Summary
We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit to certain families of finite trees. We approximate the Dirichlet form of Brownian motion on the continuum tree by adjoining one-dimensional Brownian excursions. We study the local times of the resulting diffusion. Using time-change methods, we find explicit expressions for certain hitting probabilities and the mean occupation density of the process.
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Albeverio, S., Røckner, M.: Classical Dirichlet forms on topological vector spaces-closability and a Cameron-Martin formula. J. Funct. Anal.88, 395–436 (1990)
Aldous, D.J.: The continuum random tree I. Ann. probab.19, 1–28 (1991)
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
Aldous, D.J.: The continuum random tree III. Ann. Probab.21 248–289 (1993)
Barlow, M., Bass, R.: The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. Henri Poincaré.25, 225–257 (1989)
Barlow, M., Bass, R.: Local times for Brownian motion on the Sierpinski carpet. Probab. Theory Relat. Fields.85, 91–104 (1990)
Barlow, M., Bass, R.: Transition densities for Brownian motion on the Sierpinski carpet (1991)
Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpinski Gasket. Probab. Theory Relat. Fields.9, 543–623 (1987)
Fukushima, M.: Dirichlet forms and Markov processes. New York: North-Holland 1980
Hamby, B.M.: Brownian motion on a homogeneous random fractal. Probab. Theory Relat. Fields94, 1–38 (1992)
Lindstrøm, T.: Brownian motion on nested fractals. Memoirs am. Math. Soc.420, (1990)
Marcus, M.B., Pisier, G.: Random Fourier series with applications to harmonic analysis. (Ann. math. Studies, Vol. 101) Princeton: Princeton University Press.
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)
Robinson, D.W.: The Thermodynamic Pressure in Quantum Statistical Mechanics. Berlin Heidelberg New York: Springer, 1971
Sharpe, M. (1988) General Theory of Markov Processes. Academic Press, San Diego.
Silverstein, M.L. (1973) Dirichlet Spaces and Random Time Change. Illinois J. Math.16, pp. 1–72.