The Transition Density of Brownian Motion Killed on a Bounded Set

Kôhei Uchiyama ()
Additional contact information
Kôhei Uchiyama: Tokyo Institute of Technology

Journal of Theoretical Probability, 2018, vol. 31, issue 3, 1380-1410

Abstract: Abstract 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 $$p^A_t(\mathbf{x},\mathbf{y})$$ p t A ( x , y ) , for large times t and for $$\mathbf{x}$$ x and $$\mathbf{y}$$ y in the exterior of A valid uniformly under the constraint $$|\mathbf{x}|\vee |\mathbf{y}| =O(t)$$ | x | ∨ | y | = O ( t ) . Within the parabolic regime $$|\mathbf{x}|\vee |\mathbf{y}| = O(\sqrt{t})$$ | x | ∨ | y | = O ( t ) in particular $$p^A_t(\mathbf{x},\mathbf{y})$$ p t A ( x , y ) is shown to behave like $$4e_A(\mathbf{x})e_A(\mathbf{y}) (\lg t)^{-2} p_t(\mathbf{y}-\mathbf{x})$$ 4 e A ( x ) e A ( y ) ( lg t ) - 2 p t ( y - x ) for large t, where $$p_t(\mathbf{y}-\mathbf{x})$$ p t ( y - x ) is the transition kernel of the Brownian motion (without killing) and $$e_A$$ e A is the Green function for the ‘exterior of A’ with a pole at infinity normalized so that $$e_A(\mathbf{x}) \sim \lg |\mathbf{x}|$$ e A ( x ) ∼ lg | x | . We also provide fairly accurate upper and lower bounds of $$p^A_t(\mathbf{x},\mathbf{y})$$ p t A ( x , y ) for the case $$|\mathbf{x}|\vee |\mathbf{y}|>t$$ | x | ∨ | y | > t as well as corresponding results for the higher dimensions.

Keywords: Heat kernel; Exterior domain; Transition probability; Primary 60J65; Secondary 35K20 (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-017-0758-0 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:31:y:2018:i:3:d:10.1007_s10959-017-0758-0

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-017-0758-0

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().