The Annals of Probability

Unbiased shifts of Brownian motion

Günter Last, Peter Mörters, and Hermann Thorisson

Full-text: Open access


Let $B=(B_{t})_{t\in\mathbb{R}}$ be a two-sided standard Brownian motion. An unbiased shift of $B$ is a random time $T$, which is a measurable function of $B$, such that $(B_{T+t}-B_{T})_{t\in\mathbb{R}}$ is a Brownian motion independent of $B_{T}$. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of $B$. For any probability distribution $\nu$ on $\mathbb{R}$ we construct a stopping time $T\ge0$ with the above properties such that $B_{T}$ has distribution $\nu$. We also study moment and minimality properties of unbiased shifts. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on $\mathbb{R}$. Another new result is an analogue for diffuse random measures on $\mathbb{R}$ of the cycle-stationarity characterisation of Palm versions of stationary simple point processes.

Article information

Ann. Probab., Volume 42, Number 2 (2014), 431-463.

First available in Project Euclid: 24 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 60G57: Random measures 60G55: Point processes

Brownian motion local time unbiased shift allocation rule Palm measure random measure Skorokhod embedding


Last, Günter; Mörters, Peter; Thorisson, Hermann. Unbiased shifts of Brownian motion. Ann. Probab. 42 (2014), no. 2, 431--463. doi:10.1214/13-AOP832.

Export citation


  • [1] Barlow, M. T., Perkins, E. A. and Taylor, S. J. (1986). Two uniform intrinsic constructions for the local time of a class of Lévy processes. Illinois J. Math. 30 19–65.
  • [2] Barlow, M. T. and Yor, M. (1981). (Semi-) martingale inequalities and local times. Z. Wahrsch. Verw. Gebiete 55 237–254.
  • [3] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
  • [4] Bertoin, J. and Le Jan, Y. (1992). Representation of measures by balayage from a regular recurrent point. Ann. Probab. 20 538–548.
  • [5] Bretagnolle, J. (1971). Résultats de Kesten sur les processus à accroissements indépendants. In Séminaire de Probabilités, V (Univ. Strasbourg, Année Universitaire 19691970). Lecture Notes in Math. 191 21–36. Springer, Berlin.
  • [6] Cox, A. M. G. and Hobson, D. G. (2006). Skorokhod embeddings, minimality and non-centred target distributions. Probab. Theory Related Fields 135 395–414.
  • [7] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge.
  • [8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II., 2nd ed. Wiley, New York.
  • [9] Geman, D. and Horowitz, J. (1973). Occupation times for smooth stationary processes. Ann. Probab. 1 131–137.
  • [10] Getoor, R. K. and Kesten, H. (1972). Continuity of local times for Markov processes. Compos. Math. 24 277–303.
  • [11] Holroyd, A. E., Pemantle, R., Peres, Y. and Schramm, O. (2009). Poisson matching. Ann. Inst. Henri Poincaré Probab. Stat. 45 266–287.
  • [12] Holroyd, A. E. and Peres, Y. (2005). Extra heads and invariant allocations. Ann. Probab. 33 31–52.
  • [13] Jain, N. C. and Pruitt, W. E. (1975). The other law of the iterated logarithm. Ann. Probab. 3 1046–1049.
  • [14] Kagan, Y. A. and Vere-Jones, D. (1996). Athens Conference on Applied Probability and Time Series Analysis. Vol. I, (C. C. Heyde, Y. V. Prohorov, R. Pyke and S. T. Rachev, eds.). Lecture Notes in Statistics 114. Springer, New York.
  • [15] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [16] Last, G. (2010). Modern random measures: Palm theory and related models. In New Perspectives in Stochastic Geometry (W. Kendall and I. Molchanov, eds.) 77–110. Oxford Univ. Press, Oxford.
  • [17] Last, G. and Thorisson, H. (2009). Invariant transports of stationary random measures and mass-stationarity. Ann. Probab. 37 790–813.
  • [18] Liggett, T. M. (2002). Tagged particle distributions or how to choose a head at random. In In and Out of Equilibrium (Mambucaba, 2000) (V. Sidoravicius, ed.). Progress in Probability 51 133–162. Birkhäuser, Boston, MA.
  • [19] Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman and Co., San Francisco, CA.
  • [20] Mecke, J. (1967). Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. Verw. Gebiete 9 36–58.
  • [21] Monroe, I. (1972). On embedding right continuous martingales in Brownian motion. Ann. Math. Statist. 43 1293–1311.
  • [22] Monroe, I. (1972). Using additive functionals to embed preassigned distributions in symmetric stable processes. Trans. Amer. Math. Soc. 163 131–146.
  • [23] Mörters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Univ. Press, Cambridge.
  • [24] Obłój, J. (2004). The Skorokhod embedding problem and its offspring. Probab. Surv. 1 321–390.
  • [25] Perkins, E. (1981). A global intrinsic characterization of Brownian local time. Ann. Probab. 9 800–817.
  • [26] Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. Oxford Univ. Press, New York.
  • [27] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
  • [28] Spitzer, F. (1960). A Tauberian theorem and its probability interpretation. Trans. Amer. Math. Soc. 94 150–169.
  • [29] Thorisson, H. (1996). Transforming random elements and shifting random fields. Ann. Probab. 24 2057–2064.
  • [30] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.
  • [31] Zähle, U. (1991). Self-similar random measures. III. Self-similar random processes. Math. Nachr. 151 121–148.