On a Brownian motion conditioned to stay in an open set

G. V.  Riabov

doi:10.37863/umzh.v72i9.6281

G. V. Riabov Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv https://orcid.org/0000-0003-2770-9629

DOI: https://doi.org/10.37863/umzh.v72i9.6281

Abstract

UDC 519.21

Distribution of a Brownian motion conditioned to start from the boundary of an open set $G$ and to stay in $G$ for a finite period of time is studied.
Characterizations of such distributions in terms of certain singular stochastic differential equations are obtained.
Results are applied to the study of boundaries of clusters in some coalescing stochastic flows on $\mathbb{R}.$

References

R. Garbit, Brownian motion conditioned to stay in a cone, J. Math. Kyoto Univ., 49, No. 3, 573 – 592 (2009), https://doi.org/10.1215/kjm/1260975039 DOI: https://doi.org/10.1215/kjm/1260975039

Y. Le Jan, O. Raimond, Flows, coalescence and noise, Ann. Probab., 32, No. 2, 1247 – 1315 (2004), https://doi.org/10.1214/009117904000000207 DOI: https://doi.org/10.1214/009117904000000207

G. V. Riabov, Random dynamical systems generated by coalescing stochastic flows on $R$, Stoch. and Dyn., 18, No. 04, Article 185003, 24 pp. (2018), https://doi.org/10.1142/S0219493718500314 DOI: https://doi.org/10.1142/S0219493718500314

G. V. Riabov, Duality for coalescing stochastic flows on the real line, Theory Stoch. Process., 23, No. 2, 55 – 74 (2018)

A. A. Dorogovtsev, G. V. Riabov, B. Schmalfuß, Stationary points in coalescing stochastic flows on $mathbb{R}$, Stoch. Process. and Appl., 130, No. 8, 4910 – 4926 (2020), https://doi.org/10.1016/j.spa.2020.02.005 DOI: https://doi.org/10.1016/j.spa.2020.02.005

R. A. Arratia, Coalescing Brownian motions on the line, Ph. D Thesis, Univ. Wisconsin, Madison, 134 pp. (1979).

R. A. Arratia, Coalescing Brownian motions and the voter model on $Z$ , unpublished partial manuscript (circa 1981), available from rarratia@math.usc.edu.

B. Tóth, W. Werner, ´ The true self-repelling motion , Probab. Theory and Relat. Fields, 111, No. 3, 375 – 452 (1998), https://doi.org/10.1007/s004400050172 DOI: https://doi.org/10.1007/s004400050172

R. Tribe, O. Zaboronski, Pfaffian formulae for one dimensional coalescing and annihilating systems , Electron. J. Probab., 16, no. 76, 2080 – 2103 (2011), https://doi.org/10.1214/EJP.v16-942 DOI: https://doi.org/10.1214/EJP.v16-942

V. Fomichov, The distribution of the number of clusters in the Arratia flow , Commun. Stoch. Anal., 10, No. 3, 257 – 270 (2016), https://doi.org/10.31390/cosa.10.3.01 DOI: https://doi.org/10.31390/cosa.10.3.01

J.-P. Imhof, Density factorizations for Brownian motion, meander and the three-dimensional Bessel process and applications, J. Appl. Probab., 21, No. 3, 500 – 510 (1984), https://doi.org/10.2307/3213612 DOI: https://doi.org/10.2307/3213612

R. T. Durrett, D. L. Iglehart, D. R. Miller, Weak convergence to Brownian meander and Brownian excursion , Ann. Probab., 5, No. 1, 117 – 129 (1977), https://doi.org/10.1214/aop/1176995895 DOI: https://doi.org/10.1214/aop/1176995895

A. N. Shiryaev, M. Yor, On the problem of stochastic integral representations of functionals of the Brownian motion, I , Theory Probab. and Appl., 48, No. 2, 304 – 313 (2004), https://doi.org/10.1137/S0040585X97980427 DOI: https://doi.org/10.1137/S0040585X97980427

D. Revuz, M. Yor, Continuous martingales and Brownian motion , Vol. 293, Springer Sci. & Business Media (2013), https://doi.org/10.1007/978-3-642-31898-6 DOI: https://doi.org/10.1007/978-3-642-31898-6

O. Kallenberg, Foundations of modern probability, Springer Sci. & Business Media (2006).

H. Scheffé,´ A useful convergence theorem for probability distributions , Ann. Math. Stat., 18, No. 3, 434 – 438 (1947), https://doi.org/10.1214/aoms/1177730390 DOI: https://doi.org/10.1214/aoms/1177730390

A. S. Cherny, On the strong and weak solutions of stochastic differential equations governing Bessel processes , Stochastics, 70, № 3-4, 213 – 219 (2000), https://doi.org/10.1080/17442500008834252 DOI: https://doi.org/10.1080/17442500008834252

K. Burdzy, Brownian excursions from hyperplanes and smooth surfaces , Trans. Amer. Math. Soc., 295, No. 1, 35 – 57 (1986), https://doi.org/10.2307/2000144 DOI: https://doi.org/10.2307/2000144

H. J. Brascamp, E. Lieb, On extensions of the Brunn – Minkowski and Prékopa-Leindler theorems, including ´inequalities for log concave functions, and with an application to the diffusion equation , J. Funct. Anal., 22, No. 4, 366 – 389 (1976), https://doi.org/10.1016/0022-1236(76)90004-5 DOI: https://doi.org/10.1016/0022-1236(76)90004-5