Open Access
2001 Linear Stochastic Parabolic Equations, Degenerating on the Boundary of a Domain
Sergey Lototsky
Author Affiliations +
Electron. J. Probab. 6: 1-14 (2001). DOI: 10.1214/EJP.v6-97


A class of linear degenerate second-order parabolic equations is considered in arbitrary domains. It is shown that these equations are solvable using special weighted Sobolev spaces in essentially the same way as the non-degenerate equations in $R^d$ are solved using the usual Sobolev spaces. The main advantages of this Sobolev-space approach are less restrictive conditions on the coefficients of the equation and near-optimal space-time regularity of the solution. Unlike previous works on degenerate equations, the results cover both classical and distribution solutions and allow the domain to be bounded or unbounded without any smoothness assumptions about the boundary. An application to nonlinear filtering of diffusion processes is discussed.


Download Citation

Sergey Lototsky. "Linear Stochastic Parabolic Equations, Degenerating on the Boundary of a Domain." Electron. J. Probab. 6 1 - 14, 2001.


Accepted: 17 October 2001; Published: 2001
First available in Project Euclid: 19 April 2016

zbMATH: 1008.60078
MathSciNet: MR1873301
Digital Object Identifier: 10.1214/EJP.v6-97

Primary: 60H15
Secondary: 35R60

Keywords: $L_p$ estimates , Nonlinear filtering , weighted spaces

Vol.6 • 2001
Back to Top