Alternative probabilistic representations of Barenblatt-type solutions
Volume 7, Issue 1 (2020), pp. 97–112
Pub. online: 23 March 2020
Type: Research Article
Open Access
Open Access
Received
29 November 2019
29 November 2019
Revised
4 March 2020
4 March 2020
Accepted
13 March 2020
13 March 2020
Published
23 March 2020
23 March 2020
Abstract
A general class of probability density functions
\[ u(x,t)=C{t^{-\alpha d}}{\left(1-{\left(\frac{\| x\| }{c{t^{\alpha }}}\right)^{\beta }}\right)_{+}^{\gamma }},\hspace{1em}x\in {\mathbb{R}^{d}},t>0,\]
is considered, containing as particular case the Barenblatt solutions arising, for instance, in the study of nonlinear heat equations. Alternative probabilistic representations of the Barenblatt-type solutions $u(x,t)$ are proposed. In the one-dimensional case, by means of this approach, $u(x,t)$ can be connected with the wave propagation.References
Baeumer, B., Meerschaert, M.M., Nane, E.: Space-time duality for fractional diffusion. J. Appl. Probab. 46(4), 1100–1115 (2009). MR2582709. https://doi.org/10.1239/jap/1261670691
Benachour, S., Chassaing, P., Roynette, B., Vallois, P.: Processus associés a l’équation des milieux poreux. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 4, 793–832 (1996). MR1469575
Biler, P., Imbert, C., Karch, G.: Barenblatt profiles for a non local porous medium equation. C. R. Acad. Sci. Paris, Ser. I 349, 641–645 (2011). MR2817383. https://doi.org/10.1016/j.crma.2011.06.003
Biler, P., Imbert, C., Karch, G.: The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak Solutions. Arch. Ration. Mech. Anal. 215, 497–529 (2015). MR3294409. https://doi.org/10.1007/s00205-014-0786-1
Bresters, D.W.: On the equation of Euler-Poisson-Darboux. SIAM J. Math. Anal. 4(1), 31–41 (1973). MR0324235. https://doi.org/10.1137/0504005
Chen, Z.Q., Meerschaert, M.M., Nane, E.: Space-time fractional diffusion on bounded domains. J. Math. Anal. Appl. 393(2), 479–488 (2012). MR2921690. https://doi.org/10.1016/j.jmaa.2012.04.032
De Gregorio, A., Orsingher, E.: Flying randomly in ${\mathbb{R}^{d}}$ with Dirichlet displacements. Stoch. Process. Appl. 122, 676–713 (2012). MR2868936. https://doi.org/10.1016/j.spa.2011.10.009
De Gregorio, A.: Stochastic models associated to a Nonlocal Porous Medium Equation. Mod. Stoch. Theory Appl. 5, 457–470 (2018). MR3914725. https://doi.org/10.15559/18-vmsta112
De Gregorio, A., Orsingher, E.: Random flights connecting Porous Medium and Euler-Poisson-Darboux equations (2017). arXiv:1709.07663, 20 pp.
Erdélyi, A.: On the Euler-Poisson-Darboux equation. J. Anal. Math. 23(1), 89–102 (1970). MR0285807. https://doi.org/10.1007/BF02795492
Feng, S., Iscoe, I., Seppäläinen, T.: A microscopic mechanism for the porous medium equation. Stoch. Process. Appl. 66, 147–182 (1997). MR1440397. https://doi.org/10.1016/S0304-4149(96)00121-4
Garra, R., Orsingher, E.: Random Flights Related to the Euler-Poisson-Darboux Equation. Markov Process. Relat. Fields 22, 87–110 (2016). MR3523980
Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90 (1961). MR0137148. https://doi.org/10.2307/1993412
Gradshteyn, I.S., Ryzhik, I.M.: Tables of Integrals, Series, and Products, 4th edn. Academic Press, New York (1980). MR0669666
Heyde, C.C., Leonenko, N.N.: Student processes. Adv. Appl. Probab. 37, 342–365 (2005). MR2144557. https://doi.org/10.1239/aap/1118858629
Inoue, M.: A Markov process associated with a porous medium equation. Proc. Jpn. Acad. 60, Ser. A, 157–160 (1989). MR0758056. https://doi.org/10.3792/pjaa.60.157
Inoue, M.: Construction of diffusion processes associated with a porous medium equation. Hiroshima Math. J. 19, 281–297 (1989). MR1027932. https://doi.org/10.32917/hmj/1206129389
Inoue, M.: Derivation of a porous medium equation from many Markovian particles and the propagation of chaos. Hiroshima Math. J. 21, 85–110 (1991). MR1091433. https://doi.org/10.32917/hmj/1206128924
Jourdain, B.: Probabilistic approximation for a porous medium equation. Stoch. Process. Appl. 89, 81–99 (2000). MR1775228. https://doi.org/10.1016/S0304-4149(00)00014-4
Kamin, S., Vazquez, J.L.: Fundamental solutions and asymptotic behaviour for the p-Laplacian equation. Rev. Mat. Iberoam. 4, 339–354 (1988). MR1028745. https://doi.org/10.4171/RMI/77
Kellendong, J., Richard, S.: Weber-Schafheitlin-type integrals with exponent 1. Integral Transforms Spec. Funct. 20, 147–153 (2009). MR2492212. https://doi.org/10.1080/10652460802321485
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. In: Vol., vol. 204. Elsevier Science Limited (2006). MR2218073
Lee, K., Petrosyan, A., Vazquez, J.L.: Large-time geometric properties of solutions of the evolution p-Laplacian equation. J. Differ. Equ. 229, 389–411 (2006). MR2263560. https://doi.org/10.1016/j.jde.2005.07.028
Meerschaert, M.M., Nane, E., Vellaisamy, P.: Fractional Cauchy problems on bounded domains. Ann. Probab. 37(3), 979–1007 (2009). MR2537547. https://doi.org/10.1214/08-AOP426
Mendes, R.S., Lenzi, E.K., Malacarne, L.C., Picoli, S., Jauregui, M.: Random walks associated with nonlinear Fokker-Planck equations. Entropy 19 (2017), Paper No. 155, 11 pp. MR3653180. https://doi.org/10.3390/e19040155
Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series, vol. 335. Cambridge Univ. Press, Cambridge (2006). MR2266879. https://doi.org/10.1017/CBO9780511735127
Orsingher, E., Beghin, L.: Fractional diffusion equations and processes with randomly varying time. Ann. Probab. 37(1), 206–249 (2009). MR2489164. https://doi.org/10.1214/08-AOP401
Pagnini, G.: Erdélyi-Kober fractional diffusion. Fract. Calc. Appl. Anal. 15, 117–127 (2012). MR2872114. https://doi.org/10.2478/s13540-012-0008-1
Philipowski, R.: Interacting diffusions approximating the porous medium equation and propagation of chaos. Stoch. Process. Appl. 117, 526–538 (2007). MR2305385. https://doi.org/10.1016/j.spa.2006.09.003
Plociniczak, L., Świtala, M.: Compactly supported solution of the time-fractional porous medium equation on the half-line (2018). arXiv:1803.03016. MR3924620. https://doi.org/10.1137/18M1192561
Rosencrans, S.I.: Diffusion Transforms. J. Differ. Equ. 13, 457–467 (1973). MR0331533. https://doi.org/10.1016/0022-0396(73)90004-1
Tarasov, V.E., Tarasova, S.: Probabilistic Interpretation of Kober Fractional Integral of Non-Integer Order. Prog. Fract. Differ. Appl. 5(1), 1–5 (2019). https://doi.org/10.18576/pfda/050101
Vazquez, J.L.: The Porous Medium Equation. Mathematical Theory. Oxford Math. Monogr. Oxford Univ. Press, Oxford (2007). MR2286292
Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104, 201–220 (1991). MR1094052. https://doi.org/10.1007/BF01245072
Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables: A Noncommutative Probability Approach to Free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups. CRM Monograph Series, vol. 1. Amer. Math. Soc., Providence, RI (1992). MR1217253
