Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains

Authors:
Richard F. Bass and Edwin A. Perkins

Journal:
Trans. Amer. Math. Soc. **355** (2003), 373-405

MSC (2000):
Primary 60H10

DOI:
https://doi.org/10.1090/S0002-9947-02-03120-3

Published electronically:
September 6, 2002

MathSciNet review:
1928092

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the operator \[ \sum _{i,j=1}^d \sqrt {x_ix_j}\gamma _{ij}(x) \frac {\partial ^2}{\partial x_i \partial x_j}+\sum _{i=1}^d b_i(x) \frac {\partial }{\partial x_i}\] acting on functions in $C_b^2(\mathbb {R}^d_+)$. We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on $\gamma _{ij}$ and $b_i$. In contrast to previous work, the $b_i$ need only be nonnegative on the boundary rather than strictly positive, at the expense of the $\gamma _{ij}$ and $b_i$ being Hölder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhan’s perturbation argument, but the underlying function space is now a weighted Hölder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.

- S. R. Athreya, M. T. Barlow, R. F. Bass, and E. A. Perkins, Degenerate stochastic differential equations and super-Markov chains.
*Probab. Theory Related Fields*, to appear. - Richard F. Bass,
*Diffusions and elliptic operators*, Probability and its Applications (New York), Springer-Verlag, New York, 1998. MR**1483890** - Piermarco Cannarsa and Giuseppe Da Prato,
*Infinite-dimensional elliptic equations with Hölder-continuous coefficients*, Adv. Differential Equations**1**(1996), no. 3, 425–452. MR**1401401** - Mihai Gradinaru, Bernard Roynette, Pierre Vallois, and Marc Yor,
*Abel transform and integrals of Bessel local times*, Ann. Inst. H. Poincaré Probab. Statist.**35**(1999), no. 4, 531–572 (English, with English and French summaries). MR**1702241**, DOI https://doi.org/10.1016/S0246-0203%2899%2900105-3 - L. C. G. Rogers and David Williams,
*Diffusions, Markov processes, and martingales. Vol. 2*, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987. Itô calculus. MR**921238** - Elias M. Stein,
*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095** - Daniel W. Stroock and S. R. Srinivasa Varadhan,
*Multidimensional diffusion processes*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR**532498** - Roger Tribe,
*The behavior of superprocesses near extinction*, Ann. Probab.**20**(1992), no. 1, 286–311. MR**1143421**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
60H10

Retrieve articles in all journals with MSC (2000): 60H10

Additional Information

**Richard F. Bass**

Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Email:
bass@math.uconn.edu

**Edwin A. Perkins**

Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

Email:
perkins@math.ubc.ca

Keywords:
Stochastic differential equations,
margingale problem,
elliptic operators,
degenerate operators,
diffusions,
Bessel processes,
superprocesses,
Hölder norms

Received by editor(s):
February 1, 2002

Received by editor(s) in revised form:
June 6, 2002

Published electronically:
September 6, 2002

Additional Notes:
The first author’s research was supported in part by NSF grant DMS9988496

The second author’s research was supported in part by an NSERC grant

Article copyright:
© Copyright 2002
American Mathematical Society