# American Institute of Mathematical Sciences

January  2015, 14(1): 269-284. doi: 10.3934/cpaa.2015.14.269

## Statistical exponential formulas for homogeneous diffusion

 1 Department of Mathematics, Sewanee: The University of the South, Sewanee, TN 37383

Received  March 2014 Revised  April 2014 Published  September 2014

Let $\Delta^{1}_{p}$ denote the $1$-homogeneous $p$-Laplacian, for $1 \leq p \leq \infty$. This paper proves that the unique bounded, continuous viscosity solution $u$ of the Cauchy problem \begin{eqnarray} u_{t} - ( \frac{p}{ N + p - 2 } ) \Delta^{1}_{p} u = 0 \quad \mbox{for} \quad x \in R^N \quad \mbox{and} \quad t > 0 , \\ \\ u(\cdot,0) = u_0 \in BUC(R^N). \end{eqnarray} is given by the exponential formula \begin{eqnarray} u(t) := \lim_{n \to \infty}{ ( M^{t/n}_{p} )^{n} u_{0} } \ , \end{eqnarray} where the statistical operator $M^h_p \colon BUC( R^{N} ) \to BUC( R^{N} )$ is defined by \begin{eqnarray} (M^{h}_{p} \varphi)(x) := (1-q) median_{\partial B(x,\sqrt{2h})}{ \{ \varphi \} } + q \int_{\partial B(x,\sqrt{2h})}{ \varphi ds } \end{eqnarray} when $1 \leq p \leq 2$, with $q := \frac{ N ( p - 1 ) }{ N + p - 2 }$, and by \begin{eqnarray} (M^{h}_{p} \varphi )(x) := ( 1 - q ) midrange_{\partial B(x,\sqrt{2h})}{ \{ \varphi\} } + q \int_{\partial B(x,\sqrt{2h})}{ \varphi ds } \end{eqnarray} when $p \geq 2$, with $q = \frac{ N }{ N + p - 2 }$. Possible extensions to problems with Dirichlet boundary conditions are mentioned briefly.
Citation: Matthew B. Rudd. Statistical exponential formulas for homogeneous diffusion. Communications on Pure and Applied Analysis, 2015, 14 (1) : 269-284. doi: 10.3934/cpaa.2015.14.269
##### References:
 [1] G. Akagi, P. Juutinen and R. Kajikiya, Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian, Math. Ann., 343 (2009), 921-953. doi: 10.1007/s00208-008-0297-1. [2] G. Akagi and K. Suzuki, Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian, Calc. Var. Partial Differential Equations, 31 (2008), 457-471. doi: 10.1007/s00526-007-0117-6. [3] L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal., 123 (1993), 199-257. doi: 10.1007/BF00375127. [4] M. Bardi, M. G. Crandall, L. C. Evans, H. M. Soner and P. E. Souganidis, Viscosity solutions and applications, vol. 1660 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1997, Lectures given at the 2nd C.I.M.E. Session held in Montecatini Terme, June 12-20, 1995, Edited by I. Capuzzo Dolcetta and P. L. Lions, Fondazione C.I.M.E.. [C.I.M.E. Foundation]. [5] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283. [6] G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500. doi: 10.1137/0732020. [7] F. Catté, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets, SIAM J. Numer. Anal., 32 (1995), 1895-1909. doi: 10.1137/0732085. [8] M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. [9] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [10] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [11] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681. [12] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. II, Trans. Amer. Math. Soc., 330 (1992), 321-332. doi: 10.2307/2154167. [13] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. III, J. Geom. Anal., 2 (1992), 121-150. doi: 10.1007/BF02921385. [14] L. C. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533-557. doi: 10.1512/iumj.1993.42.42024. [15] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. IV, J. Geom. Anal., 5 (1995), 77-114. doi: 10.1007/BF02926443. [16] Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470. doi: 10.1512/iumj.1991.40.40023. [17] Y. Giga, Surface Evolution Equations, vol. 99 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 2006, A level set approach. [18] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1985. [19] A. Grigor'yan, Heat Kernel and Analysis on Manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics, American Mathematical Society, Providence, RI, 2009. [20] D. Hartenstine and M. Rudd, Asymptotic statistical characterizations of $p$-harmonic functions of two variables, Rocky Mountain J. Math., 41 (2011), 493-504. doi: 10.1216/RMJ-2011-41-2-493. [21] D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions, Adv. Nonlinear Stud., 13 (2013), 191-207. [22] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1993, Oxford Science Publications. [23] T. Ilmanen, P. Sternberg and W. P. Ziemer, Equilibrium solutions to generalized motion by mean curvature, J. Geom. Anal., 8 (1998), 845-858. Dedicated to the memory of Fred Almgren. doi: 10.1007/BF02922673. [24] H. Ishii, G. E. Pires and P. E. Souganidis, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan, 51 (1999), 267-308. doi: 10.2969/jmsj/05120267. [25] V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation, Comm. Partial Differential Equations, 37 (2012), 934-946. doi: 10.1080/03605302.2011.615878. [26] P. Juutinen, $p$-harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1030. doi: 10.1512/iumj.2005.54.2658. [27] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851. doi: 10.1007/s00208-006-0766-3. [28] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179. [29] B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations, Comm. Partial Differential Equations, 32 (2007), 1209-1224. doi: 10.1080/03605300601113043. [30] B. Kawohl, Variations on the $p$-Laplacian, in Nonlinear elliptic partial differential equations, vol. 540 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2011, 35-46. doi: 10.1090/conm/540/10657. [31] B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188. doi: 10.1016/j.matpur.2011.07.001. [32] B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations, Funkcial. Ekvac., 43 (2000), 241-253. [33] R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407. doi: 10.1002/cpa.20101. [34] G. F. Lawler, Random Walk and the Heat Equation, vol. 55 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2010. [35] P. D. Lax, Functional Analysis, Pure and Applied Mathematics, Wiley-Interscience, John Wiley & Sons, New York, 2002. [36] E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29-55. doi: 10.1007/s00030-006-4030-z. [37] E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292. doi: 10.1137/S0036141095294067. [38] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [39] P. Lindqvist, Notes on the $p$-Laplace equation, vol. 102 of Report, University of Jyväskylä Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, 2006. [40] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal., 42 (2010), 2058-2081. doi: 10.1137/100782073. [41] B. Merriman, J. K. Bence and S. J. Osher, Motion of multiple functions: a level set approach, J. Comput. Phys., 112 (1994), 334-363. doi: 10.1006/jcph.1994.1105. [42] A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math., 99 (2004), 365-379. doi: 10.1007/s00211-004-0566-1. [43] A. M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), 1217-1230. doi: 10.1090/S0025-5718-04-01688-6. [44] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [45] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [46] Y. Peres and S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048. [47] S. J. Ruuth and B. Merriman, Convolution-generated motion and generalized Huygens' principles for interface motion, SIAM J. Appl. Math., 60 (2000), 868-890. doi: 10.1137/S003613999833397X. [48] P. Sternberg and W. P. Ziemer, Generalized motion by curvature with a Dirichlet condition, J. Differential Equations, 114 (1994), 580-600. doi: 10.1006/jdeq.1994.1162. [49] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, vol. 100 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1992. [50] W. P. Ziemer, Weakly Differentiable Functions, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.

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##### References:
 [1] G. Akagi, P. Juutinen and R. Kajikiya, Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian, Math. Ann., 343 (2009), 921-953. doi: 10.1007/s00208-008-0297-1. [2] G. Akagi and K. Suzuki, Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian, Calc. Var. Partial Differential Equations, 31 (2008), 457-471. doi: 10.1007/s00526-007-0117-6. [3] L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing, Arch. Rational Mech. Anal., 123 (1993), 199-257. doi: 10.1007/BF00375127. [4] M. Bardi, M. G. Crandall, L. C. Evans, H. M. Soner and P. E. Souganidis, Viscosity solutions and applications, vol. 1660 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1997, Lectures given at the 2nd C.I.M.E. Session held in Montecatini Terme, June 12-20, 1995, Edited by I. Capuzzo Dolcetta and P. L. Lions, Fondazione C.I.M.E.. [C.I.M.E. Foundation]. [5] G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271-283. [6] G. Barles and C. Georgelin, A simple proof of convergence for an approximation scheme for computing motions by mean curvature, SIAM J. Numer. Anal., 32 (1995), 484-500. doi: 10.1137/0732020. [7] F. Catté, F. Dibos and G. Koepfler, A morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets, SIAM J. Numer. Anal., 32 (1995), 1895-1909. doi: 10.1137/0732085. [8] M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93 (1971), 265-298. [9] M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [10] E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [11] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681. [12] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. II, Trans. Amer. Math. Soc., 330 (1992), 321-332. doi: 10.2307/2154167. [13] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. III, J. Geom. Anal., 2 (1992), 121-150. doi: 10.1007/BF02921385. [14] L. C. Evans, Convergence of an algorithm for mean curvature motion, Indiana Univ. Math. J., 42 (1993), 533-557. doi: 10.1512/iumj.1993.42.42024. [15] L. C. Evans and J. Spruck, Motion of level sets by mean curvature. IV, J. Geom. Anal., 5 (1995), 77-114. doi: 10.1007/BF02926443. [16] Y. Giga, S. Goto, H. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470. doi: 10.1512/iumj.1991.40.40023. [17] Y. Giga, Surface Evolution Equations, vol. 99 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 2006, A level set approach. [18] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1985. [19] A. Grigor'yan, Heat Kernel and Analysis on Manifolds, vol. 47 of AMS/IP Studies in Advanced Mathematics, American Mathematical Society, Providence, RI, 2009. [20] D. Hartenstine and M. Rudd, Asymptotic statistical characterizations of $p$-harmonic functions of two variables, Rocky Mountain J. Math., 41 (2011), 493-504. doi: 10.1216/RMJ-2011-41-2-493. [21] D. Hartenstine and M. Rudd, Statistical functional equations and $p$-harmonious functions, Adv. Nonlinear Stud., 13 (2013), 191-207. [22] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1993, Oxford Science Publications. [23] T. Ilmanen, P. Sternberg and W. P. Ziemer, Equilibrium solutions to generalized motion by mean curvature, J. Geom. Anal., 8 (1998), 845-858. Dedicated to the memory of Fred Almgren. doi: 10.1007/BF02922673. [24] H. Ishii, G. E. Pires and P. E. Souganidis, Threshold dynamics type approximation schemes for propagating fronts, J. Math. Soc. Japan, 51 (1999), 267-308. doi: 10.2969/jmsj/05120267. [25] V. Julin and P. Juutinen, A new proof for the equivalence of weak and viscosity solutions for the $p$-Laplace equation, Comm. Partial Differential Equations, 37 (2012), 934-946. doi: 10.1080/03605302.2011.615878. [26] P. Juutinen, $p$-harmonic approximation of functions of least gradient, Indiana Univ. Math. J., 54 (2005), 1015-1030. doi: 10.1512/iumj.2005.54.2658. [27] P. Juutinen and B. Kawohl, On the evolution governed by the infinity Laplacian, Math. Ann., 335 (2006), 819-851. doi: 10.1007/s00208-006-0766-3. [28] P. Juutinen, P. Lindqvist and J. J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation, SIAM J. Math. Anal., 33 (2001), 699-717. doi: 10.1137/S0036141000372179. [29] B. Kawohl and N. Kutev, Comparison principle for viscosity solutions of fully nonlinear, degenerate elliptic equations, Comm. Partial Differential Equations, 32 (2007), 1209-1224. doi: 10.1080/03605300601113043. [30] B. Kawohl, Variations on the $p$-Laplacian, in Nonlinear elliptic partial differential equations, vol. 540 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2011, 35-46. doi: 10.1090/conm/540/10657. [31] B. Kawohl, J. Manfredi and M. Parviainen, Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97 (2012), 173-188. doi: 10.1016/j.matpur.2011.07.001. [32] B. Kawohl and N. Kutev, Comparison principle and Lipschitz regularity for viscosity solutions of some classes of nonlinear partial differential equations, Funkcial. Ekvac., 43 (2000), 241-253. [33] R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407. doi: 10.1002/cpa.20101. [34] G. F. Lawler, Random Walk and the Heat Equation, vol. 55 of Student Mathematical Library, American Mathematical Society, Providence, RI, 2010. [35] P. D. Lax, Functional Analysis, Pure and Applied Mathematics, Wiley-Interscience, John Wiley & Sons, New York, 2002. [36] E. Le Gruyer, On absolutely minimizing Lipschitz extensions and PDE $\Delta_\infty(u)=0$, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 29-55. doi: 10.1007/s00030-006-4030-z. [37] E. Le Gruyer and J. C. Archer, Harmonious extensions, SIAM J. Math. Anal., 29 (1998), 279-292. doi: 10.1137/S0036141095294067. [38] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [39] P. Lindqvist, Notes on the $p$-Laplace equation, vol. 102 of Report, University of Jyväskylä Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, 2006. [40] J. J. Manfredi, M. Parviainen and J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games, SIAM J. Math. Anal., 42 (2010), 2058-2081. doi: 10.1137/100782073. [41] B. Merriman, J. K. Bence and S. J. Osher, Motion of multiple functions: a level set approach, J. Comput. Phys., 112 (1994), 334-363. doi: 10.1006/jcph.1994.1105. [42] A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numer. Math., 99 (2004), 365-379. doi: 10.1007/s00211-004-0566-1. [43] A. M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp., 74 (2005), 1217-1230. doi: 10.1090/S0025-5718-04-01688-6. [44] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. [45] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [46] Y. Peres and S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$-Laplacian, Duke Math. J., 145 (2008), 91-120. doi: 10.1215/00127094-2008-048. [47] S. J. Ruuth and B. Merriman, Convolution-generated motion and generalized Huygens' principles for interface motion, SIAM J. Appl. Math., 60 (2000), 868-890. doi: 10.1137/S003613999833397X. [48] P. Sternberg and W. P. Ziemer, Generalized motion by curvature with a Dirichlet condition, J. Differential Equations, 114 (1994), 580-600. doi: 10.1006/jdeq.1994.1162. [49] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, vol. 100 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1992. [50] W. P. Ziemer, Weakly Differentiable Functions, vol. 120 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.
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