# zbMATH — the first resource for mathematics

Brownian motion on the continuum tree. (English) Zbl 0822.60069
The author constructs a so-called Brownian motion on the continuum tree, that is a continuous symmetric Markov process with further special properties of hitting times of points. The elegant construction relies on the theory of Dirichlet forms. Typically, a continuous Markov process $$X^ 0$$ is first obtained as the diffusion associated to some regular local Dirichlet form; and then the Brownian motion $$X$$ is defined by a time-change of $$X^ 0$$ based on an adequate additive functional. Moreover, one can establish the existence of a jointly continuous version of the local times of $$X^ 0$$, using recent results of M. B. Marcus and J. Rosen [Ann. Probab. 20, No. 4, 1603-1684 (1992; Zbl 0762.60068)].
Reviewer: J.Bertoin (Paris)

##### MSC:
 60J65 Brownian motion 31C25 Dirichlet forms 60J55 Local time and additive functionals
##### Keywords:
Brownian motion; continuum tree; Dirichlet form; local times
Full Text:
##### References:
 [1] Albeverio, S., R?ckner, M.: Classical Dirichlet forms on topological vector spaces-closability and a Cameron-Martin formula. J. Funct. Anal.88, 395-436 (1990) · Zbl 0737.46036 [2] Aldous, D.J.: The continuum random tree I. Ann. probab.19, 1-28 (1991) · Zbl 0722.60013 [3] Aldous, D.J.: The continuum random tree II: an overview. In: Stochastic Anal. (eds.) Cambridge University Press, Cambridge, New York: Barlow, M.T., Bingham, N.H. 1992 · Zbl 0791.60008 [4] Aldous, D.J.: The continuum random tree III. Ann. Probab.21 248-289 (1993) · Zbl 0791.60009 [5] Barlow, M., Bass, R.: The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. Henri Poincar?.25, 225-257 (1989) · Zbl 0691.60070 [6] Barlow, M., Bass, R.: Local times for Brownian motion on the Sierpinski carpet. Probab. Theory Relat. Fields.85, 91-104 (1990) · Zbl 0685.60076 [7] Barlow, M., Bass, R.: Transition densities for Brownian motion on the Sierpinski carpet (1991) · Zbl 0739.60071 [8] Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpinski Gasket. Probab. Theory Relat. Fields.9, 543-623 (1987) · Zbl 0635.60090 [9] Fukushima, M.: Dirichlet forms and Markov processes. New York: North-Holland 1980 · Zbl 0422.31007 [10] Hamby, B.M.: Brownian motion on a homogeneous random fractal. Probab. Theory Relat. Fields94, 1-38 (1992) · Zbl 0767.60075 [11] Lindstr?m, T.: Brownian motion on nested fractals. Memoirs am. Math. Soc.420, (1990) · Zbl 0688.60065 [12] Marcus, M.B., Pisier, G.: Random Fourier series with applications to harmonic analysis. (Ann. math. Studies, Vol. 101) Princeton: Princeton University Press. · Zbl 0474.43004 [13] Marcus, M.B., Rosen, J.: Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab.20, 1603-1684 (1992) · Zbl 0762.60068 [14] Robinson, D.W.: The Thermodynamic Pressure in Quantum Statistical Mechanics. Berlin Heidelberg New York: Springer, 1971 [15] Sharpe, M. (1988) General Theory of Markov Processes. Academic Press, San Diego. · Zbl 0649.60079 [16] Silverstein, M.L. (1973) Dirichlet Spaces and Random Time Change. Illinois J. Math.16, pp. 1-72. · Zbl 0275.60092
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.