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Fractional equations via convergence of forms. (English) Zbl 1476.60106

In this paper, the authors investigate time-changed Lévy processes and relate these processes to Dirichlet forms. As a main result they show under which conditions the convergence of the Dirchlet forms and the convergence of the time changed processes are equivalent. The result is applied to skew diffusions and reflected Brownian motion.

MSC:

60H20 Stochastic integral equations
60B10 Convergence of probability measures
60H30 Applications of stochastic analysis (to PDEs, etc.)
31C25 Dirichlet forms
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[1] E. Acerbi, G. Buttazzo, Reinforcement problems in the calculus of variations. Ann. Inst. H. Poincaré Anal. Non Linear3, No 4 (1986), 273-284.; Acerbi, E.; Buttazzo, G., Reinforcement problems in the calculus of variations, Ann. Inst. H. Poincaré Anal. Non Linear, 3, 4, 273-284 (1986) · Zbl 0607.73018
[2] Y. Achdou, T. Deheuvels, A transmission problem across a fractal self-similar interface. Multiscale Model. Simul. 14, No 2 (2016), 708-736.; Achdou, Y.; Deheuvels, T., A transmission problem across a fractal self-similar interface, Multiscale Model. Simul., 14, 2, 708-736 (2016) · Zbl 1375.35135
[3] M. Allen, L. Caffarelli, A. Vasseur, A parabolic problem with a fractional time derivative. Arch. Ration. Mech. Anal. 221, No 2 (2016), 603-630.; Allen, M.; Caffarelli, L.; Vasseur, A., A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221, 2, 603-630 (2016) · Zbl 1338.35428
[4] H. Allouba, Brownian-time processes: the PDE connection. II. And the corresponding Feynman-Kac formula. Trans. Amer. Math. Soc. 354 No 11 (2002), 4627-4637.; Allouba, H., Brownian-time processes: the PDE connection. II. And the corresponding Feynman-Kac formula, Trans. Amer. Math. Soc., 354, 11, 4627-4637 (2002) · Zbl 1006.60063
[5] H. Allouba, W. Zheng, Brownian-time processes: the PDE connection and the half-derivative generator. Ann. Probab. 29, No 4 (2001), 1780-1795.; Allouba, H.; Zheng, W., Brownian-time processes: the PDE connection and the half-derivative generator, Ann. Probab., 29, 4, 1780-1795 (2001) · Zbl 1018.60066
[6] H. Attouch, Variational Convergence for Functions and Operators. Applicable Math. Ser. Pitman (Advanced Publ. Program), Boston, MA, 1984.; Attouch, H., Variational Convergence for Functions and Operators. (1984) · Zbl 0561.49012
[7] B. Baeumer, M.M. Meerschaert, Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, No 4 (2001), 481-500.; Baeumer, B.; Meerschaert, M. M., Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal., 4, 4, 481-500 (2001) · Zbl 1057.35102
[8] M.T. Barlow, Diffusions on Fractals. Lectures on Probability Theory and Statistics, Part of Ser. Lecture Notes in Mathematics (LNM), Volume 1690, Springer, 2006.; Barlow, M. T., Diffusions on Fractals., 1690 (2006)
[9] M.T. Barlow, R. Bass, Z-Q. Chen, M. Kassmann, Non-local Dirichlet forms and symmetric jump processes. Trans. Amer. Math. Soc. 361, No 4 (2009), 1963-1999.; Barlow, M. T.; Bass, R.; Chen, Z-Q.; Kassmann, M., Non-local Dirichlet forms and symmetric jump processes, Trans. Amer. Math. Soc., 361, 4, 1963-1999 (2009) · Zbl 1166.60045
[10] E.G. Bazhlekova, Subordination principle for fractional evolution equations. Fract. Calc. Appl. Anal. 3, No 3 (2000), 213-230.; Bazhlekova, E. G., Subordination principle for fractional evolution equations, Fract. Calc. Appl. Anal., 3, 3, 213-230 (2000) · Zbl 1041.34046
[11] L. Beghin, E. Orsingher, The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation. Fract. Calc. Appl. Anal. 6, No 2 (2003), 187-204.; Beghin, L.; Orsingher, E., The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation, Fract. Calc. Appl. Anal., 6, 2, 187-204 (2003) · Zbl 1083.60039
[12] L. Beghin, E. Orsingher, Iterated elastic Brownian motions and fractional diffusion equations. Stoch. Proc. Appl. 119 (2009), 1975-2003.; Beghin, L.; Orsingher, E., Iterated elastic Brownian motions and fractional diffusion equations, Stoch. Proc. Appl., 119, 1975-2003 (2009) · Zbl 1226.60107
[13] J. Bertoin, J.L. Bretagnolle, R.A. Doney, I.A. Ibragimov, J. Jacod, Lévy Processes at Saint-Flour. Probability at Saint-Flour, Springer, 2012.; Bertoin, J.; Bretagnolle, J. L.; Doney, R. A.; Ibragimov, I. A.; Jacod, J., Lévy Processes at Saint-Flour. (2012) · Zbl 1254.60004
[14] S. Bonaccorsi, M. D’Ovidio, S. Mazzucchi, Probabilistic representation formula for the solution of fractional high order heat-type equations. J. of Evolution Equations19, No 2 (2019), 523-558.; Bonaccorsi, S.; D’Ovidio, M.; Mazzucchi, S., Probabilistic representation formula for the solution of fractional high order heat-type equations, J. of Evolution Equations, 19, 2, 523-558 (2019) · Zbl 1419.35201
[15] B. Böttecher, R. Schilling, J. Wang, Lévy Matters III, Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Springer, New York, 2013.; Böttecher, B.; Schilling, R.; Wang, J., Lévy Matters III, Lévy-Type Processes: Construction, Approximation and Sample Path Properties. (2013) · Zbl 1384.60004
[16] R.M. Blumenthal, R.K. Getoor, Markov Processes and Potential Theory. Academic Press, New York, 1968.; Blumenthal, R. M.; Getoor, R. K., Markov Processes and Potential Theory. (1968) · Zbl 0169.49204
[17] A. Braides, Γ-Convergence for Beginners. Oxford University Press, Oxford, 2002.; Braides, A., Γ-Convergence for Beginners. (2002)
[18] H. Brezis, L. A. Caffarelli, A. Friedman, Reinforcement problems for elliptic equations and variational inequalities.Ann. Mat. Pura Appl. (4)123 (1980), 219-246.; Brezis, H.; Caffarelli, L. A.; Friedman, A., Reinforcement problems for elliptic equations and variational inequalities, Ann. Mat. Pura Appl. (4), 123, 219-246 (1980) · Zbl 0434.35079
[19] G. Buttazzo, G. Dal Maso, U. Mosco, Asymptotic behaviour for Dirichlet problems in domains bounded by thin layers. In: Partial Differential Equations and the Calculus of Variations, Vol. I. Ser. Progr. Nonlinear Differential Equations Appl., 1, 193-249, Birkhäuser Boston, Boston, MA, 1989.; Buttazzo, G.; Dal Maso, G.; Mosco, U., Partial Differential Equations and the Calculus of Variations. Ser. Progr. Nonlinear Differential Equations Appl., 1, 193-249 (1989) · Zbl 0699.35062
[20] R. Capitanelli, Asymptotics for mixed Dirichlet-Robin problems in irregular domains. J. Math. Anal. Appl. 362, No 2 (2010), 450-459.; Capitanelli, R., Asymptotics for mixed Dirichlet-Robin problems in irregular domains, J. Math. Anal. Appl., 362, 2, 450-459 (2010) · Zbl 1179.35106
[21] R. Capitanelli, M. D’Ovidio, Asymptotics for time-changed diffusions. Probability Theory and Mathematical Statistics95 (2017), 41-58.; Capitanelli, R.; D’Ovidio, M., Asymptotics for time-changed diffusions, Probability Theory and Mathematical Statistics, 95, 41-58 (2017) · Zbl 1390.60295
[22] R. Capitanelli, M. D’Ovidio, Skew Brownian diffusions across Koch interfaces. Potential Analysis46, No 3, (2017), 431-461.; Capitanelli, R.; D’Ovidio, M., Skew Brownian diffusions across Koch interfaces, Potential Analysis, 46, 3, 431-461 (2017) · Zbl 1367.60102
[23] R. Capitanelli, M.R. Lancia, M.A. Vivaldi, Insulating layers of fractal type. Differential Integral Equations26, No 9-10 (2013), 1055-1076.; Capitanelli, R.; Lancia, M. R.; Vivaldi, M. A., Insulating layers of fractal type, Differential Integral Equations, 26, 9-10, 1055-1076 (2013) · Zbl 1299.35028
[24] R. Capitanelli, M.A. Vivaldi, Insulating layers and Robin problems on Koch mixtures. J. Differential Equations251, No 4-5 (2011), 1332-1353.; Capitanelli, R.; Vivaldi, M. A., Insulating layers and Robin problems on Koch mixtures, J. Differential Equations, 251, 4-5, 1332-1353 (2011) · Zbl 1225.28006
[25] R. Capitanelli, M.A. Vivaldi, On the Laplacean transfer across fractal mixtures. Asymptot. Anal. 83, No 1-2 (2013), 1-33.; Capitanelli, R.; Vivaldi, M. A., On the Laplacean transfer across fractal mixtures, Asymptot. Anal., 83, 1-2, 1-33 (2013) · Zbl 1282.35144
[26] R. Capitanelli, M.A. Vivaldi, Reinforcement problems for variational inequalities on fractal sets. Calc. Var. Partial Differential Equations. 54, No 3 (2015), 2751-2783.; Capitanelli, R.; Vivaldi, M. A., Reinforcement problems for variational inequalities on fractal sets, Calc. Var. Partial Differential Equations., 54, 3, 2751-2783 (2015) · Zbl 1333.35078
[27] R. Capitanelli, M.A. Vivaldi, Dynamical quasi-filling fractal layers. SIAM J. Math. Anal. 48, No 6 (2016), 3931-3961.; Capitanelli, R.; Vivaldi, M. A., Dynamical quasi-filling fractal layers, SIAM J. Math. Anal., 48, 6, 3931-3961 (2016) · Zbl 1353.28003
[28] J. A. van Casteren, On martingales and feller semigroups. Results in Mathematics21 (1992), 274-288.; van Casteren, J. A., On martingales and feller semigroups, Results in Mathematics, 21, 274-288 (1992) · Zbl 0753.60068
[29] Z.-Q. Chen, Time fractional equations and probabilistic representation. Chaos, Solitons and Fractals102 (2017), 168-174.; Chen, Z.-Q., Time fractional equations and probabilistic representation, Chaos, Solitons and Fractals, 102, 168-174 (2017) · Zbl 1374.60122
[30] Z.-Q. Chen, M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory. London Math. Soc. Monographs, Princeton University Press, 2012.; Chen, Z.-Q.; Fukushima, M., Symmetric Markov Processes, Time Change, and Boundary Theory. (2012) · Zbl 1253.60002
[31] Z.-Q. Chen, P. Kim, T. Kumagai, Discrete approximation of symmetric jump processes on metric measure spaces. Probab. Theory Related Fields. 155, No 3-4 (2013), 703-749.; Chen, Z.-Q.; Kim, P.; Kumagai, T., Discrete approximation of symmetric jump processes on metric measure spaces, Probab. Theory Related Fields., 155, 3-4, 703-749 (2013) · Zbl 1274.60238
[32] Z.-Q. Chen, P. Kim, T. Kumagai, J. Wang, Heat kernel estimates for time fractional equations. Forum Mathematicum30, No 5 (2018), 1163-1192.; Chen, Z.-Q.; Kim, P.; Kumagai, T.; Wang, J., Heat kernel estimates for time fractional equations, Forum Mathematicum, 30, 5, 1163-1192 (2018) · Zbl 1401.60088
[33] Z.-Q. Chen, R. Song, Continuity of eigenvalues of subordinate processes in domains. Mathematische Zeitschrift252, No 1 (2006), 71-89.; Chen, Z.-Q.; Song, R., Continuity of eigenvalues of subordinate processes in domains, Mathematische Zeitschrift, 252, 1, 71-89 (2006) · Zbl 1090.60070
[34] G. Dal Maso, An Introduction to Γ-Convergence. Birkhäuser Boston, Boston, MA, 1993.; Dal Maso, G., An Introduction to Γ-Convergence. (1993) · Zbl 0816.49001
[35] E. De Giorgi, T. Franzoni, Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)58, No 6 (1975), 842-850.; De Giorgi, E.; Franzoni, T., Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 58, 6, 842-850 (1975) · Zbl 0339.49005
[36] M. D’Ovidio, On the fractional counterpart of the higher-order equations. Statistics and Probability Letters81 (2011), 1929-1939.; D’Ovidio, M., On the fractional counterpart of the higher-order equations, Statistics and Probability Letters, 81, 1929-1939 (2011) · Zbl 1236.60065
[37] M. D’Ovidio, From Sturm-Liouville problems to fractional and anomalous diffusions. Stochastic Processes and their Applications122 (2012), 3513-3544.; D’Ovidio, M., From Sturm-Liouville problems to fractional and anomalous diffusions, Stochastic Processes and their Applications, 122, 3513-3544 (2012) · Zbl 1260.60159
[38] M. D’Ovidio, Coordinates changed random fields on the sphere. J. of Statistical Phys. 154 (2014), 1153-1176.; D’Ovidio, M., Coordinates changed random fields on the sphere, J. of Statistical Phys., 154, 1153-1176 (2014) · Zbl 1296.60133
[39] M. D’Ovidio, E. Nane, Fractional Cauchy problems on compact manifolds. Stochastic Anal. and Appl. 34, No 2 (2016), 232-257.; D’Ovidio, M.; Nane, E., Fractional Cauchy problems on compact manifolds, Stochastic Anal. and Appl., 34, 2, 232-257 (2016) · Zbl 1341.60068
[40] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter & Co, New York, 1994.; Fukushima, M.; Oshima, Y.; Takeda, M., Dirichlet Forms and Symmetric Markov Processes. (1994) · Zbl 0838.31001
[41] I. Golding, E.C. Cox, Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96 (2006), Art. 098102.; Golding, I.; Cox, E. C., Physical nature of bacterial cytoplasm, Phys. Rev. Lett., 96 (2006)
[42] R. Gorenflo, A. Iskenderov, Yu. Luchko, Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Anal. 3, No 1 (2000), 75-86.; Gorenflo, R.; Iskenderov, A.; Luchko, Yu., Mapping between solutions of fractional diffusion-wave equations, Fract. Calc. Appl. Anal., 3, 1, 75-86 (2000) · Zbl 1033.35161
[43] M. Giona, H. Roman, Fractional diffusion equation on fractals: one-dimensional case and asymptotic behavior. J. Phys. A25 (1992), 2093-2105.; Giona, M.; Roman, H., Fractional diffusion equation on fractals: one-dimensional case and asymptotic behavior, J. Phys. A, 25, 2093-2105 (1992) · Zbl 0755.60067
[44] M.E. Hernández-Hernández, V.N. Kolokoltsov, L. Toniazzi, Generalised fractional evolution equations of Caputo type. Chaos, Solitons and Fractals102 (2017), 184-196.; Hernández-Hernández, M. E.; Kolokoltsov, V. N.; Toniazzi, L., Generalised fractional evolution equations of Caputo type, Chaos, Solitons and Fractals, 102, 184-196 (2017) · Zbl 1374.34009
[45] R. Hilfer, Fractional diffusion based on Riemann-Liouville fractional derivatives. J. Phys. Chem. B. 104 (2000), 3914-3917.; Hilfer, R., Fractional diffusion based on Riemann-Liouville fractional derivatives, J. Phys. Chem. B., 104, 3914-3917 (2000) · Zbl 0994.34050
[46] O. Kallenberg, Foundations of Modern Probability. Springer-Verlag New York, Inc., New York, 1997.; Kallenberg, O., Foundations of Modern Probability. (1997) · Zbl 0892.60001
[47] T. Kato, Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York, 1966.; Kato, T., Perturbation Theory for Linear Operators. (1966)
[48] T. Kazuaki, Semigroups, Boundary Value Problems and Markov Processes. Second Edition, Springer New York, 2014.; Kazuaki, T., Semigroups, Boundary Value Problems and Markov Processes. (2014) · Zbl 1314.47001
[49] V. Keyantuo, C. Lizama, On a connection between powers of operators and fractional Cauchy problems. J. Evol. Equ. 12 (2012), 245-265.; Keyantuo, V.; Lizama, C., On a connection between powers of operators and fractional Cauchy problems, J. Evol. Equ., 12, 245-265 (2012) · Zbl 1336.47046
[50] V. Keyantuo, C. Lizama, M. Warma, Existence, regularity and representation of solutions of time fractional diffusion equations. Adv. Differential Equations21, No 9-10 (2016), 837-886.; Keyantuo, V.; Lizama, C.; Warma, M., Existence, regularity and representation of solutions of time fractional diffusion equations, Adv. Differential Equations, 21, 9-10, 837-886 (2016) · Zbl 1375.47034
[51] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Math. Studies # 204, Elsevier, Amsterdam, 2006.; Kilbas, A.; Srivastava, H.; Trujillo, J., Theory and Applications of Fractional Differential Equations. (2006) · Zbl 1092.45003
[52] A.N. Kochubei, The Cauchy problem for evolution equations of fractional order. Differential Equations25 (1989), 967-974.; Kochubei, A. N., The Cauchy problem for evolution equations of fractional order, Differential Equations, 25, 967-974 (1989) · Zbl 0696.34047
[53] A.N. Kochubei, Diffusion of fractional order. Lecture Notes in Physics26 (1990), 485-492.; Kochubei, A. N., Diffusion of fractional order, Lecture Notes in Physics, 26, 485-492 (1990) · Zbl 0729.35064
[54] A.N. Kochubei, General fractional calculus, evolution equations, and renewal processes. Integr. Equations and Operator Theory71, No 4 (2011), 583-600.; Kochubei, A. N., General fractional calculus, evolution equations, and renewal processes, Integr. Equations and Operator Theory, 71, 4, 583-600 (2011) · Zbl 1250.26006
[55] A.V. Kolesnikov, Mosco convergence of Dirichlet forms in infinite dimensions with changing reference measures. J. Funct. Anal. 230 (2006), 382-418.; Kolesnikov, A. V., Mosco convergence of Dirichlet forms in infinite dimensions with changing reference measures, J. Funct. Anal., 230, 382-418 (2006) · Zbl 1093.31007
[56] V.N. Kolokoltsov, Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Theory of Probability and its Applications53, No 4 (2009), 594-609.; Kolokoltsov, V. N., Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics, Theory of Probability and its Applications, 53, 4, 594-609 (2009) · Zbl 1193.60046
[57] T.G. Kurtz, Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Probab. 19, No 3 (1991), 1010-1034.; Kurtz, T. G., Random time changes and convergence in distribution under the Meyer-Zheng conditions, Ann. Probab., 19, 3, 1010-1034 (1991) · Zbl 0742.60036
[58] K. Kuwae, T. Shioya, Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry. Commun. Anal. Geom. 11 (2003), 599-673.; Kuwae, K.; Shioya, T., Convergence of spectral structures: A functional analytic theory and its applications to spectral geometry, Commun. Anal. Geom., 11, 599-673 (2003) · Zbl 1092.53026
[59] M.R. Lancia, V.R. Durante, P. Vernole, Asymptotics for Venttsel’ problems for operators in non divergence form in irregular domains. Discrete Contin. Dyn. Syst. Ser. S9, No 5 (2016), 1493-1520.; Lancia, M. R.; Durante, V. R.; Vernole, P., Asymptotics for Venttsel’ problems for operators in non divergence form in irregular domains, Discrete Contin. Dyn. Syst. Ser. S, 9, 5, 1493-1520 (2016) · Zbl 1355.35087
[60] M.R. Lancia, P. Vernole, Venttsel’ problems in fractal domains. J. Evol. Equ. 14, No 3 (2014), 681-712.; Lancia, M. R.; Vernole, P., Venttsel’ problems in fractal domains, J. Evol. Equ., 14, 3, 681-712 (2014) · Zbl 1298.31013
[61] M.R. Lancia, M.A. Vivaldi, Asymptotic convergence for energy forms. Adv. Math. Sci. Appl. 13 (2003), 315-341.; Lancia, M. R.; Vivaldi, M. A., Asymptotic convergence for energy forms, Adv. Math. Sci. Appl., 13, 315-341 (2003) · Zbl 1205.35129
[62] M. Magdziarz, R.L. Schilling, Asymptotic properties of Brownian motion delayed by inverse subordinators. Proc. Amer. Math. Soc. 143 (2015), 4485-4501.; Magdziarz, M.; Schilling, R. L., Asymptotic properties of Brownian motion delayed by inverse subordinators, Proc. Amer. Math. Soc., 143, 4485-4501 (2015) · Zbl 1329.60291
[63] R.L. Magin, Modeling the cardiac tissue electrode interface using fractional calculus. J. of Vibration and Control14 (2008), 1431-1442.; Magin, R. L., Modeling the cardiac tissue electrode interface using fractional calculus, J. of Vibration and Control, 14, 1431-1442 (2008) · Zbl 1229.92018
[64] F. Mainardi, On the Initial Value Problem for the Fractional Diffusion-Wave Equation. In: Waves and Stability in Continuous Media (Bologna, 1993), 246-251, Ser. Adv. Math. Appl. Sci. 23, World Sci. Publ., River Edge, NJ, 1994.; Mainardi, F., On the Initial Value Problem for the Fractional Diffusion-Wave Equation., 23 (1994) · Zbl 0879.35036
[65] F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, No 2 (2001), 153-192.; Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4, 2, 153-192 (2001) · Zbl 1054.35156
[66] F. Mainardi, A. Mura, G. Pagnini, The M-Wright function in time-fractional diffusion processes: A tutorial survey. Int. J. Differ. Equ. (2010).; Mainardi, F.; Mura, A.; Pagnini, G., The M-Wright function in time-fractional diffusion processes: A tutorial survey, Int. J. Differ. Equ. (2010) · Zbl 1222.60060
[67] F. Mainardi, G. Pagnini, R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. and Computing187 (2007), 295-305.; Mainardi, F.; Pagnini, G.; Gorenflo, R., Some aspects of fractional diffusion equations of single and distributed order, Appl. Math. and Computing, 187, 295-305 (2007) · Zbl 1122.26004
[68] F. Mainardi, G. Pagnini, R.K. Saxena, Fox H-functions in fractional diffusion. J. of Comput. and Appl. Math. 178 (2005), 321-331.; Mainardi, F.; Pagnini, G.; Saxena, R. K., Fox H-functions in fractional diffusion, J. of Comput. and Appl. Math., 178, 321-331 (2005) · Zbl 1061.33012
[69] M.M. Meerschaert, E. Nane, P. Vellaisamy, Fractional Cauchy problems on bounded domains. Ann. Probab. 37, No 3 (2009), 979-1007.; Meerschaert, M. M.; Nane, E.; Vellaisamy, P., Fractional Cauchy problems on bounded domains, Ann. Probab., 37, 3, 979-1007 (2009) · Zbl 1247.60078
[70] M.M. Meerschaert, H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41 (2004), 623-638.; Meerschaert, M. M.; Scheffler, H. P., Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab., 41, 623-638 (2004) · Zbl 1065.60042
[71] M.M. Meerschaert, H.P. Scheffler, Triangular array limits for continuous time random walks. Stochastic Processes and their Applications118 (2008), 1606-1633.; Meerschaert, M. M.; Scheffler, H. P., Triangular array limits for continuous time random walks, Stochastic Processes and their Applications, 118, 1606-1633 (2008) · Zbl 1153.60023
[72] M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus. De Gruyter Studies in Math. # 43, Walter de Gruyter, Berlin/Boston, 2012.; Meerschaert, M. M.; Sikorskii, A., Stochastic Models for Fractional Calculus. (2012) · Zbl 1247.60003
[73] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Physics Reports339 (2000), 1-77.; Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339, 1-77 (2000) · Zbl 0984.82032
[74] R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations. Physica A278 (2000), 107-125.; Metzler, R.; Klafter, J., Boundary value problems for fractional diffusion equations, Physica A, 278, 107-125 (2000) · Zbl 0984.82032
[75] U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Adv. in Math. 3 (1969), 510-585.; Mosco, U., Convergence of convex sets and of solutions of variational inequalities, Adv. in Math., 3, 510-585 (1969) · Zbl 0192.49101
[76] U. Mosco, Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, No 2 (1994), 368-421.; Mosco, U., Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123, 2, 368-421 (1994) · Zbl 0808.46042
[77] U. Mosco, M.A. Vivaldi, Layered fractal fibers and potentials. J. Math. Pures Appl. 9 (2015), 1198-1227.; Mosco, U.; Vivaldi, M. A., Layered fractal fibers and potentials, J. Math. Pures Appl., 9, 1198-1227 (2015) · Zbl 1319.35137
[78] U. Mosco, M.A. Vivaldi, Thin fractal fibers. Math. Methods Appl. Sci. 36, (2013), 2048-2068.; Mosco, U.; Vivaldi, M. A., Thin fractal fibers, Math. Methods Appl. Sci., 36, 2048-2068 (2013) · Zbl 1278.74142
[79] E. Nane, Higher order PDE’s and iterated processes. Trans. Amer. Math. Soc. 360 (2008), 2681-2692.; Nane, E., Higher order PDE’s and iterated processes, Trans. Amer. Math. Soc., 360, 2681-2692 (2008) · Zbl 1157.60071
[80] R. Nigmatullin, The realization of the generalized transfer in a medium with fractal geometry. Phys. Status Solidi B133 (1986), 425-430.; Nigmatullin, R., The realization of the generalized transfer in a medium with fractal geometry, Phys. Status Solidi B, 133, 425-430 (1986)
[81] E. Orsingher, L. Beghin, Fractional diffusion equations and processes with randomly varying time. Ann. Probab. 37 (2009), 206-249.; Orsingher, E.; Beghin, L., Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 37, 206-249 (2009) · Zbl 1173.60027
[82] H. Roman, P. Alemany, Continuous-time random walks and the fractional diffusion equation. J. Phys. A. 27 (1994), 3407-3410.; Roman, H.; Alemany, P., Continuous-time random walks and the fractional diffusion equation, J. Phys. A., 27, 3407-3410 (1994) · Zbl 0827.60057
[83] A. Saichev, G. Zaslavsky, Fractional kinetic equations: solutions and applications. Chaos. 7 (1997), 753-764.; Saichev, A.; Zaslavsky, G., Fractional kinetic equations: solutions and applications, Chaos., 7, 753-764 (1997) · Zbl 0933.37029
[84] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance. Physica A284 (2000), 376-384.; Scalas, E.; Gorenflo, R.; Mainardi, F., Fractional calculus and continuous-time finance, Physica A, 284, 376-384 (2000) · Zbl 1138.91444
[85] R.L. Schilling, R. Song, Z. Vondracek, Bernstein Functions, Theory and Applications. Ser. De Gruyter Studies in Math. # 37, Berlin, 2010.; Schilling, R. L.; Song, R.; Vondracek, Z., Bernstein Functions, Theory and Applications. (2010) · Zbl 1197.33002
[86] W. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134-144.; Schneider, W.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 134-144 (1989) · Zbl 0692.45004
[87] E. Soczkiewicz, Application of fractional calculus in the theory of viscoelasticity. Molecular and Quantum Acoustics23 (2002), 397-404.; Soczkiewicz, E., Application of fractional calculus in the theory of viscoelasticity, Molecular and Quantum Acoustics, 23, 397-404 (2002) · Zbl 0424.76075
[88] R. Song, Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain. Probab. Theory Relat. Fields125 (2013), 578-592.; Song, R.; Vondraček, Z., Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Relat. Fields, 125, 578-592 (2013) · Zbl 1022.60078
[89] H. Sun, M.M. Meerschaert, Y. Zhang, J. Zhu, W. Chen, A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Advances in Water Resources52 (2013), 292-295.; Sun, H.; Meerschaert, M. M.; Zhang, Y.; Zhu, J.; Chen, W., A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media, Advances in Water Resources, 52, 292-295 (2013)
[90] B. Toaldo, Convolution-type derivatives, hitting-times of subordinators and time-changed \(C_∞\)-semigroups. Potential Analysis42 (2015), 115-140.; Toaldo, B., Convolution-type derivatives, hitting-times of subordinators and time-changed \(C_∞\)-semigroups, Potential Analysis, 42, 115-140 (2015) · Zbl 1315.60050
[91] W. Wyss, The fractional diffusion equations. J. Math. Phys. 27 (1986), 2782-2785.; Wyss, W., The fractional diffusion equations, J. Math. Phys., 27, 2782-2785 (1986) · Zbl 0632.35031
[92] G. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos. Phys. D76 (1994), 110-122.; Zaslavsky, G., Fractional kinetic equation for Hamiltonian chaos, Phys. D, 76, 110-122 (1994) · Zbl 1194.37163
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