Electronic Journal of Probability

On countably skewed Brownian motion with accumulation point

Gerald Trutnau, Youssef Ouknine, and Francesco Russo

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In this work we connect the theory of symmetric Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and possible applications to advection-diffusion in layered media.

Article information

Electron. J. Probab. Volume 20 (2015), paper no. 82, 27 pp.

Accepted: 7 August 2015
First available in Project Euclid: 4 June 2016

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Zentralblatt MATH identifier

Primary: 31C25: Dirichlet spaces 60J60: Diffusion processes [See also 58J65] 60J55: Local time and additive functionals
Secondary: 31C15: Potentials and capacities 60B10: Convergence of probability measures

Skew Brownian motion local time strong existence pathwise uniqueness transience recurrence positive recurrence

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Trutnau, Gerald; Ouknine, Youssef; Russo, Francesco. On countably skewed Brownian motion with accumulation point. Electron. J. Probab. 20 (2015), paper no. 82, 27 pp. doi:10.1214/EJP.v20-3640. https://projecteuclid.org/euclid.ejp/1465067189

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  • Albeverio, Sergio; Høegh-Krohn, Raphael; Streit, Ludwig. Energy forms, Hamiltonians, and distorted Brownian paths. J. Mathematical Phys. 18 (1977), no. 5, 907–917.
  • Bass, Richard F.; Chen, Zhen-Qing. One-dimensional stochastic differential equations with singular and degenerate coefficients. Sankhyā 67 (2005), no. 1, 19–45.
  • Durrett, Richard. Stochastic calculus. A practical introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. x+341 pp. ISBN: 0-8493-8071-5.
  • Engelbert, H. J.; Schmidt, W. On one-dimensional stochastic differential equations with generalized drift. Stochastic differential systems (Marseille-Luminy, 1984), 143–155, Lecture Notes in Control and Inform. Sci., 69, Springer, Berlin, 1985.
  • Flandoli, Franco; Russo, Francesco; Wolf, Jochen. Some SDEs with distributional drift. I. General calculus. Osaka J. Math. 40 (2003), no. 2, 493–542.
  • Flandoli, Franco; Russo, Francesco; Wolf, Jochen. Some SDEs with distributional drift. II. Lyons-Zheng structure, Itô's formula and semimartingale characterization. Random Oper. Stochastic Equations 12 (2004), no. 2, 145–184.
  • Fukushima, Masatoshi; Ōshima, Yoichi; Takeda, Masayoshi. Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. x+392 pp. ISBN: 3-11-011626-X.
  • Fukushima, M. On a stochastic calculus related to Dirichlet forms and distorted Brownian motions. New stochasitic methods in physics. Phys. Rep. 77 (1981), no. 3, 255–262.
  • Gim, M.; Trutnau, G. Explicit recurrence criteria for symmetric gradient type Dirichlet forms satisfying a Hamza type condition. Math. Rep. (Bucur.) 15(65) (2013), no. 4, 397–410.
  • Harrison, J. M.; Shepp, L. A. On skew Brownian motion. Ann. Probab. 9 (1981), no. 2, 309–313.
  • Itô, Kiyosi; McKean, Henry P., Jr. Diffusion processes and their sample paths. Second printing, corrected. Die Grundlehren der mathematischen Wissenschaften, Band 125. Springer-Verlag, Berlin-New York, 1974. xv+321 pp.
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8.
  • Le Gall, J.-F. Applications du temps local aux équations différentielles stochastiques unidimensionnelles. (French) [Local time applications to one-dimensional stochastic differential equations] Seminar on probability, XVII, 15–31, Lecture Notes in Math., 986, Springer, Berlin, 1983.
  • Le Gall, J.-F. One-dimensional stochastic differential equations involving the local times of the unknown process. Stochastic analysis and applications (Swansea, 1983), 51–82, Lecture Notes in Math., 1095, Springer, Berlin, 1984.
  • Lejay, Antoine. On the constructions of the skew Brownian motion. Probab. Surv. 3 (2006), 413–466.
  • Mandl, Petr. Analytical treatment of one-dimensional Markov processes. Die Grundlehren der mathematischen Wissenschaften, Band 151 Academia Publishing House of the Czechoslovak Academy of Sciences, Prague; Springer-Verlag New York Inc., New York 1968 xx+192 pp.
  • Nakao, Shintaro. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka J. Math. 9 (1972), 513–518.
  • Ōshima, Yoichi. On conservativeness and recurrence criteria for Markov processes. Potential Anal. 1 (1992), no. 2, 115–131.
  • Ouknine, Y. Le "Skew-Brownian motion” et les processus qui en dérivent. [Skew-Brownian motion and associated processes] (French) Teor. Veroyatnost. i Primenen. 35 (1990), no. 1, 173–179; translation in Theory Probab. Appl. 35 (1990), no. 1, 163–169 (1991)
  • Ouknine, Youssef; Rutkowski, Marek. Local times of functions of continuous semimartingales. Stochastic Anal. Appl. 13 (1995), no. 2, 211–231.
  • Ramirez, Jorge M. Multi-skewed Brownian motion and diffusion in layered media. Proc. Amer. Math. Soc. 139 (2011), no. 10, 3739–3752.
  • Ramirez, Jorge M.; Thomann, Enrique A.; Waymire, Edward C.; Haggerty, Roy; Wood, Brian. A generalized Taylor-Aris formula and skew diffusion. Multiscale Model. Simul. 5 (2006), no. 3, 786–801.
  • Revuz, D,. Yor, M.: Continuous Martingales and Brownian Motion, Springer Verlag, 2005.
  • Russo, Francesco; Trutnau, Gerald. About a construction and some analysis of time inhomogeneous diffusions on monotonely moving domains. J. Funct. Anal. 221 (2005), no. 1, 37–82.
  • Russo, Francesco; Trutnau, Gerald. Some parabolic PDEs whose drift is an irregular random noise in space. Ann. Probab. 35 (2007), no. 6, 2213–2262.
  • Rutkowski, Marek. Strong solutions of stochastic differential equations involving local times. Stochastics 22 (1987), no. 3-4, 201–218.
  • Rutkowski, Marek. Stochastic differential equations with singular drift. Statist. Probab. Lett. 10 (1990), no. 3, 225–229.
  • Stannat, Wilhelm. (Nonsymmetric) Dirichlet operators on $L^ 1$: existence, uniqueness and associated Markov processes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 1, 99–140.
  • Sturm, Karl-Theodor. Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^p$-Liouville properties. J. Reine Angew. Math. 456 (1994), 173–196.
  • Takanobu, Satoshi. On the existence of solutions of stochastic differential equations with singular drifts. Probab. Theory Related Fields 74 (1987), no. 2, 295–315.
  • Takanobu, Satoshi. On the uniqueness of solutions of stochastic differential equations with singular drifts. Publ. Res. Inst. Math. Sci. 22 (1986), no. 5, 813–848.
  • Takeda, Masayoshi. On a martingale method for symmetric diffusion processes and its applications. Osaka J. Math. 26 (1989), no. 3, 605–623.
  • Trutnau, Gerald. On a class of non-symmetric diffusions containing fully nonsymmetric distorted Brownian motions. Forum Math. 15 (2003), no. 3, 409–437.
  • Trutnau, Gerald. A short note on Lyons-Zheng decomposition in the non-sectorial case. Proceedings of RIMS Workshop on Stochastic Analysis and Applications, 237–245, RIMS Kôkyûroku Bessatsu, B6, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008.