{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T00:01:11Z","timestamp":1756771271062,"version":"3.44.0"},"reference-count":74,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this study, we explore the applicability of the Feynman\u2013Kac (FK) path integral formula to space-time fractional Schr\u00f6dinger equations.\nIn this work, a FK method based on the L\u00e9vy measure has been proposed for solving the Cauchy problems associated\nwith the space-time fractional Schr\u00f6dinger equations arising in interacting quantum systems. Application of an uncoupled Continuous Time Random Walk (CTRW) model with an exponentially distributed waiting time makes the underlying stochastic process a L\u00e9vy process which is basically a generalized Wiener process. Since these processes are Markovian in nature we can adopt\nthe classical FK approach to simulate the CTRW model for solving the space-time fractional diffusion process with comparable simplicity and convergence rate as in the case of standard diffusion processes. Our findings underscore the viability of employing the Fractional Feynman\u2013Kac path integral technique as an effective numerical method for solving space-time diffusion equations, thereby offering a promising alternative to traditional fractional calculus approaches.<\/jats:p>","DOI":"10.1515\/mcma-2025-2011","type":"journal-article","created":{"date-parts":[[2025,4,25]],"date-time":"2025-04-25T11:00:31Z","timestamp":1745578831000},"page":"189-205","source":"Crossref","is-referenced-by-count":0,"title":["On the applicability of Feynman\u2013Kac path integral simulation to space-time fractional Schr\u00f6dinger equations"],"prefix":"10.1515","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7673-2126","authenticated-orcid":false,"given":"Sumita","family":"Datta","sequence":"first","affiliation":[{"name":"Department of Pure and Applied Mathematics , Alliance University , Bengaluru 562106 , India ; and Department of Physics, The University of Texas at Arlington, Texas 76019, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7671-7065","authenticated-orcid":false,"given":"Radhika Prosad","family":"Datta","sequence":"additional","affiliation":[{"name":"Indian Institute of Foreign Trade (IIFT) , Kolkata Campus , Kolkata 700107 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-2104-6257","authenticated-orcid":false,"given":"James M.","family":"Rejcek","sequence":"additional","affiliation":[{"name":"Department of Physics , The University of Texas at Arlington , Arlington , TX 76019 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2025,4,26]]},"reference":[{"key":"2025090108241552846_j_mcma-2025-2011_ref_001","doi-asserted-by":"crossref","unstructured":"E. A.  Abdel-Rehim,\nFrom power laws to fractional diffusion processes with and without external forces, the non-direct way,\nFract. Calc. Appl. Anal. 22 (2019), no. 1, 60\u201377.","DOI":"10.1515\/fca-2019-0004"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_002","doi-asserted-by":"crossref","unstructured":"E. A.  Abdel-Rehim,\nFrom the space-time fractional integral of the continuous time random walk to the space-time fractional diffusion equations, a short proof and simulation,\nPhys. A 531 (2019), Article ID 121547.","DOI":"10.1016\/j.physa.2019.121547"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_003","doi-asserted-by":"crossref","unstructured":"E. A.  Abdel-Rehim and R.  Gorenflo,\nSimulation of the continuous time random walk of the space-fractional diffusion equations,\nJ. Comput. Appl. Math. 222 (2008), no. 2, 274\u2013283.","DOI":"10.1016\/j.cam.2007.10.052"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_004","doi-asserted-by":"crossref","unstructured":"X.  Antoine, Q.  Tang and J.  Zhang,\nOn the numerical solution and dynamical laws of nonlinear fractional Schr\u00f6dinger\/Gross\u2013Pitaevskii equations,\nInt. J. Comput. Math. 95 (2018), no. 6\u20137, 1423\u20131443.","DOI":"10.1080\/00207160.2018.1437911"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_005","unstructured":"B. C.  Arnold,\nPareto Distributions,\nChapman and Hall\/CRC, New York, 1983."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_006","doi-asserted-by":"crossref","unstructured":"S. \u015e.  Bay\u0131n,\nTime fractional Schr\u00f6dinger equation: Fox\u2019s H-functions and the effective potential,\nJ. Math. Phys. 54 (2013), no. 1, Article ID 012103.","DOI":"10.1063\/1.4773100"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_007","unstructured":"C.  Bender, M.  Bormann and Y. A.  Butko,\nSubordination principle and Feynman\u2013Kac formulae for generalized time-fractional evolution equations,\npreprint (2022), https:\/\/arxiv.org\/abs\/2202.01655v2."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_008","unstructured":"P.  Billingsley,\nConvergence of Probability Measures,\nJohn Wiley & Sons, New York, 1968."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_009","doi-asserted-by":"crossref","unstructured":"P. L.  Butzer and U.  Westphal,\nAn introduction to fractional calculus,\nApplications of Fractional Calculus in Physics,\nWorld Scientific, River Edge (2000), 1\u201385.","DOI":"10.1142\/9789812817747_0001"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_010","doi-asserted-by":"crossref","unstructured":"M.  Caffarel and P.  Claverie,\nDevelopment of a pure diffusion quantum Monte Carlo method using a full generalized Feynman\u2013Kac formula. II. Applications to simple systems,\nJ. Chem. Phys. 88 (1988), no. 2, 1100\u20131109.","DOI":"10.1063\/1.454228"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_011","doi-asserted-by":"crossref","unstructured":"M.  Caputo,\nLinear models of dissipation whose Q is almost frequency independent II,\nGeophys. J. R. Astron. Soc. 13 (1967), 529\u2013539.","DOI":"10.1111\/j.1365-246X.1967.tb02303.x"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_012","doi-asserted-by":"crossref","unstructured":"S.  Datta,\nComputing quantum correlation functions by importance sampling method based on path integrals,\nInt. J. Mod. Phys. B 37 (2023), Article ID 2350024.","DOI":"10.1142\/S0217979223500248"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_013","doi-asserted-by":"crossref","unstructured":"S.  Datta, V.  Dunjko and M.  Olshanii,\nPath integral estimates of the quantum fluctuations of the relative soliton-soliton velocity in a Gross\u2013Pitaevskii breather,\nPhysics 4 (2022), 12\u201320.","DOI":"10.3390\/physics4010002"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_014","doi-asserted-by":"crossref","unstructured":"S.  Datta, J. L.  Fry, N. G.  Fazleev, S. A.  Alexander and R. L.  Coldwell,\nFeynman\u2013Kac path-integral calculations with high quality trial wave functions,\nPhys. Rev. A 61 (2000), Article ID 030502.","DOI":"10.1103\/PhysRevA.61.030502"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_015","unstructured":"J.  Dong,\nLevy path integral approach to the solution of the fractional Schr\u00f6dinger equation with infinite square well,\npreprint (2013), https:\/\/arxiv.org\/abs\/1301.3009."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_016","doi-asserted-by":"crossref","unstructured":"J.  Dong and M.  Xu,\nSpace-time fractional Schr\u00f6dinger equation with time-independent potentials,\nJ. Math. Anal. Appl. 344 (2008), no. 2, 1005\u20131017.","DOI":"10.1016\/j.jmaa.2008.03.061"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_017","doi-asserted-by":"crossref","unstructured":"M. D.  Donsker and M.  Kac,\nA sampling method for determining the lowest eigenvalue and the principal eigenfunction of Schr\u00f6dinger\u2019s equation,\nJ. Res. Nat. Bur. Standards 44 (1950), 551\u2013557.","DOI":"10.6028\/jres.044.050"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_018","unstructured":"M. D.  Donsker and S. R.  Varadhan,\nAsymptotic evaluation of certain Wiener integrals for large time,\nFunctional Integration and its Applications,\nOxford University, Oxford (1975), 15\u201333."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_019","doi-asserted-by":"crossref","unstructured":"R. P.  Feynman,\nSpace-time approach to non-relativistic quantum mechanics,\nRev. Modern Physics 20 (1948), 367\u2013387.","DOI":"10.1103\/RevModPhys.20.367"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_020","unstructured":"R. P.  Feynman and A. R.  Hibbs,\nQuantum Mechanics and Path Integrals,\nMcGraw-Hill, New York, 1965."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_021","doi-asserted-by":"crossref","unstructured":"C.  Fox,\nThe G and H functions as symmetrical Fourier kernels,\nTrans. Amer. Math. Soc. 98 (1961), 395\u2013429.","DOI":"10.2307\/1993339"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_022","doi-asserted-by":"crossref","unstructured":"D.  Fulger, E.  Scalas and G.  Germano,\nMonte Carlo simulation of uncoupled continuous-time random walks yielding a stochatic solution of the space-time fractional diffusion equation,\nPhys. Rev. E 77 (2008), Article ID 021122.","DOI":"10.1103\/PhysRevE.77.021122"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_023","doi-asserted-by":"crossref","unstructured":"D.  Fulger, E.  Scalas and G.  Germano,\nRandom numbers from the tails of probability distributions using the transformation method,\nFract. Calc. Appl. Anal. 16 (2013), no. 2, 332\u2013353.","DOI":"10.2478\/s13540-013-0021-z"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_024","unstructured":"B. V.  Gnedenko and A. N.  Kolmogorov,\nLimit Distributions for Sums of Independent Random Variables,\nAddison-Wesley, Reading, 1954."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_025","unstructured":"R.  Gorenflo and E. A.  Abdel-Rehim,\nFrom power laws to fractional diffusion: The direct way,\nVietnam J. Math. 32 (2004), 65\u201375."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_026","doi-asserted-by":"crossref","unstructured":"R.  Gorenflo, F.  Mainardi, D.  Moretti and P.  Paradisi,\nTime fractional diffusion: A discrete random walk approach,\nNonlinear Dyn. 29 (2002), 129\u2013143.","DOI":"10.1023\/A:1016547232119"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_027","unstructured":"M.  Gutowski,\nL\u00e9vy flights as an underlying mechanism for global optimization algorithms,\npreprint (2001), https:\/\/arxiv.org\/abs\/math-ph\/0106003."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_028","doi-asserted-by":"crossref","unstructured":"H. E.  Hurst,\nLong-term storage capacity of resorvoirs,\nTrans. Amer. Soc. Civ. Eng. 116 (1951), 770\u2013799.","DOI":"10.1061\/TACEAT.0006518"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_029","doi-asserted-by":"crossref","unstructured":"A.  Iomin,\nFractional-time quantum dynamics,\nPhys. Rev. E 80 (2009), Article ID 022103.","DOI":"10.1103\/PhysRevE.80.022103"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_030","doi-asserted-by":"crossref","unstructured":"A.  Iomin,\nL\u00e9vy flights in a box,\nChaos Solitons Fractals 71 (2015), 73\u201377.","DOI":"10.1016\/j.chaos.2014.12.010"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_031","unstructured":"M.  Kac,\nProbability and Related Topics in Physical Sciences,\nLect. Appl. Math.,\nInterscience, London, 1959."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_032","unstructured":"I.  Karatzas and S. E.  Shreve,\nBrownian Motion and Stochastic Calculus,\nGrad. Texts in Math. 113,\nSpringer, New York, 1991."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_033","doi-asserted-by":"crossref","unstructured":"T.  Kato,\nOn the eigenfunctions of many-particle systems in quantum mechanics,\nComm. Pure Appl. Math. 10 (1957), 151\u2013177.","DOI":"10.1002\/cpa.3160100201"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_034","doi-asserted-by":"crossref","unstructured":"H.  Kleinert,\nStrong-coupling \n                  \n                     \n                        \n                           \u03d5\n                           4\n                        \n                     \n                     \n                     {{\\phi}^{4}}\n                  \n               -theory in \n                  \n                     \n                        \n                           4\n                           -\n                           \u03f5\n                        \n                     \n                     \n                     {4-\\epsilon}\n                  \n                dimensions and critical exponents,\nPhys. Rev. D 57 (1998), 2264\u20132278.","DOI":"10.1103\/PhysRevD.57.2264"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_035","doi-asserted-by":"crossref","unstructured":"H.  Kleinert,\nFractional quantum field theory, path integral, and stochastic differential equation for strongly correlated interacting many-particle systems,\nEurophys. Lett. 100 (2012), Article ID 10001.","DOI":"10.1209\/0295-5075\/100\/10001"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_036","doi-asserted-by":"crossref","unstructured":"H.  Kleinert,\nFractional field equtions for highly improbable events,\nJ. Phys. Conf. Ser. 442 (2013), Article ID 012019.","DOI":"10.1088\/1742-6596\/442\/1\/012019"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_037","doi-asserted-by":"crossref","unstructured":"D.  Kleinhans and R.  Friedrich,\nContinuous time random walks: Simulation of continuous trajectories,\nPhys. Rev. E 76 (2007), Article ID 061102.","DOI":"10.1103\/PhysRevE.76.061102"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_038","doi-asserted-by":"crossref","unstructured":"A.  Korzeniowski,\nOn computer simulation of Feynman\u2013Kac path-integrals,\nJ. Comput. Appl. Math. 66 (1996), 333\u2013336.","DOI":"10.1016\/0377-0427(95)00170-0"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_039","doi-asserted-by":"crossref","unstructured":"A.  Korzeniowski, J. L.  Fry, D. E.  Orr and N. G.  Fazleev,\nFeynman\u2013Kac path integral calculation of the ground state energies of atoms,\nPhys. Rev. Lett. 69 (1992), 893\u2013896.","DOI":"10.1103\/PhysRevLett.69.893"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_040","doi-asserted-by":"crossref","unstructured":"N.  Laskin,\nFractional quantum mechanics,\nPhys. Rev. E (3) 62 (2000), no. 5, 3135\u20133145.","DOI":"10.1103\/PhysRevE.62.3135"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_041","doi-asserted-by":"crossref","unstructured":"N.  Laskin,\nFractional Schr\u00f6dinger equation,\nPhys. Rev. E (3) 66 (2002), no. 5, Article ID 056108.","DOI":"10.1103\/PhysRevE.66.056108"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_042","unstructured":"N.  Laskin,\nPrinciples of fractional quantum mechanics,\npreprint (2010), https:\/\/arxiv.org\/abs\/1009.5533."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_043","doi-asserted-by":"crossref","unstructured":"N.  Laskin,\nTime fractional quantum mechanics,\nChaos Solitons Fractals 102 (2017), 16\u201328.","DOI":"10.1016\/j.chaos.2017.04.010"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_044","unstructured":"P. L.  L\u00e9vy,\nCalcul des probabilit\u00e9s,\nGauthier-Villars, Paris, 1925."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_045","unstructured":"F.  Mainardi, Y.  Luchko and G.  Pagnini,\nThe fundamental solution of the space-time fractional diffusion equation,\nFract. Calc. Appl. Anal. 4 (2001), no. 2, 153\u2013192."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_046","doi-asserted-by":"crossref","unstructured":"F.  Mainardi, G.  Pagnini and R. K.  Saxena,\nFox H functions in fractional diffusion,\nJ. Comput. Appl. Math. 178 (2005), no. 1\u20132, 321\u2013331.","DOI":"10.1016\/j.cam.2004.08.006"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_047","doi-asserted-by":"crossref","unstructured":"M. M.  Meerschaert and H.-P.  Scheffler,\nLimit theorems for continuous-time random walks with infinite mean waiting times,\nJ. Appl. Probab. 41 (2004), no. 3, 623\u2013638.","DOI":"10.1239\/jap\/1091543414"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_048","doi-asserted-by":"crossref","unstructured":"M.  Meerschaert and H. P.  Scheffler,\nContinuous time random walks and space-time fractional differential equations,\nHandbook of Fractional Calculus with Applications,\nDe Gruyter, Berlin (2019), https:\/\/doi.org\/10.1515\/9783110571622-0161.","DOI":"10.1515\/9783110571622-016"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_049","doi-asserted-by":"crossref","unstructured":"N.  Metropolis and S.  Ulam,\nThe Monte Carlo method,\nJ. Amer. Statist. Assoc. 44 (1949), 335\u2013341.","DOI":"10.1080\/01621459.1949.10483310"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_050","unstructured":"F.  Mies, M.  Sadr and M.  Torrilhon,\nAn efficient jump-diffusion approximation of the Boltzmann equation,\npreprint (2021), https:\/\/arxiv.org\/abs\/2112.08362."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_051","doi-asserted-by":"crossref","unstructured":"J. B.  Mijena and E.  Nane,\nSpace-time fractional stochastic partial differential equations,\nStochastic Process. Appl. 125 (2015), no. 9, 3301\u20133326.","DOI":"10.1016\/j.spa.2015.04.008"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_052","unstructured":"M. G.  Mittag-Leffler,\nSur la nouvelle fonction \n                  \n                     \n                        \n                           \n                              E\n                              \u03b1\n                           \n                           \u2062\n                           \n                              (\n                              x\n                              )\n                           \n                        \n                     \n                     \n                     E_{\\alpha}(x)\n                  \n               ,\nC. R. Acad. Sci, Paris 137 (1903), 554\u2013558."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_053","doi-asserted-by":"crossref","unstructured":"M.  Montero and J.  Masoliver,\nNonindependent continuous-time random walks,\nPhys. Rev. E (3) 76 (2007), no. 6, Article ID 061115.","DOI":"10.1103\/PhysRevE.76.061115"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_054","doi-asserted-by":"crossref","unstructured":"E. W.  Montroll and G. H.  Weiss,\nRandom walks on lattices. II,\nJ. Math. Phys. 6 (1965), 167\u2013181.","DOI":"10.1063\/1.1704269"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_055","doi-asserted-by":"crossref","unstructured":"M.  Naber,\nTime fractional Schr\u00f6dinger equation,\nJ. Math. Phys. 45 (2004), no. 8, 3339\u20133352.","DOI":"10.1063\/1.1769611"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_056","doi-asserted-by":"crossref","unstructured":"E. C.  Oliveira, F. S.  Costa and J.  Vaz, Jr.,\nThe fractional Schr\u00f6dinger equation for delta potentials,\nJ. Math. Phys. 51 (2010), no. 12, Article ID 123517.","DOI":"10.1063\/1.3525976"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_057","doi-asserted-by":"crossref","unstructured":"C.-K.  Peng, S. V.  Buldyrev, S.  Havlin, M.  Simons, H. E.  Stanley and A. L.  Goldberger,\nMosaic organization of DNA nucleotides,\nPhys. Rev. E 49 (1994), 1685\u20131689.","DOI":"10.1103\/PhysRevE.49.1685"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_058","doi-asserted-by":"crossref","unstructured":"C.-K.  Peng, S.  Havlin, H. E.  Stanley and A. L.  Goldberger,\nQuantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series,\nChaos 5 (1995), 82\u201387.","DOI":"10.1063\/1.166141"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_059","doi-asserted-by":"crossref","unstructured":"C.  Penland and B. D.  Ewald,\nOn modelling physical systems with stochastic models: Diffusion versus L\u00e9vy processes,\nPhilos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), no. 1875, 2457\u20132476.","DOI":"10.1098\/rsta.2008.0051"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_060","doi-asserted-by":"crossref","unstructured":"N.  Petroni and M.  Pusterla,\nL\u00e9vy processes and Schr\u00f6dinger equation,\nPhys. A 388 (2009), no. 6, 824\u2013836.","DOI":"10.1016\/j.physa.2008.11.035"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_061","unstructured":"L.  Pitaevskii and S.  Stringari,\nBose\u2013Einstein Condensation,\nInternat. Ser. Monogr. Phys. 116,\nThe Clarendon Press, Oxford, 2003."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_062","doi-asserted-by":"crossref","unstructured":"A. V.  Ponomarev, S.  Denisov and P.  H\u00e4nggi,\nL\u00e9vy distribution in many particle quantum systems,\nPhys. Rev. A 81 (2010), Article ID 043615.","DOI":"10.1103\/PhysRevA.81.043615"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_063","doi-asserted-by":"crossref","unstructured":"H. C.  Rosu and S. C.  Mancas,\nFactorization of the Riesz\u2013Feller fractional quantum harmonic oscillators,\nJ. Phys. Conf. Ser. 1540 (2020), Article ID 012005.","DOI":"10.1088\/1742-6596\/1540\/1\/012005"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_064","doi-asserted-by":"crossref","unstructured":"S.  Saberhaghparvar and H.  Panahi,\nInitial value problem for a Caputo space-time fractional Schr\u00f6dinger equation for the delta potential,\nRev. Mexicana F\u00eds. 68 (2022), no. 4, Article ID 040703.","DOI":"10.31349\/RevMexFis.68.040703"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_065","doi-asserted-by":"crossref","unstructured":"A. A.  Saberi,\nFractal structure of a three-dimensional Brownian motion on an attractive plane,\nPhys. Rev. E. 84 (2011), Article ID 021113.","DOI":"10.1103\/PhysRevE.84.021113"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_066","unstructured":"L. S.  Schulman,\nTechniques and Applications of Path Integration,\nJohn Wiley & Sons, New York, 1993."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_067","unstructured":"B.  Simon,\nFunctional Integrals and Quantum Mechanics,\nAcademic Press, New York, 1979."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_068","unstructured":"F.  Soto-Eguibar and P.  Claverie,\nStochastic Processes Applied to Physics and Other a Rueda,\nWorld Scientific, Singapore, 1983."},{"key":"2025090108241552846_j_mcma-2025-2011_ref_069","doi-asserted-by":"crossref","unstructured":"X.  Sun, L.  Xie and Y.  Xie,\nDerivative formula for the Feynman\u2013Kac semigroup of SDEs driven by rotationally invariant \u03b1-stable process,\nStatist. Probab. Lett. 158 (2020), Article ID 108664.","DOI":"10.1016\/j.spl.2019.108664"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_070","doi-asserted-by":"crossref","unstructured":"V. E.  Tarasov and G. M.  Zaslavsky,\nFractional generalization of Kac integral,\nCommun. Nonlinear Sci. Numer. Simul. 13 (2008), no. 2, 248\u2013258.","DOI":"10.1016\/j.cnsns.2007.04.020"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_071","doi-asserted-by":"crossref","unstructured":"H. F.  Trotter,\nOn the product of semi-groups of operators,\nProc. Amer. Math. Soc. 10 (1959), 545\u2013551.","DOI":"10.1090\/S0002-9939-1959-0108732-6"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_072","doi-asserted-by":"crossref","unstructured":"S.  Wang and M.  Xu,\nGeneralized fractional Schr\u00f6dinger equation with time-independent potentials,\nJ. Math. Phys. 484 (2007), Article ID 043502.","DOI":"10.1063\/1.2716203"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_073","doi-asserted-by":"crossref","unstructured":"A.  Weron and R.  Weron,\nComputer simulation of L\u00e9vy \u03b1-stable variables and processes,\nChaos\u2014The Interplay Between Stochastic and Deterministic Behaviour,\nLecture Notes in Phys. 457,\nSpringer, Berlin (1995), 379\u2013392.","DOI":"10.1007\/3-540-60188-0_67"},{"key":"2025090108241552846_j_mcma-2025-2011_ref_074","doi-asserted-by":"crossref","unstructured":"N.  Wiener,\nDifferential-space,\nJ. Math. Phys. 2 (1923), 131\u2013174.","DOI":"10.1002\/sapm192321131"}],"container-title":["Monte Carlo Methods and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/mcma-2025-2011\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/mcma-2025-2011\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,1]],"date-time":"2025-09-01T08:24:36Z","timestamp":1756715076000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/mcma-2025-2011\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,26]]},"references-count":74,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,4,26]]},"published-print":{"date-parts":[[2025,9,1]]}},"alternative-id":["10.1515\/mcma-2025-2011"],"URL":"https:\/\/doi.org\/10.1515\/mcma-2025-2011","relation":{},"ISSN":["0929-9629","1569-3961"],"issn-type":[{"type":"print","value":"0929-9629"},{"type":"electronic","value":"1569-3961"}],"subject":[],"published":{"date-parts":[[2025,4,26]]}}}