{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T22:51:52Z","timestamp":1775256712755,"version":"3.50.1"},"reference-count":41,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T00:00:00Z","timestamp":1540339200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004543","name":"China Scholarship Council","doi-asserted-by":"publisher","award":["201306160071"],"award-info":[{"award-number":["201306160071"]}],"id":[{"id":"10.13039\/501100004543","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Science Foundation","award":["DMS-1620449"],"award-info":[{"award-number":["DMS-1620449"]}]},{"name":"NNSFs of China","award":["11531006 and 11771449"],"award-info":[{"award-number":["11531006 and 11771449"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2019,4]]},"DOI":"10.1007\/s10444-018-9642-4","type":"journal-article","created":{"date-parts":[[2018,10,24]],"date-time":"2018-10-24T10:12:34Z","timestamp":1540375954000},"page":"787-811","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Fokker-Planck equation driven by asymmetric L\u00e9vy motion"],"prefix":"10.1007","volume":"45","author":[{"given":"Xiao","family":"Wang","sequence":"first","affiliation":[]},{"given":"Wenpeng","family":"Shang","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9571-7482","authenticated-orcid":false,"given":"Xiaofan","family":"Li","sequence":"additional","affiliation":[]},{"given":"Jinqiao","family":"Duan","sequence":"additional","affiliation":[]},{"given":"Yanghong","family":"Huang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,24]]},"reference":[{"issue":"3","key":"9642_CR1","first-page":"661","volume":"46","author":"H Abels","year":"2009","unstructured":"Abels, H., Kassmann, M.: The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. Osaka. J. Math 46(3), 661\u2013683 (2009)","journal-title":"Osaka. J. Math"},{"issue":"2","key":"9642_CR2","doi-asserted-by":"publisher","first-page":"472","DOI":"10.1137\/15M1033952","volume":"55","author":"G Acosta","year":"2017","unstructured":"Acosta, G., Borthagaray, J.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55(2), 472\u2013495 (2017)","journal-title":"SIAM J. Numer. Anal."},{"key":"9642_CR3","doi-asserted-by":"publisher","first-page":"1821","DOI":"10.1090\/mcom\/3276","volume":"87","author":"G Acosta","year":"2018","unstructured":"Acosta, G., Borthagaray, J., Bruno, O., Maas, M.: Regularity theory and high order numerical methods for the (1D)-fractional Laplacian. Math. Comput. 87, 1821\u20131857 (2018)","journal-title":"Math. Comput."},{"key":"9642_CR4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511809781","volume-title":"L\u00e9vy processes and stochastic calculus","author":"D Applebaum","year":"2009","unstructured":"Applebaum, D.: L\u00e9vy processes and stochastic calculus, 2nd edn. Cambridge University Press, Cambridge (2009)","edition":"2nd edn."},{"issue":"2","key":"9642_CR5","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1016\/j.physa.2006.08.071","volume":"374","author":"A Cartea","year":"2007","unstructured":"Cartea, A., del Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Physica A 374(2), 749\u2013763 (2007)","journal-title":"Physica A"},{"issue":"10","key":"9642_CR6","doi-asserted-by":"publisher","first-page":"6576","DOI":"10.1016\/j.jde.2017.07.023","volume":"263","author":"Z Chen","year":"2017","unstructured":"Chen, Z., Hu, E., Xie, L., Zhang, X.: Heat kernels for non-symmetric diffusion operators with jumps. J. Differ. Equations 263(10), 6576\u20136634 (2017)","journal-title":"J. Differ. Equations"},{"issue":"1","key":"9642_CR7","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/s00440-015-0631-y","volume":"165","author":"Z Chen","year":"2016","unstructured":"Chen, Z., Zhang, X.: Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Rel. 165(1), 267\u2013312 (2016)","journal-title":"Probab. Theory Rel."},{"issue":"2","key":"9642_CR8","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1007\/s10231-016-0586-3","volume":"196","author":"M Cozzi","year":"2017","unstructured":"Cozzi, M.: Interior regularity of solutions of non-local equations in Sobolev and Nikol\u2019skii spaces. Ann. Mat. Pura. Appl. 196(2), 555\u2013578 (2017)","journal-title":"Ann. Mat. Pura. Appl."},{"issue":"7","key":"9642_CR9","doi-asserted-by":"publisher","first-page":"1245","DOI":"10.1016\/j.camwa.2013.07.022","volume":"66","author":"M D\u2019Elia","year":"2013","unstructured":"D\u2019Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66(7), 1245\u20131260 (2013)","journal-title":"Comput. Math. Appl."},{"issue":"4","key":"9642_CR10","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1137\/110833294","volume":"54","author":"Q Du","year":"2012","unstructured":"Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54(4), 667\u2013696 (2012)","journal-title":"SIAM Rev."},{"key":"9642_CR11","volume-title":"An introduction to stochastic dynamics","author":"J Duan","year":"2015","unstructured":"Duan, J.: An introduction to stochastic dynamics. Cambridge University Press, New York (2015)"},{"issue":"3","key":"9642_CR12","doi-asserted-by":"publisher","first-page":"A887","DOI":"10.1137\/120897262","volume":"36","author":"T Gao","year":"2014","unstructured":"Gao, T., Duan, J., Li, X., Song, R.: Mean exit time and escape probability for dynamical systems driven by L\u00e9vy noise. SIAM J. Sci. Comput. 36(3), A887\u2013A906 (2014)","journal-title":"SIAM J. Sci. Comput."},{"key":"9642_CR13","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-05389-8","volume-title":"Handbook of stochastic methods","author":"CW Gardiner","year":"2004","unstructured":"Gardiner, C.W.: Handbook of stochastic methods, 3rd edn. Springer, Berlin (2004)","edition":"3rd edn."},{"key":"9642_CR14","doi-asserted-by":"crossref","DOI":"10.56021\/9781421407944","volume-title":"Matrix computations","author":"GH Golub","year":"2012","unstructured":"Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. JHU Press, Baltimore (2012)","edition":"4th edn."},{"key":"9642_CR15","doi-asserted-by":"publisher","first-page":"478","DOI":"10.1016\/j.aim.2014.09.018","volume":"268","author":"G Grubb","year":"2015","unstructured":"Grubb, G.: Fractional Laplacians on domains, a development of Hormander\u2019s theory of u-transmission pseudodifferential operators. Adv. Math. 268, 478\u2013528 (2015)","journal-title":"Adv. Math."},{"key":"9642_CR16","doi-asserted-by":"publisher","first-page":"4428","DOI":"10.1016\/j.apm.2014.02.026","volume":"38","author":"M Hao","year":"2014","unstructured":"Hao, M., Duan, J., Song, R., Xu, W.: Asymmetric non-Gaussian effects in a tumor growth model with immunization. Appl. Math. Model. 38, 4428\u20134444 (2014)","journal-title":"Appl. Math. Model."},{"key":"9642_CR17","first-page":"137","volume":"8","author":"C Hein","year":"2009","unstructured":"Hein, C., Imkeller, P., Pavlyukevich, I.: Limit theorems for p-variations of solutions of SDEs driven by additive stable L\u00e9vy noise and model selection for paleo-climatic data. Interdiscip. Math. Sci. 8, 137\u2013150 (2009)","journal-title":"Interdiscip. Math. Sci."},{"key":"9642_CR18","doi-asserted-by":"publisher","unstructured":"Huang, Q., Duan, J., Wu, J.: Maximum principles for nonlocal parabolic Waldenfels operators. Bull. Math. Sci., published online. https:\/\/doi.org\/10.1007\/s13373-018-0126-0 (2018)","DOI":"10.1007\/s13373-018-0126-0"},{"issue":"6","key":"9642_CR19","doi-asserted-by":"publisher","first-page":"3056","DOI":"10.1137\/140954040","volume":"52","author":"Y Huang","year":"2014","unstructured":"Huang, Y., Oberman, A.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52(6), 3056\u20133084 (2014)","journal-title":"SIAM J. Numer. Anal."},{"issue":"7301","key":"9642_CR20","doi-asserted-by":"publisher","first-page":"1066","DOI":"10.1038\/nature09116","volume":"465","author":"NE Humphries","year":"2010","unstructured":"Humphries, N.E., Queiroz, N., Dyer, J.R., Pade, N.G., Musyl, M.K., Schaefer, K.M., Fuller, D.W., Brunnschweiler, J.M., Doyle, T.K., Houghton, J.D., et al.: Environmental context explains L\u00e9vy and Brownian movement patterns of marine predators. Nature 465(7301), 1066\u20131069 (2010)","journal-title":"Nature"},{"key":"9642_CR21","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1016\/j.physa.2006.12.039","volume":"379","author":"T Koren","year":"2007","unstructured":"Koren, T., Chechkin, A., Klafter, J.: On the first passage time and leapover properties of L\u00e9vy mmotion. Physica A 379, 10\u201322 (2007)","journal-title":"Physica A"},{"issue":"1","key":"9642_CR22","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1515\/fca-2017-0002","volume":"20","author":"M Kwasnicki","year":"2017","unstructured":"Kwasnicki, M.: Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20(1), 7\u201351 (2017)","journal-title":"Fract. Calc. Appl. Anal."},{"key":"9642_CR23","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1016\/j.cam.2003.09.028","volume":"166","author":"F Liu","year":"2004","unstructured":"Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comput. Appl. Math. 166, 209\u2013219 (2004)","journal-title":"J. Comput. Appl. Math."},{"key":"9642_CR24","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1016\/j.jcp.2015.11.047","volume":"307","author":"Z Mao","year":"2016","unstructured":"Mao, Z., Shen, J.: Efficient Spectral-Galerkin method for fractional partial differential equations with variable coefficients. J. Comput. Phys. 307, 243\u2013261 (2016)","journal-title":"J. Comput. Phys."},{"issue":"5","key":"9642_CR25","doi-asserted-by":"publisher","first-page":"A1928","DOI":"10.1137\/16M1097109","volume":"39","author":"Z Mao","year":"2017","unstructured":"Mao, Z., Shen, J.: Hermite spectral methods for fractional PDEs in unbounded domains. SIAM J. Sci. Comput. 39(5), A1928\u2013A1950 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"9642_CR26","doi-asserted-by":"crossref","unstructured":"Riabiz, M., Godsill, S.: Approximate simulation of linear continuous time models driven by asymmetric stable L\u00e9vy processes. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), New Orleans, LA, 2017 pp. 4676\u20134680 (2017)","DOI":"10.1109\/ICASSP.2017.7953043"},{"key":"9642_CR27","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61544-3","volume-title":"The Fokker-Planck equation methods of solution and applications","author":"H Risken","year":"1996","unstructured":"Risken, H.: The Fokker-Planck equation methods of solution and applications, 2nd edn. Springer, Berlin (1996)","edition":"2nd edn."},{"issue":"3","key":"9642_CR28","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/j.matpur.2013.06.003","volume":"101","author":"X Ros-Oton","year":"2014","unstructured":"Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary. J. Math. Pure. Appl. 101(3), 275\u2013302 (2014)","journal-title":"J. Math. Pure. Appl."},{"key":"9642_CR29","unstructured":"Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian random process. Chapman & Hall\/CRC (1994)"},{"key":"9642_CR30","volume-title":"L\u00e9vy processes and infinitely divisible distributions","author":"KI Sato","year":"1999","unstructured":"Sato, K. I.: L\u00e9vy processes and infinitely divisible distributions. Cambridge University Press, Cambridge (1999)"},{"issue":"1","key":"9642_CR31","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1063\/1.1318734","volume":"42","author":"D Schertzer","year":"2001","unstructured":"Schertzer, D., Larcheveque, M., Duan, J, Yanovsky, V.V., Lovejoy, S.: Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian L\u00e9vy stable noises. J. Math. Phys. 42(1), 200\u2013212 (2001)","journal-title":"J. Math. Phys."},{"issue":"2","key":"9642_CR32","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/BF01061258","volume":"3","author":"A Sidi","year":"1998","unstructured":"Sidi, A., Israeli, M.: Quadrature methods for periodic singular and weakly singular Fredholm integral equtaions. J. Sci. Comput. 3(2), 201\u2013231 (1998)","journal-title":"J. Sci. Comput."},{"key":"9642_CR33","unstructured":"Song, R., Xie, L. arXiv: 1806.09033 (2018)"},{"issue":"5","key":"9642_CR34","doi-asserted-by":"publisher","first-page":"P05,024","DOI":"10.1088\/1742-5468\/2014\/05\/P05024","volume":"2014","author":"T Srokowski","year":"2014","unstructured":"Srokowski, T.: Asymmetric L\u00e9vy flights in nonhomogeneous environments. J. Stat. Mech.-Theory E. 2014(5), P05,024 (2014)","journal-title":"J. Stat. Mech.-Theory E."},{"issue":"2","key":"9642_CR35","doi-asserted-by":"publisher","first-page":"762","DOI":"10.1137\/140978831","volume":"53","author":"X Tian","year":"2015","unstructured":"Tian, X., Du, Q.: Nonconforming discontinuous Galerkin methods for nonlocal variational problems. SIAM J. Numer. Anal. 53(2), 762\u2013781 (2015)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"9642_CR36","doi-asserted-by":"publisher","first-page":"228","DOI":"10.1016\/j.jmaa.2016.12.013","volume":"449","author":"M Wang","year":"2017","unstructured":"Wang, M., Duan, J.: Existence and regularity of a linear nonlocal Fokker-Planck equation with growing drift. J. Math. Anal. Appl. 449(1), 228\u2013243 (2017)","journal-title":"J. Math. Anal. Appl."},{"key":"9642_CR37","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/j.amc.2015.01.117","volume":"258","author":"X Wang","year":"2015","unstructured":"Wang, X., Duan, J., Li, X., Luan, Y.: Numerical methods for the mean exit time and escape probability of two-dimensional stochastic dynamical systems with non-Gaussian noises. Appl. Math. Comput. 258, 282\u2013295 (2015)","journal-title":"Appl. Math. Comput."},{"key":"9642_CR38","doi-asserted-by":"publisher","first-page":"618","DOI":"10.1016\/j.cam.2018.01.003","volume":"337","author":"X Wang","year":"2018","unstructured":"Wang, X., Duan, J., Li, X., Song, R.: Numerical algorithms for mean exit time and escape probability of stochastic systems with asymmetric L\u00e9vy motion. Appl. Math. Comput. 337, 618\u2013634 (2018)","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"9642_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1063\/1.4916286","volume":"56","author":"J Wei","year":"2015","unstructured":"Wei, J., Tian, R.: Well-posedness for the fractional Fokker-Planck equations. J. Math. Phys. 56(3), 1\u201312 (2015)","journal-title":"J. Math. Phys."},{"issue":"1","key":"9642_CR40","doi-asserted-by":"publisher","first-page":"013,110","DOI":"10.1063\/1.4775758","volume":"23","author":"Y Xu","year":"2013","unstructured":"Xu, Y., Feng, J., Li, J., Zhang, H.: L\u00e9vy noise induced switch in the gene transcriptional regulatory system. Chaos 23(1), 013,110 (2013)","journal-title":"Chaos"},{"key":"9642_CR41","doi-asserted-by":"publisher","first-page":"5128","DOI":"10.1016\/j.physa.2010.07.032","volume":"389","author":"L Zeng","year":"2010","unstructured":"Zeng, L., Xu, B.: Effects of asymmetric L\u00e9vy noise in parameter-induced aperiodic stochastic resonance. Physica A 389, 5128\u20135136 (2010)","journal-title":"Physica A"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-018-9642-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-018-9642-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-018-9642-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T21:29:06Z","timestamp":1775251746000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-018-9642-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,24]]},"references-count":41,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,4]]}},"alternative-id":["9642"],"URL":"https:\/\/doi.org\/10.1007\/s10444-018-9642-4","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,24]]},"assertion":[{"value":"30 May 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 October 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 October 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}