Abstract
The Brownian reference measure on periodic functions provides a framework for investigating more general circular processes. These include a significant class of periodic diffusions. We illustrate by proposing simple analytic criteria for finiteness and absolute continuity of the intrinsic circular measure associated to drifting Brownian motion. Our approach exploits a property of approximate bridges.
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References
A. Borodin, P. Salminen, Handbook of Brownian Motion – Facts and Formulae (Birkhaüser, Basel, 2002)
S. Cambronero, Some Ornstein-Uhlenbeck potentials for the one-dimensional Schrödinger operator part II: position-dependent drift. Revista de Mathemática: Teoria y Applicationes 9(2), 31–38 (2002)
S. Cambronero, H.P. McKean, The ground state eigenvalue of Hill’s equation with white noise potential. Commun. Pure Appl. Math. 52(10), 1277–1294 (1999)
K. Itô, H.P. McKean, Diffusion Processes and Their Sample Paths, 2nd edn. (Springer, Berlin, 1974)
Th. Jeulin, Application de la théorie du grossissement à l’étude des temps locaux browniens, in Grossissement de filtrations: exemples et applications. Lecture Notes in Mathematics, vol. 1118 (Springer, Berlin, 1985), pp. 197–304
N.H. Kuiper, Tests concerning random points on a circle. Indag. Math. 22, 32–37, 38–47 (1960)
H.P. McKean, K.T. Vaninsky, Statistical mechanics of nonlinear wave equations, in Trends and Perspectives in Applied Mathematics, ed. by L. Sirovich (Springer, Berlin, 1994), pp. 239–264
A. Nahmod, L. Rey-Bellet, S. Sheffield, G. Staffilani, Absolute continuity of Brownian bridges under certain gauge transformations. Math. Res. Lett. 18(5), 875–887 (2011)
J. Quastel, B. Valko, KdV preserves white noise. Commun. Math. Phys. 277(3), 707–714 (2008)
D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)
W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966)
A.K. Zvonkin, A transformation of the phase space of a process that removes the drift. Math. USSR Sb. 22, 129–149 (1974)
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McGill, P. (2018). On Drifting Brownian Motion Made Periodic. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIX. Lecture Notes in Mathematics(), vol 2215. Springer, Cham. https://doi.org/10.1007/978-3-319-92420-5_9
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DOI: https://doi.org/10.1007/978-3-319-92420-5_9
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