## The Annals of Probability

### The Adjoint Process of Killed Reflected Brownian Motion in a Cone and Applications

R. Dante DeBlassie

#### Abstract

Let $X_t$ be reflected Brownian motion (RBM) in a cone with radially homogeneous reflection, killed upon reaching the vertex of the cone. We determine the adjoint process and use it to find the Martin boundary of the killed RBM together with all the corresponding positive harmonic functions. Then we can identify and prove uniqueness (up to positive scalar multiples) of the invariant measure for killed RBM and RBM without killing. Along the way, we prove the strong Feller property of the resolvent of RBM (no killing).

#### Article information

Source
Ann. Probab., Volume 27, Number 4 (1999), 1679-1737.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022874812

Digital Object Identifier
doi:10.1214/aop/1022874812

Mathematical Reviews number (MathSciNet)
MR1742885

Zentralblatt MATH identifier
0965.60069

#### Citation

DeBlassie, R. Dante. The Adjoint Process of Killed Reflected Brownian Motion in a Cone and Applications. Ann. Probab. 27 (1999), no. 4, 1679--1737. doi:10.1214/aop/1022874812. https://projecteuclid.org/euclid.aop/1022874812

#### References

• Abramowitz, M. and Stegun, I. A. (1972). Handbook of Mathematical Functions. Dover, New York.
• Azema, J., Kaplan-Duflo, M. and Revuz, D. (1967). Measure invariant sur les classes r´ecurrentes des processus de Markov,Wahrsch. Verw. Gebieze 8 157-181.
• Bass, R. F. and Pardoux, E. (1987). Uniqueness for diffusions with piecewise constant coefficients. Probab. Theory Related Fields 76 557-572.
• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Borodin, A. N. and Salminen, P. (1996). Handbook of Brownian Motion. Facts and Formulae. Birkh¨auser, Basel.
• DeBlassie, R. D. (1994). Invariant measures for transient reflected Brownian motion in a wedge: existence and uniqueness. J. Multivariate Anal. 48 203-227.
• DeBlassie, R. D., Hobson, D., Housworth, E. D. and Toby, E. H. (1996). Escape rates for transient reflected Brownian motion in wedges and cones. Stochastic, Stochastics Rep. 57 199-211.
• Doob, J. L. (1984). Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin.
• Gilbarg, D. and Trudinger, N. (1983). Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, New York.
• Helgason, S. (1962). Differential Geometry and Symmetric Spaces. Academic Press, New York.
• Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam.
• Kunita, H. and Watanabe, T. (1965). Markov processes and Martin boundaries I. Illinois J. Math. 9 485-526.
• Kwon, y. (1992). The submartingale problem for Brownian motion in a cone with non-constant oblique reflection. Probab. Theory Related Fields 92 351-391.
• Kwon, Y. and Williams, R. F. (1991). Reflected Brownian motion in a cone with radially homogeneous reflection. Trans. Amer. Math. Soc. 327 739-790.
• Lions, P. L. and Sznitman, A. S. (1984). Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511-537.
• Nagasawa, M. (1961). The adjoint process of a diffusion with reflecting barrier. Kodai Math. Sem. Rep. 13 235-248.
• Port, S. C. and Stone, C. J. (1978). Brownian Motion and Classical Potential Theory. Academic Press, New York.
• Stroock, D. W. and Varadhan, S. R. S. (1971). Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24 147-225.
• Taira, K. (1988). Diffusion Processes and Partial Differential Equations. Academic Press, Boston.
• Williams, R. J. (1985). Recurrence classification and invariant measure for reflected Brownian motion in a wedge. Ann. Probab. 13 758-778.