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

Анотація

УДК 519.21

Про умовний розподiл броунiвського руху, що не виходить з вiдкритої множини

Досліджується розподіл броунівського руху, що стартував із граничної точки відкритої множини $G$ і залишається в $G$ протягом скінченного інтервалу часу. Отримано характеризацію таких розподілів у термінах певних сингулярних стохастичних диференціальних рівнянь. Отримані результати застосовано до вивчення меж кластерів у деяких стохастичних потоках зі склеюванням на $\mathbb{R}.$

Посилання

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R. A. Arratia, Coalescing Brownian motions and the voter model on $Z$ , unpublished partial manuscript (circa 1981), available from rarratia@math.usc.edu.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