Monotonicity properties of the Neumann heat kernel in the ball☆☆The authors kindly acknowledge the support from CNCSIS - UEFISCSU research grant PNII - IDEI 209/2007.

Atar, 2004, On Neumann eigenfunctions in lip domains, J. Amer. Math. Soc., 17, 243, 10.1090/S0894-0347-04-00453-9 Atar, 2008, Mirror couplings and Neumann eigenfunctions, Indiana Univ. Math. J., 57, 1317, 10.1512/iumj.2008.57.3222 Bañuelos, 1999, On the “Hot Spots” conjecture of J. Rauch, J. Funct. Anal., 164, 1, 10.1006/jfan.1999.3397 Bañuelos, 2009, Neumann heat kernel monotonicity, Potential Anal., 3, 65, 10.1007/s11118-008-9106-4 Bass, 2000, Pathwise uniqueness for reflecting Brownian motion in Euclidean domains, Probab. Theory Related Fields, 117, 183, 10.1007/s004400050003 Burdzy, 2000, Efficient Markovian couplings: Examples and counterexamples, Ann. Appl. Probab., 10, 362, 10.1214/aoap/1019487348 Jerison, 2000, The “hot spots” conjecture for domains with two axes of symmetry, J. Amer. Math. Soc., 13, 741, 10.1090/S0894-0347-00-00346-5 Kendall, 1986, Nonnegative Ricci curvature and the Brownian coupling property, Stochastics, 19, 111, 10.1080/17442508608833419 Laugesen, 1998, Extremals for eigenvalues of Laplacians under conformal mapping, J. Funct. Anal., 155, 64, 10.1006/jfan.1997.3222 Pascu Pascu, 2002, Scaling coupling and the Hot spots conjecture, Trans. Amer. Math. Soc., 354, 4681, 10.1090/S0002-9947-02-03020-9 Pascu, 2009, On a discrete version of the Laugesen–Morpurgo conjecture, Statist. Probab. Lett., 79, 797, 10.1016/j.spl.2008.10.034 Pascu, 2007, Monotonicity properties of reflecting Brownian motion, 109 Pascu, 2008, Brownian motion on the circle and applications, Bull. Transilv. Univ. of Braşov Ser. III, 1, 469 Wang, 1994, Application of coupling methods to the Neumann eigenvalue problem, Probab. Theory Related Fields, 98, 299, 10.1007/BF01192256