Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces

Authors: Siva R. Athreya, Richard F. Bass and Edwin A. Perkins
Journal: Trans. Amer. Math. Soc. 357 (2005), 5001-5029
MSC (2000): Primary 35J15; Secondary 35R15, 47D07, 60J35
Published electronically: March 10, 2005
MathSciNet review: 2165395
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new method for proving the estimate

\begin{displaymath}\left\Vert\frac{\partial^2 u}{\partial x_i \partial x_j} \right\Vert_{C^\alpha}\leq c\Vert f\Vert _{C^\alpha},\end{displaymath}

where $u$ solves the equation $\Delta u-\lambda u=f$. The method can be applied to the Laplacian on $\mathbb{R}^\infty$. It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.

References [Enhancements On Off] (What's this?)

  • [ABGP] S. Athreya, R.F. Bass, M. Gordina and E.A. Perkins, Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type, in preparation.
  • [Ba] R.F. Bass, Probabilistic techniques in analysis. Springer-Verlag, New York, 1995. MR 1329542 (96e:60001)
  • [BP] R.F. Bass and E.A. Perkins, Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains. Trans. Amer. Math. Soc. 355 (2003) 373-405. MR 1928092 (2003m:60144)
  • [CD] P. Cannarsa and G. Da Prato, Infinite-dimensional elliptic equations with Hölder-continuous coefficients. Adv. Differential Equations 1 (1996) 425-452. MR 1401401 (97g:35174)
  • [D] G. Da Prato, Some results on elliptic and parabolic equations in Hilbert spaces. Rend. Mat. Acc. Lincei 7 (1996) 181-199. MR 1454413 (98g:35206)
  • [DZ] G. Da Prato and J. Zabczyk, Second order partial differential equations in Hilbert spaces. Cambridge University Press, Cambridge, 2002. MR 1985790 (2004e:47058)
  • [GT] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Second edition. Springer-Verlag, Berlin, 1983. MR 0737190 (86c:35035)
  • [KX] G. K. Kallianpur and J. Xiong, Stochastic Differential Equations in Infinite Dimensional Spaces. IMS Lecture Notes-Monograph Series, Vol. 26, IMS, Hayward, 1995. MR 1465436 (98h:60001)
  • [L] A. Lunardi, An interpolation method to characterize domains of generators of semigroups. Semigroup Forum 53 (1996) 321-329. MR 1406778 (98a:47040)
  • [M] P.A. Meyer, Probability and Potentials, Blaisdell, Waltham, Mass., 1966. MR 0205288 (34:5119)
  • [RN] F. Riesz, B. Sz.-Nagy, Functional Analysis, Ungar, New York, 1955. MR 0071727 (17:175i)
  • [S] E.M. Stein, Singular integrals and differentiability properties of functions. Princeton, Princeton Univ. Press, 1970. MR 0290095 (44:7280)
  • [W] J.B. Walsh, An introduction to stochastic partial differential equations. Ecole d'été de probabilités de Saint-Flour, XIV--1984, 265-439. Springer-Verlag, Berlin, 1986. MR 0876082 (88a:60002)
  • [Z] L. Zambotti, An analytic approach to existence and uniqueness for martingale problems in infinite dimensions. Probab. Theory Related Fields 118 (2000) 147-168. MR 1790079 (2001h:60116)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35J15, 35R15, 47D07, 60J35

Retrieve articles in all journals with MSC (2000): 35J15, 35R15, 47D07, 60J35

Additional Information

Siva R. Athreya
Affiliation: Indian Statistical Institute, 8th Mile Mysore Road, Bangalore 560059, India

Richard F. Bass
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Edwin A. Perkins
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2

Keywords: Semigroups, Schauder estimates, H\"older spaces, perturbations, resolvents, elliptic operators, Laplacian, Ornstein-Uhlenbeck processes, infinite-dimensional stochastic differential equations
Received by editor(s): October 24, 2003
Received by editor(s) in revised form: February 13, 2004
Published electronically: March 10, 2005
Additional Notes: The first author’s research was supported in part by an NBHM travel grant.
The second author’s research was supported in part by NSF grant DMS0244737.
The third author’s research was supported in part by an NSERC Research Grant
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society