Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1380–1410 | Cite as

The Transition Density of Brownian Motion Killed on a Bounded Set

  • Kôhei UchiyamaEmail author


We study the transition density of a standard two-dimensional Brownian motion killed when hitting a bounded Borel set A. We derive the asymptotic form of the density, say \(p^A_t(\mathbf{x},\mathbf{y})\), for large times t and for \(\mathbf{x}\) and \(\mathbf{y}\) in the exterior of A valid uniformly under the constraint \(|\mathbf{x}|\vee |\mathbf{y}| =O(t)\). Within the parabolic regime \(|\mathbf{x}|\vee |\mathbf{y}| = O(\sqrt{t})\) in particular \(p^A_t(\mathbf{x},\mathbf{y})\) is shown to behave like \(4e_A(\mathbf{x})e_A(\mathbf{y}) (\lg t)^{-2} p_t(\mathbf{y}-\mathbf{x})\) for large t, where \(p_t(\mathbf{y}-\mathbf{x})\) is the transition kernel of the Brownian motion (without killing) and \(e_A\) is the Green function for the ‘exterior of A’ with a pole at infinity normalized so that \(e_A(\mathbf{x}) \sim \lg |\mathbf{x}|\). We also provide fairly accurate upper and lower bounds of \(p^A_t(\mathbf{x},\mathbf{y})\) for the case \(|\mathbf{x}|\vee |\mathbf{y}|>t\) as well as corresponding results for the higher dimensions.


Heat kernel Exterior domain Transition probability 

Mathematics Subject Classification (2010)

Primary 60J65 Secondary 35K20 


  1. 1.
    Bass, R.F.: Probabilistic Techniques in Analysis. Springer, Berlin (1995)zbMATHGoogle Scholar
  2. 2.
    Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York (1968)zbMATHGoogle Scholar
  3. 3.
    Chung, K.L., Walsh, J.B.: Markov Processes, Brownian Motion, and Time Symmetry, 2nd edn. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    Collet, P., Martinez, S., Martin, J.: Asymptotic behaviour of a Brownian motion on exterior domains. Probab. Theory Relat. Fields 116, 303–316 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carslaw, H.S., Jaeger, J.C.: Conduction of Heat in Solids. Clarrendon press, Oxford (1992)zbMATHGoogle Scholar
  6. 6.
    Erdélyi, A.: c, vol. I. McGraw-Hill Inc, New York (1954)Google Scholar
  7. 7.
    Lebedev, N.N.: Special Functions and Their Applications. Prentice-Hall Inc, Englewood Cliffs (1965)zbMATHGoogle Scholar
  8. 8.
    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, BErlin (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    Uchiyama, K.: Asymptotic estimates of the distribution of Brownian hitting time of a disc. J. Theor. Probab. 25(3), 450–463 (2012). (Erratum, J. Theor. Probab. 25 (2012), issue 3, 910–911)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Uchiyama, K.: Asymptotics of the densities of the first passage time distributions of Bessel diffusion. Trans. Am. Math. Soc. 367, 2719–2742 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Uchiyama, K.: Density of space–time distribution of Brownian first hitting of a disc and a ball. Potential Anal. 44, 495–541 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Uchiyama, K.: The Brownian hitting distributions in space-time of bounded sets and the expected volume of the Wiener sausage for a Brownian bridge. arxiv:1406.1307v3
  13. 13.
    Uchiyama, K.: Asymptotic behaviour of a random walk killed on a finite set. Potential Anal. (2016). doi: 10.1007/s11118-016-9598-2 Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsTokyo Institute of TechnologyTokyoJapan

Personalised recommendations