## Electronic Communications in Probability

### Discrete approximations to local times for reflected diffusions

Wai-Tong Louis Fan

#### Abstract

We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from random walks on lattices. We establish weak convergence of the joint law of the discrete local time and the associated random walks as the lattice size decreases to zero. A cornerstone of the proof is the local central limit theorem for reflected diffusions developed in [7]. Applications of the join convergence result to PDE problems are illustrated.

#### Article information

Source
Electron. Commun. Probab. Volume 21 (2016), paper no. 16, 12 pp.

Dates
Accepted: 19 February 2016
First available in Project Euclid: 23 February 2016

https://projecteuclid.org/euclid.ecp/1456238572

Digital Object Identifier
doi:10.1214/16-ECP4694

Mathematical Reviews number (MathSciNet)
MR3485385

Zentralblatt MATH identifier
1336.60068

#### Citation

Fan, Wai-Tong Louis. Discrete approximations to local times for reflected diffusions. Electron. Commun. Probab. 21 (2016), paper no. 16, 12 pp. doi:10.1214/16-ECP4694. https://projecteuclid.org/euclid.ecp/1456238572

#### References

• [1] R. F. Bass, K. Burdzy and Z.-Q. Chen. Uniqueness for reflecting Brownian motion in lip domains. Annales de l’Institut Henri Poincare (B) Probability and Statistics. 41 (2005), 197-235.
• [2] R. F. Bass and P. Hsu. Some potential theory for reflecting Brownian motion in Hölder and lipschitz domains. Ann. Probab. 19 (1991), 486-508.
• [3] R. F. Bass and and T. Kumagai. Symmetric Markov chains on $\mathbb{Z}^d$ with unbounded range. Trans. Amer. Math. Soc. 360 (2008), 2041-2075.
• [4] K. Burdzy and Z.-Q. Chen. Discrete approximations to reflected Brownian motion. Ann. Probab. 36 (2008), 698-727.
• [5] K. Burdzy and Z.-Q. Chen. Reflected random walk in fractal domains. Ann. Probab. 41 (2011), 2791-2819.
• [6] Z.-Q. Chen. On reflecting diffusion processes and Skorokhod decompositions. Probab. Theory Relat. Fields. 94 (1993), 281-316.
• [7] Z.-Q. Chen and W.-T. Fan. Hydrodynamic limits and propagation of chaos for interacting random walks in domains. Preprint, arXiv:1311.2325.
• [8] Z.-Q. Chen and W.-T. Fan. Systems of interacting diffusions with partial annihilations through membranes. Ann. of Probab. To appear.
• [9] Z.-Q. Chen and M. Fukushima. Symmetric Markov Processes, Time Change and Boundary Theory. Princeton. University Press, 2012.
• [10] S.N. Ethier and T.G. Kurtz. Markov processes. Characterization and Convergence. Wiley, New York, 1986.
• [11] W.-T. Fan. Interacting particle systems with partial annihilation through membranes. PhD thesis, University of Washington, 2014.
• [12] P. Gyrya and L. Saloff-Coste. Neumann and Dirichlet Heat Kernels in Inner Uniform Domains. Astérisque 336 (2011), viii+144 pp.
• [13] J.M. Harrison. Brownian motion and stochastic flow systems. Wiley, New York, 1985
• [14] G. N. Milstein and M.V. Tretyakov. Stochastic numerics for mathematical physics. Springer, 2004
• [15] V. G. Papanicolaou. The probabilistic solution of the third boundary value problem for second order elliptic equations. Probab. Theory Relat. Fields. 87 (1990), 27-77.
• [16] A. Singer, Z. Schuss, A. Osipov, and D. Holcman. Partially reflected diffusion. SIAM Journal on Applied Mathematics. 68 (2008), 844-868.
• [17] D.W. Stroock and W. Zheng. Markov chain approximations to symmetric diffusions. Ann. Inst. Henri. Poincaré-Probab. Statist. 33 (1997), 619-649.