Abstract
In this paper, we study time changes of Brownian motions by positive continuous additive functionals. Under a certain regularity condition on the associated Revuz measures, we prove that the resolvents of the time-changed Brownian motions are locally Hölder continuous in the spatial components. We also obtain lower bounds for the indices of the Hölder continuity.
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References
R.F. Bass, Probabilistic Techniques in Analysis, Probability and its Applications (New York) (Springer, New York, 1995)
R.F. Bass, Diffusions and Elliptic Operators Probability and Its Applications (New York). (Springer, New York, 1998)
R.F. Bass, M. Kassmann, T. Kumagai, Symmetric jump processes: localization, heat kernels and convergence. Ann. Inst. Henri Poincaré Probab. Stat. 46, 59–71 (2010)
Z.-Q. Chen, M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, London Mathematical Society Monographs Series, vol. 35. (Princeton University Press, Princeton, NJ, 2012)
M. Fukushima, Y. Oshima, M. Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, second revised and extended edition (Walter de Gruyter & Co., Berlin, 2011)
M. Fukushima, T. Uemura, Capacitary bounds of measures and ultracontractivity of time changed processes. J. Math. Pures Appl. 82(9), 553–572 (2003)
C. Garban, R. Rhodes, V. Vargas, Liouville Brownian motion. Ann. Probab. 44, 3076–3110 (2016)
C. Garban, R. Rhodes, V. Vargas, On the heat kernel and the Dirichlet form of Liouville Brownian motion. Electron. J. Probab. 19(96), 25 (2014)
O. Kallenberg, Foundations of Modern Probability, 2nd edn, Probability and Its Applications (Springer, New York, 2002)
T. Lindvall, L.C.G. Rogers, Coupling of multidimensional diffusions by reflection. Ann. Probab. 14, 860–872 (1986)
D.W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, Séminaire de Probabilités, XXII, Lecture Notes in Mathematics, vol. 1321 (Springer, Berlin, 1988), pp. 316–347
Acknowledgements
The author expresses his gratitude to Professor Yuichi Shiozawa for very careful reading of an earlier manuscript. This work was supported by JSPS KAKENHI Grant number 20K22299.
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Matsuura, K. (2022). Hölder Estimates for Resolvents of Time-Changed Brownian Motions. In: Chen, ZQ., Takeda, M., Uemura, T. (eds) Dirichlet Forms and Related Topics. IWDFRT 2022. Springer Proceedings in Mathematics & Statistics, vol 394. Springer, Singapore. https://doi.org/10.1007/978-981-19-4672-1_19
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DOI: https://doi.org/10.1007/978-981-19-4672-1_19
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