Stability and approximations of symmetric diffusion semigroups and kernels. (English) Zbl 0907.47036
The authors consider diffusion semigroups and estimate the operator norm of the difference of two such semigroups in terms of the norm distance of the diffusion matrices. This allows to conclude uniform convergence of diffusion semigroups from the convergence of the diffusion coefficients.
Reviewer: R.Nagel (Tübingen)
MSC:
| 47D06 | One-parameter semigroups and linear evolution equations |
| 35K57 | Reaction-diffusion equations |
References:
| [1] | Brenner, P.; Thomée, V.; Wahlbin, L. B., Besov Spaces and Applications to Difference Methods for Initial Problems. Besov Spaces and Applications to Difference Methods for Initial Problems, Lecture Notes in Mathematics, 434 (1975), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0294.35002 |
| [2] | Barbatis, G., Spectral stability under\(L^p\)-perturbation of the second-order coefficients, J. Differential Equations, 124, 302-323 (1996) · Zbl 0848.35082 |
| [3] | Barbatis, G., Stability of weighted Laplace-Beltrami operators under\(L^p\)-perturbation of the Riemannian metric, J. Anal. Math., 68, 253-276 (1996) · Zbl 0868.35025 |
| [4] | Davies, E. B., One-Parameter Semigroups (1980), Academic Press: Academic Press London · Zbl 0457.47030 |
| [5] | Davies, E. B., Heat Kernels and Spectral Theory (1989), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0699.35006 |
| [6] | Ethier, S. N.; Kurtz, T., Markov Processes—Characterization and Convergence (1986), Wiley: Wiley New York · Zbl 0592.60049 |
| [7] | Fukushima, M.; Oshima, Y.; Takeda, M., Dirichlet Forms and Symmetric Markov Processes (1994), De Gruyter: De Gruyter Berlin · Zbl 0838.31001 |
| [8] | Forsythe, G. E.; Wasow, W. R., Finite-Difference Methods for Partial Differential Equations (1960), Wiley: Wiley New York · Zbl 0099.11103 |
| [9] | Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0691.35001 |
| [10] | Holden, H.; Hu, Y., Finite difference approximation of pressure equation for fluid flow in a stochastic medium, Comm. Partial Differential Equations, 21, 1367-1388 (1996) · Zbl 0859.76045 |
| [11] | Hu, Y., On the differentiability of functions of an operator, Séminaire de Probabilités XXIX. Séminaire de Probabilités XXIX, Lecture Notes in Mathematics, 1613 (1995), Springer-Verlag: Springer-Verlag Berlin/New York, p. 218-219 · Zbl 0837.47013 |
| [12] | Hu, Y.; Watanabe, S., Donsker delta function and approximation of heat kernel, J. Math. Kyoto Univ., 36, 499-518 (1996) · Zbl 0883.60056 |
| [14] | Kato, T., Perturbation Theory for Linear Operators (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0148.12601 |
| [15] | Kunita, H., Stochastic differential equations and stochastic flows of diffeomorphisms, École d’été de probabilités de Saint-Flour, XII, 1982. École d’été de probabilités de Saint-Flour, XII, 1982, Lecture Notes in Mathematics, 1097 (1984), Springer-Verlag: Springer-Verlag Berlin/New York, p. 143-303 · Zbl 0554.60066 |
| [16] | Kushner, H. J.; Dupuis, P. G., Numerical Methods for Stochastic Control Problems in Continuous Time (1992), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0754.65068 |
| [17] | Lyons, T. J.; Zheng, W. A., A crossing estimate for the canonical process on a Dirichlet space and a tightness result, Colloque Paul Levy sur let Processus Stochastiques. Colloque Paul Levy sur let Processus Stochastiques, Asterisque, 157-158 (1988), p. 249-271 · Zbl 0654.60059 |
| [18] | Lyons, T. J.; Zhang, T. S., Note on convergence of Dirichlet processes, Bull. London Math. Soc., 25, 353-356 (1993) · Zbl 0793.31006 |
| [19] | Meyers, N. G., An\(L^p\)-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Pisa, 17, 189-206 (1963) · Zbl 0127.31904 |
| [20] | Moser, J., On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math., 14, 577-591 (1961) · Zbl 0111.09302 |
| [21] | Nash, J., Continuity of solutions of parabolic and elliptic equations, Am. J. Math., 80, 931-954 (1958) · Zbl 0096.06902 |
| [22] | Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations (1989), Wadsworth: Wadsworth Belmont · Zbl 0681.65064 |
| [23] | Stroock, D. W., Diffusion semigroups corresponding to uniformly elliptic divergence form operator. Diffusion semigroups corresponding to uniformly elliptic divergence form operator, Lecture Notes in Mathematics, 1321 (1988), Springer-Verlag: Springer-Verlag Berlin/New York, p. 316-347 · Zbl 0651.47031 |
| [25] | Takeda, M., Tightness property for symmetric diffusion processes, Proc. J. Acad. Ser. A Math. Sci., 64 (1988) · Zbl 0667.60077 |
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