Linear Stochastic Parabolic Equations, Degenerating on the Boundary of a Domain

Sergey Lototsky

doi:10.1214/ejp.v6-97

Abstract

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.

Highlights

Sobolev spaces Hpγ are very convenient to study parabolic equations in all of Rd : ∂u ∂t = aij DiDj u + biDiu + cu + f, (1.1)where summation over the repeated indices is assumed from 1 to d
If the initial condition is in Hpγ+1−2/p and the right hand side is in Hpγ−1, p ≥ 2, u ∈ Hpγ+1 as long as the coefficients are bounded and sufficiently smooth, and the matrix is uniformly positive definite
It was shown in [4] that a similar result holds for the Ito stochastic parabolic equations du =dt +dwk as long as the stochastic right hand side gk is in Hpγ and the matrix (aij − (1/2)σikσjk) is uniformly positive definite

Summary

Introduction

Sobolev spaces Hpγ are very convenient to study parabolic equations in all of Rd :. If the initial condition is in Hpγ+1−2/p and the right hand side is in Hpγ−1, p ≥ 2, u ∈ Hpγ+1 as long as the coefficients are bounded and sufficiently smooth, and the matrix (aij) is uniformly positive definite. It was shown in [4] that a similar result holds for the Ito stochastic parabolic equations du = (aijDiDju + biDiu + cu + f )dt + (σikDiu + νku + gk)dwk (1.2).

Definition of the spaces

Main result

Application to nonlinear filtering of diffusion processes