Abstract
We consider the degenerate elliptic operator acting on \({C^2_b}\) functions on [0,∞)d:
where the a i are continuous functions that are bounded above and below by positive constants, the b i are bounded and measurable, and the \({\alpha_i\in (0,1)}\) . We impose Neumann boundary conditions on the boundary of [0,∞)d. There will not be uniqueness for the submartingale problem corresponding to \({\mathcal{L}}\) . If we consider, however, only those solutions to the submartingale problem for which the process spends 0 time on the boundary, then existence and uniqueness for the submartingale problem for \({\mathcal{L}}\) holds within this class. Our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations
where \({W_t^i}\) are independent Brownian motions and \({L^{X_i}_t}\) is a local time at 0 for X i.
References
Athreya S.R., Barlow M.T., Bass R.F. and Perkins E.A. (2002). Degenerate stochastic differential equations and super-Markov chains. Probab. Theory Relat. Fields 123: 484–520
Bass R.F. (1995). Probabilistic Techniques in Analysis. Springer, Berlin Heidelberg New York
Bass R.F. (1997). Diffusions and Elliptic Operators. Springer, Berlin Heidelberg New York
Bass, R.F., Burdzy, K., Chen, Z.-Q.: Pathwise uniqueness for a degenerate stochastic differential equation, preprint
Bass R.F. and Perkins E.A. (2003). Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains. Trans. Am. Math. Soc. 355: 373–405
Dupuis P. and Ishii H. (1993). SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21: 554–580
Ikeda N. and Watanabe S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd edn. Elsevier North-Holland, Amsterdam
Krylov N.V. (1971). certain estimate from the theory of stochastic integrals. Theor. Probab. Appl. 16: 438–448
Lebedev N.N. (1972). Special Functions and their Applications. Dover, New York
Lions P.-L. and Sznitman A.-S. (1984). Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37: 511–537
Meyer P.A. (1976). Démonstration probabiliste de certaines inégalités de Littlewood-Paley. I. Les inégalités classiques. Séminaire de Probabilités, X. Springer, Berlin Heidelberg New york
Rudin W. (1973). Functional Analysis. McGraw-Hill, New York
Revuz D. and Yor M. (1999). Continuous martingales and Brownian motion, 3rd edn. Springer, Berlin Heidelberg New york
Stroock D.W. and Varadhan S.R.S. (1971). Diffusion processes with boundary conditions. Comm. Pure Appl. Math. 24: 147–225
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by NSF grant DMS-0244737.
Rights and permissions
About this article
Cite this article
Bass, R.F., Lavrentiev, A. The submartingale problem for a class of degenerate elliptic operators. Probab. Theory Relat. Fields 139, 415–449 (2007). https://doi.org/10.1007/s00440-006-0047-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-006-0047-9
Keywords
- Martingale problem
- Stochastic differential equations
- Degenerate elliptic operators
- Speed measure
- Perturbation
- Bessel process
- Littlewood-Paley
Mathematics Subject Classification (2000)
- Primary 60H10
- Secondary 60H30