Abstract
We establish several comparison theorems for the transition probability density p b (x,t,y) of Brownian motion with drift b, and deduce explicit, sharp lower and upper bounds for p b (x,t,y) in terms of the norms of the vector field b. The main results are obtained through carefully estimating the mixed moments of Bessel processes. All constants are explicit in our lower and upper bounds, which is different from most of the previous estimates, and is important in many applications for example in statistical inferences for diffusion processes.
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Azencott, R.: Grandes déviations et applications. Ecoles d’Eté de Probabilités de Saint-Flour VIII-1978, P. L. Hennequin (ed.), Lecture Notes in Math. 774, Springer, Berlin, pp. 1–176, 1980
Aronson, D.G.: Non-negative solutions of linear parabolic equation. Ann. Sci. Norm. Sup. Pisa 22, 607–694 (1967)
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Math. 92, Cambridge University Press, 1989
Gradinaru, M., Herrmann, S., Roynette, B.: A singular large deviations phenomenon. Ann. Inst. H. Poincaré Probab. Stat. 37(5), 555–580 (2001)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer-Verlag, 1988
Kendall, D.: Pole-seeking Brownian motion and bird navigation. J. Royal Stat. Soc., Series B 36(3), 365–417 (1974)
Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. Translations of Mathematical Monographs, Volume 23, AMS, 1968
Lyons, T.J., Zheng, W.A.: On conditional diffusion processes. Proc. Royal Soc. Edinburgh 115A, 243–255 (1990)
Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Norris, J.R.: Long time behavior of heat flow: global estimates and exact asymptotics. Arch. Rational Mech. Anal. 140, 161–195 (1997)
Norris, J.R., Stroock, D.W.: Estimates on the fundamental solution to heat flows with uniformly elliptic coefficients. Proc. London Math. Soc. 62(3), 373–402 (1991)
Pitman, J., Yor, M.: Bessel processes and infinitely divisible laws. In: ‘‘Stochastic Integrals’’, Lecture Notes in Math. D. Williams (ed.), 851, Springer, 1981
Prakasa Rao, B.L.S.: Statistical Inference for Diffusion Type Processes. Arnold, Oxford University Press Inc., New York, 1998
Qian, Z., Zheng, W.: Sharp bounds for transition probability densities of a class of diffusions. C. R. Acad. Sci. Paris, Ser. I 335, 953–957 (2002)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer-Verlag, 1991
Shiga, T., Watanabe, S.: Bessel diffusions as a one-parameter family of diffusion processes. Z. F. W. 27, 37–46 (1973)
Yor, M.: On square root boundaries for Bessel processes and pole seeking Brownian motion. In: ‘‘Stochastic Analysis and Applications’’, Lecture Notes in Math. 1095, Springer, 1984
Yor, M.: Local Times and Excursions for Brownian Motion: a concise introduction. Leccions en Matemáticas, Número 1, 1996, Universidad Central de Venezuela, 1995
Yor, M.: Some Aspects of Brownian Motion Part I: Some Special Functionals. Birkhäuser Verlag, Basel, Boston, Berlin, 1992
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Research partially supported by N.S.F. Grants DMS-0203823, and by Doctoral Program Fundation of the Ministry of Education of China, Grant No. 20020269015.
Mathematics Subject Classification (2000): Primary: 60H10, 60H30; Secondary: 35K05
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Qian, Z., Russo, F. & Zheng, W. Comparison theorem and estimates for transition probability densities of diffusion processes. Probab. Theory Relat. Fields 127, 388–406 (2003). https://doi.org/10.1007/s00440-003-0291-1
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DOI: https://doi.org/10.1007/s00440-003-0291-1