\phi(d(x,y)))\hat P_y(\hat\zeta>\phi(d(x,y)))\bar G(x,y)$, and a proof that the two factorizations are equivalent. The boundary exponent is extracted from the half-space identity $\mathrm{p.v.}\int_{\mathbb{R}^d_+}(y_d^q-x_d^q)/|x-y|^{d+\alpha}\,dy=C(d,\alpha,q)x_d^{q-\alpha}$, which matches the critical killing rate to the boundary decay. A regularized distance $\rho$ with controlled gradient modulus and Hessian converts this half-space computation into estimates on $D$, and barrier functions built from $\rho^q$ force the survival probabilities $P_x(\zeta>t)$ to decay like $(\delta_D(x)/t^{1/\alpha})^q$.

What would settle it

Take the $C^{1,\mathrm{Dini}}$ domain $D=\{(x_1,x_2):x_2>x_1(1+|\log x_1|)^{-k}\}$ with $k>1$, fix $x=(0,t^{1/\alpha})$ and $y=x$, and estimate the killed $\alpha$-stable heat kernel $p(t,x,x)$ as $t\downarrow0$. If it is not comparable to $t^{-d/\alpha}(\delta_D(x)/t^{1/\alpha})^{\alpha/2}$, the boundary-exponent claim fails.

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Extended reading notes

Core claim

The central claim is Theorem 5.7: if $D$ is a $C^{1,\mathrm{Dini}}$ open set, the jump coefficient is bounded and Hölder near the diagonal, and the killing measure $\kappa$ belongs to the critical class $\mathcal{K}_\alpha(q)$, then the heat kernel of the Feynman–Kac process satisfies $p(t,x,y)\asymp (1\wedge \delta_D(x)/t^{1/\alpha})^q(1\wedge \delta_D(y)/t^{1/\alpha})^q(t^{-d/\alpha}\wedge t/|x-y|^{d+\alpha})$ on $(0,T]$. The boundary exponent $q$ is determined by the constant $C(d,\alpha,q)$ in the killing rate $\kappa(x)\approx C(d,\alpha,q)K(x,x)\delta_D(x)^{-\alpha}$. As corollaries, killed stable-like processes have $q=\alpha/2$ and censored stable-like processes have $q=\alpha-1$; non-symmetric stable processes instead have exponents $\gamma(n_{Q_x})$ and $\alpha-\gamma(n_{Q_y})$ set by the directional asymmetry of the Lévy measure.

Load-bearing premise

The argument needs enough control of the region where the domain differs from its tangent half-space, obtained from a Dini boundary modulus that is made power-comparable; if that boundary control is absent, the error integrals in the proof do not close and the boundary exponent would not follow.

Editorial extensions

If this is right

  • Killed $\alpha$-stable-like processes in $C^{1,\mathrm{Dini}}$ domains have explicit heat kernel boundary decay $(\delta_D(x)/t^{1/\alpha})^{\alpha/2}$ for every $\alpha\in(0,2)$.
  • Censored $\alpha$-stable processes in $C^{1,\mathrm{Dini}}$ domains have boundary exponent $\alpha-1$ for every $\alpha\in(1,2)$.
  • For non-symmetric stable processes, heat kernel boundary decay is asymmetric: the factor at $x$ decays like $\delta_D(x)^{\gamma(n_{Q_x})}$ while the factor at $y$ decays like $\delta_D(y)^{\alpha-\gamma(n_{Q_y})}$.
  • In bounded domains the same estimates give Green-function bounds $G(x,y)\asymp(1\wedge\delta_D(x)/|x-y|)^q(1\wedge\delta_D(y)/|x-y|)^q|x-y|^{\alpha-d}$ and large-time spectral decay $e^{-\lambda_1 t}\delta_D(x)^q\delta_D(y)^q$.
  • Because heat-kernel and Green-function factorizations are equivalent under the paper's conditions, future Green-function estimates in this framework automatically yield heat-kernel estimates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof regularizes the boundary modulus into one satisfying a power-comparability condition; a natural test is whether the same boundary exponents survive for $C^{1,\mathrm{Dini}}$ domains whose original modulus decays slower than every power, such as the example in Remark 5.3.
  • The direction-dependent exponents for non-symmetric stable processes suggest that variable-coefficient versions will have boundary exponents depending on position through the local Lévy measure, not merely through the normal direction.
  • The factorization route is likely to transfer to other critical-killing Feynman–Kac processes with different jump kernels, producing boundary Harnack principles and Green-function asymptotics whenever the half-space killing identity can be matched.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper develops an approximate factorization for the heat kernel and Green function of a purely discontinuous Markov process with critical killings in η-fat open sets, and then applies it to three families of processes: stable-like processes with critical killings in C^{1,Dini} domains, killed stable-like processes in the setting of Kim–Weidner, and non-symmetric stable processes in C^{1,2-Dini} domains. The main abstract factorization is Theorem 4.5, which is imported from the authors' prior work [12] with the proof omitted; the later boundary estimates are obtained through explicit calculation with barrier functions and a regularized distance. The paper derives explicit boundary exponents, in particular q=α/2 for killed α-stable processes and q=α−1 for censored α-stable processes in C^{1,Dini} domains, and exposes a dependence of the exponent on the normal direction in the non-symmetric case.

Significance. If the results are fully established, this is a substantial contribution: it significantly weakens the regularity assumptions under which sharp two-sided heat kernel and Green function estimates are known for non-local operators, and it covers non-symmetric stable processes with variable boundary exponents. The paper contains many detailed and apparently correct barrier computations, and the regularization lemmas in the appendix are effective tools. The main reservation is that the central engine, Theorem 4.5, is not proved in this manuscript and is inherited from [12] without a detailed verification that the hypotheses used in [12] are indeed dispensed with. The rest of the paper is built on this theorem, so the current version is not fully verifiable as a standalone contribution.

major comments (2)
  1. [Section 4, Theorem 4.5] The approximate factorization (4.15) is the central tool for every subsequent application, but its proof is omitted: the text states 'following the arguments in [12, Theorems 2.22 and 2.23], we obtain the next theorem. We omit the proof.' The abstract and Introduction announce that the paper removes Assumptions A and U from [12], yet those assumptions are never stated, and the reader is not told which ingredients of the proof in [12] are replaced by the survival estimates in Section 3, Lemma 4.2, Corollary 4.3, and Proposition 4.4. Since Theorems 4.8, 4.11, Corollary 4.12, and the applications in Section 5 all call on Theorem 4.5, this is a load-bearing gap. Please add at least a detailed proof sketch that explicitly identifies the differences from [12], states what Assumptions A and U are, and verifies that they are not needed under the hypotheses of Theorem 4.5.
  2. [Section 4.1, Theorem 4.8] The statement of Theorem 4.8 uses Py(ζ>1) in both the upper and lower bounds, but the proof uses bPy(bζ>1) throughout, and the later application in Theorem 4.11 also uses bPy(bζ>1). In the general nonsymmetric setting the transition density is not symmetric, so the theorem as stated appears to be false; the correct statement should involve the survival probability of the dual process at y. Please correct the statement to C−1Px(ζ>1)bPy(bζ>1)e−λ1t ≤ p(t,x,y) ≤ C Px(ζ>1)bPy(bζ>1)e−λ1t, or explain why the two factors coincide under the stated hypotheses.
minor comments (3)
  1. [Section 5, Lemma 5.5] The polynomial-scale condition (5.15) is not satisfied by every Dini modulus, so as stated Lemma 5.5 appears to require an extra assumption. The later text uses the regularized modulus from Lemma A.2, which does satisfy (5.15), but this should be stated explicitly before Lemma 5.5 is used so that the hypothesis is not read as an additional regularity restriction on the domain.
  2. [Corollary 5.14, proof] The proof states that lim_{s→0} s^{q−α}ℓ(s) ≥ lim_{s→0} s^{(q−α)/2}ℓ(1) = 0, but for q<α the exponent (q−α)/2 is negative, so the displayed limit is infinite, not zero. The subsequent choice of σ0 is still valid, but the displayed limit should be corrected.
  3. [Proposition 5.13, proof of (ii)] At the end of part (ii), the text says 'we obtain (5.40)' after proving the estimate for h_{r,σ}; the displayed estimate being proved is (5.39). Please correct the cross-reference.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: heat-kernel exponents are computed from killing and boundary data, not fitted; the main caveat is a load-bearing but non-circular omission of the proof of Theorem 4.5, delegated to the authors' prior paper [12].

full rationale

The central derivation is self-contained apart from Theorem 4.5. The approximate factorization (4.15) is imported from [12, Theorems 2.22 and 2.23] with proof omitted: 'following the arguments in [12, Theorems 2.22 and 2.23], we obtain the next theorem. We omit the proof.' [12] overlaps with the authors and is load-bearing, and the abstract's promised removal of Assumptions A and U is not verified by any proof in this paper. However, this is a verification gap, not circularity: the factorization was proved in [12], and the paper supplies the new survival estimates (Propositions 3.4 and 4.4, Lemma 4.2, Corollary 4.3) needed to fit the framework, rather than defining the target estimate into its assumptions. The boundary exponents q, alpha/2, alpha-1, gamma(n_Q), and gamma-hat(n_Q) are computed from the killing-measure asymptotics (5.25), the half-space calculation (5.29), and the Levy-exponent formula (5.56), not fitted to the heat kernel. Lemma 5.6 computes kappa_D(x) from the stable-like kernel K and the C^{1,Dini} boundary, and Corollaries 5.14 and 5.20 derive boundary survival via barrier functions. Theorem 5.22 is obtained from the Green function estimate of [22] via Corollary 4.12, an external benchmark. The Dini-modulus concern around Lemma 5.5 is mitigated by the regularization Lemma A.2. No equation reduces to its own input; the score reflects the unresolved self-cited Theorem 4.5, not circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper fits no parameters to data. The exponents q and gamma are determined by the killing coefficient and the Levy measure, respectively. The central estimates rest on stated regularity and ellipticity assumptions and on published theorems; the proof of the approximate factorization is inherited from [12] with an omitted proof. No fitted constants or invented entities were found.

assumptions (9)
  • domain assumption Metric measure space satisfies volume doubling (2.1), reverse volume doubling (2.2), and weak scaling of phi (2.4).
    Used throughout Sections 2-4 to control ball volumes and to define the heat kernel scale eq(t,x,y); standard in Dirichlet heat kernel theory and assumed, not proved.
  • domain assumption Base Hunt process Y satisfies the two-sided heat kernel estimate (A) with respect to eq.
    This is the starting point in Section 2.1; all jump kernel bounds and the existence of densities derive from it. For stable-like processes it is supplied by [10, Theorem 1.1].
  • domain assumption Killing measure kappa belongs to the Kato class K1(D) (Definition 2.2); examples allow kappa(x) up to C/phi(delta_D(x)).
    The whole factorization section assumes kappa in K1(D). Critical killings of type C delta^{-alpha} are in this class because phi(r) = r^alpha.
  • domain assumption Open set D is eta-fat, with the ball condition in Definition 4.1.
    The factorization Theorem 4.5 and Green function Theorem 4.11 require eta-fatness; C^{1,Dini} domains are eta-fat.
  • domain assumption The regularized distance rho satisfies (5.6)-(5.8), and Lemma 5.5 requires condition (5.15), i.e. the boundary modulus is controlled by a power.
    The barrier estimates in Lemmas 5.6, 5.12 and Propositions 5.13 and 5.19 rely on these estimates; Lemma A.2 provides such an ell for any Dini function.
  • domain assumption Coefficient K is symmetric, bounded, and Holder near the diagonal with exponent theta > (alpha-1)_+ (5.20), (5.22), (5.23).
    These regularity hypotheses are used to estimate the error terms when replacing K(x0,y) by K(x0,x0) in the killing rate and in the operator applied to barriers.
  • domain assumption For non-symmetric processes, the spherical measure m(dxi) has density m(xi) bounded above and below (5.54).
    This non-degeneracy gives the two-sided Green and harmonic estimates from [15] used to identify gamma(theta) and to control generator errors.
  • domain assumption Green function condition (G) in (4.23) holds for the process, for example when beta1 > alpha2.
    Needed in Theorem 4.11 and Corollary 4.12 for the upper Green bound and the equivalence with heat kernel estimates.
  • standard math External results [10, Theorem 1.1], [15, Corollary 4.7], [22, Theorem 1.1], and [13, Proposition 5.1] are used as inputs.
    The paper relies on published or preprint results for heat kernel estimates of stable-like processes, stable operator identities, Kim-Weidner Green function estimates, and generator identification. [13] is an author preprint, which increases uncertainty but is not circular by itself.

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Pith. "Pith review of Approximate factorizations for non-symmetric jump processes." pith.science (2026). https://pith.science/paper/RXYPD3MO

@misc{pith2026250414763,
  author       = {Pith},
  title        = {Pith review of: Approximate factorizations for non-symmetric jump processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RXYPD3MO}},
  note         = {Machine review of arXiv:2504.14763}
}
abstract

In this paper, we first extend the approximate factorization for purely discontinuous Markov process established in \cite{CKSV20} by getting rid of some of the conditions imposed in \cite{CKSV20}. Then we apply the approximate factorization to obtain sharp two-sided heat kernel estimates for three classes of processes: stable-like processes with critical killings in $C^{1, {\rm Dini}}$ open sets; killed stable-like processes in the setting of \cite{KW24} in $C^{1, \varepsilon}$ open sets; and non-symmetric stable processes in what we call $C^{1,2{\text - \rm Dini}}$ open sets. In particular, we obtain explicit sharp two-sided heat kernel estimates of killed $\alpha$-stable processes in $C^{1, {\rm Dini}}$ open sets for all $\alpha\in (0, 2)$ and of censored $\alpha$-stable processes in $C^{1, {\rm Dini}}$ open sets for all $\alpha\in (1, 2)$.

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Works this paper leans on

40 extracted references · 37 canonical work pages · cited by 2 Pith papers

  1. [12]

    S. Cho, P. Kim, R. Song, Z. Vondraˇ cek: Factorization and estimates of Dirichlet heat kernels for non-local operators with critical killings. J. Math. Pures Appl. 143 (2020), 208–256

    work page 2020

  2. [13]

    S. Cho, P. Kim, R. Song, Z. Vondraˇ cek: Markov processes with jump kernels decaying at the boundary. arXiv:2403.00480

    work page Pith review arXiv

  3. [14]

    S. Cho, R. Song: Abnormal boundary decay for fractional Laplacian. In preparation

    work page

  4. [1]

    Abatangelo, X

    N. Abatangelo, X. Ros-Oton: Obstacle problems for integro-differential operators: higher regularity of free boundaries. Adv. Math., 360 (2020), 106931

    work page 2020

  5. [2]

    R. F. Bass: Adding and subtracting jumps from Markov processes. Trans. Amer. Math. Soc.255 (1979), 363–376

    work page 1979

  6. [3]

    R. F. Bass, K. Burdzy: A probabilistic proof of the boundary Harnack principle. Seminar on Stochastic Processes (1989), 1-16, Birkh¨ auser Boston, 1990

    work page 1989

  7. [4]

    Benveniste, J

    A. Benveniste, J. Jacod: Syst´ emes de L´ evy des processus de Markov.Invent. Math. 21 (1973), 183–198

    work page 1973

  8. [5]

    Bogdan, K

    K. Bogdan, K. Burdzy, Z.-Q. Chen: Censored stable processes. Probab. Theory Related Fields 127 (2003), 89–152. 54

    work page 2003

Show all 40 references
  1. [6]

    Bogdan, T

    K. Bogdan, T. Grzywny, M. Ryznar: Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab., 38(5) (2010), 1901–1923

    2010

  2. [7]

    Cao, Z.-Q

    S. Cao, Z.-Q. Chen: Uniform boundary Harnack principle for non-local operators on metric measure spaces. arXiv:2410.20719

    arXiv

  3. [8]

    Z.-Q. Chen, P. Kim, R. Song: Heat kernel estimates for Dirichlet fractional Laplacian. J. European Math. Soc., 12 (2010), 1307–1329

    2010

  4. [9]

    Z.-Q. Chen, P. Kim, R. Song: Two-sided heat kernel estimates for censored stable-like processes. Probab. Theory Related Fields 146 (2010), no. 3-4, 361–399

    2010

  5. [10]

    Z.-Q. Chen, T. Kumagai: Heat kernel estimates for stable-like processes on d-sets. Stochastic Process. Appl. 108 (2003), 27–62

    2003

  6. [11]

    Z.-Q. Chen, T. Kumagai, J. Wang: Stability of heat kernel estimates for symmetric jump processes on metric measure spaces. Mem. Amer. Math. Soc. 271 (2021), no. 1330

    2021

  7. [15]

    Dipierro, X

    S. Dipierro, X. Ros-Oton, J. Serra, E. Valdinoci: Non-symmetric stable operators: regularity theory and integration by parts. Adv. Math. 401 (2022), Paper No. 108321, 100 pp

    2022

  8. [16]

    R. K. Getoor: Duality of L´ evy systems. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19 (1971), 257–270

    1971

  9. [17]

    Grubb: Fractional Laplacians on domains, a development of H¨ ormander’s theory of µ-transmission pseudodifferential operators

    G. Grubb: Fractional Laplacians on domains, a development of H¨ ormander’s theory of µ-transmission pseudodifferential operators. Adv. Math. 268 (2015), 478–528

    2015

  10. [18]

    Grube: Boundary regularity and Hopf lemma for non-degenerate stabel operators

    F. Grube: Boundary regularity and Hopf lemma for non-degenerate stabel operators. arXiv:2410.00829

    arXiv

  11. [19]

    Juszczyszyn: Decay rate of harmonic functions for non-symmetric strictly α-stable L´ evy processes

    T. Juszczyszyn: Decay rate of harmonic functions for non-symmetric strictly α-stable L´ evy processes. Studia Math. 260 (2021), 141–165

    2021

  12. [20]

    K-Y. Kim, P. Kim: Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C 1,η open sets. Stochastic processes and their applications 2014 124(9) (2014), 3055–3083

    2014

  13. [21]

    Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes inC 1,η open sets

    K-Y. Kim, P. Kim: Erratum to “Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes inC 1,η open sets” [Stochastic Process. Appl. 124(9) (2014) 3055–3083]. Stochastic processes and their applications 2014 129(6) (2019...

    2014

  14. [22]

    M. Kim, M. Weidner: Optimal boundary regularity and Green function estimates for nonlocal equations in divergence form. arXiv:2408.12987

  15. [23]

    P. Kim, R. Song, Z. Vondraˇ cek: Harnack inequality and interior regularity for Markov processes with degenerate jump kernels. J. Differential Equations 357 (2023), 138–180

    2023

  16. [24]

    G. M. Lieberman: Regularized distance and its applications. Pac. J. Math. 117 (1985), 329–352

    1985

  17. [25]

    Z.-M. Ma, M. R¨ ockner: Introduction to the theory of (nonsymmetric) Dirichlet forms. Springer-Verlag, Berlin, 1992

    1992

  18. [26]

    Masamune, T

    J. Masamune, T. Uemura: Conservation property of symmetric jump processes. Ann. Inst. Henri Poincar´ e Probab. Stat.47 (2011), 650–662

    2011

  19. [27]

    Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations

    A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983. 55

    1983

  20. [28]

    Y.-X. Ren, R. Song, R. Zhang: Central limit theorems for supercritical branching nonsymmetric Markov processes. Ann. Probab. 45 (2017), 564–623

    2017

  21. [29]

    Ros-Oton, J

    X. Ros-Oton, J. Serra: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. , 101 (2014), 275–302

    2014

  22. [30]

    Ros-Oton, J

    X. Ros-Oton, J. Serra: Boundary regularity for fully nonlinear integro-differential equations. Duke Math. J. 165 (2016), 2079–2154

    2016

  23. [31]

    Ros-Oton, J

    X. Ros-Oton, J. Serra: Regularity theory for general stable operators. J. Differential Equations , 260 (2016), 8675–8715

    2016

  24. [32]

    Ros-Oton, J

    X. Ros-Oton, J. Serra: Boundary regularity estimates for nonlocal elliptic equations in C 1 and C 1,α domains. Ann. Mat. Pura Appl. 196 (2017), 1637–1668

    2017

  25. [33]

    Ros-Oton, M

    X. Ros-Oton, M. Weidner: Optimal regularity for nonlocal elliptic equations and free boundary prob- lems. arXiv:2403.07793

    arXiv

  26. [34]

    Sato: L´ evy processes and infinitely divisible distributions

    K.-i. Sato: L´ evy processes and infinitely divisible distributions. Cambridge University Press, Cambridge, 2013

    2013

  27. [35]

    H. H. Schaefer: Banach Lattices and Positive Operators. Springer, New York. 1974

    1974

  28. [36]

    M. A. Shubin: Spectral theory of elliptic operators on noncompact manifolds. M´ ethodes semi-classiques, Vol. 1 (Nantes, 1991). Ast´ erisque No. 207 (1992), 5, 35–108

    1992

  29. [37]

    R. Song, P. Wu and S. Wu: Heat kernel estimates for regional fractional Laplacians with multi-singular critical potentials in C 1,β open sets. Stoch. Proc. Appl., 189 (2025), 104727

    2025

  30. [38]

    E. M. Stein: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, NJ, 1970

    1970

  31. [39]

    Watanabe: Asymptotic estimates of multi-dimensional stable densities and their applications

    T. Watanabe: Asymptotic estimates of multi-dimensional stable densities and their applications. Trans. Amer. Math. Soc. 359 (2007), 2851–2879

    2007

  32. [40]

    Ying, Bivariate Revuz measures and the Feynman-Kac formula

    J. Ying, Bivariate Revuz measures and the Feynman-Kac formula. Ann. Inst. H. Poincar´ e Probab. Statist., 32 (1996), 251–287. Soobin Cho: Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL 61801, U.S.A. Email: soobinc@illinois.edu Renming Song: Depa...

    1996

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