LINEAR STOCHASTIC PARABOLIC EQUATIONS, DEGENERATING ON THE BOUNDARY OF A DOMAIN

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 coe(cid:14)cients 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 (cid:12)ltering of di(cid:11)usion processes is discussed.


Introduction
Sobolev spaces H γ p are very convenient to study parabolic equations in all of R d : where summation over the repeated indices is assumed from 1 to d. Roughly speaking, if the initial condition is in H γ+1−2/p p and the right hand side is in H γ−1 p , p ≥ 2, then u ∈ H γ+1 p as long as the coefficients are bounded and sufficiently smooth, and the matrix (a ij ) is uniformly positive definite. It was shown in [4] that a similar result holds for the Ito stochastic parabolic equations as long as the stochastic right hand side g k is in H γ p and the matrix (a ij − (1/2)σ ik σ jk ) is uniformly positive definite.
The objective of this paper is to show that, if the Sobolev spaces H γ p are replaced with weighted spaces, then an analogous result holds for the Ito stochastic parabolic equations with quadratic degeneracy of the characteristic form. Let ρ = ρ(x) be a smooth function so that ρ(x) ∼ dist(x, ∂G) near the boundary. Consider a linear stochastic parabolic equation du = (ρ 2 a ij D i D j u + ρb i D i u + cu + f )dt + (ρσ ik D i u + ν k u + g k )dw k . (1.3) Equations with operators of the type ρ α ∆, α > 0, have been studied by many authors in deterministic setting [9,13,14], and operators with quadratic degeneracy of the characteristic form (α = 2) always required separate treatment. Therefore, in the stochastic setting, it is also natural to consider these operators separately.
To study equation (1.3), the Sobolev spaces H γ p are replaced with certain weighted Sobolev spaces. These weighted space H γ p,θ (G) were first introduced in [5] to study stochastic parabolic equations on the half-line, and further investigated in subsequent papers by both authors. In the notation H γ p,θ (G) of the space, the indices γ, p have the same meaning as in H γ p , and the index θ determines the boundary behavior of the functions from the space: the larger the value of θ, the faster the functions and their derivatives can blow up near the boundary of G. The advantage of solving equation (1.3) in the space H γ p,θ (G) is that existence and uniqueness results can be obtained for a wide class of linear and quasilinear equations. Unlike many related works in the deterministic setting, these results, for different values of γ and θ, cover both classical and distribution solutions and, for γ > 0, go beyond abstract solvability. Indeed, for γ > 0, embedding theorems show that the solution is a continuous function of x and t and, for sufficiently large p, has almost equal number of classical and generalized derivatives.
Since the spaces H γ p,θ (G) have been used in the analysis of the Dirichlet boundary value problem for nondegenerate parabolic equations [6,5,7], let us recall the main result.
Consider the Dirichlet problem for equation (1.2) in a sufficiently regular domain. Suppose that the coefficients are sufficiently smooth, the matrix (a ij − (1/2)σ ik σ jk ) is uniformly positive definite inside the domain, and f ∈ H γ−1 p,θ (G), g k ∈ H γ p,θ (G), u| t=0 ∈ H γ+1−2/p p,θ (G), p ≥ 2. Then, for certain values of θ, the solution will be in H γ+1 p,θ (G). Note that θ of the solution space is different from the corresponding values for the initial condition and the right hand side, and if θ is too large or too small, then the corresponding solvability result does not hold.
The results are quite different, and completely analogous to the whole space, if we consider degenerate parabolic equations. Namely, for equation (1.3), the solution will be in H γ+1 p,θ (G) as long as the coefficients are sufficiently smooth, the matrix ( with an appropriate choice of r, B, and W . Assuming that the functions ρ, B, r are globally Lipschitz continuous and bounded, the unique solvability of (1.4) implies that, if x 0 is in G, then x t will never reach the boundary of G. This is why it is natural to expect that the solvability results for (1.3) will not involve any conditions about the boundary of G.
The construction and analysis of the spaces H γ p,θ (G) for general domains are in [8]. Section 2 presents a summary of results from [8] and the construction of the necessary stochastic parabolic spaces. The main result of the paper, the statement about solvability of a second-order degenerate semi-linear stochastic parabolic equation, is in Section 3. In Section 4, an application is given to the problem of nonlinear filtering of diffusion processes, when the unobserved process evolves in a bounded region. In this paper, notation D m is used for a generic partial derivative of order m with respect to the spatial variable x = (x 1 , . . . , x d ); D i = ∂/∂x i . Summation over repeated indices is assumed.

Definition of the spaces
Let G ⊂ R d be a domain (open connected set) with non-empty boundary ∂G. Denote by ρ G (x), x ∈ G, the distance from x to ∂G. For n ∈ Z define the subsets G n of G by (2.1) Let {ζ n , n ∈ Z} be a collection of non-negative functions with the following properties: The function ζ n (x) can be constructed by mollifying the characteristic (indicator) function of G n . If G n is an empty set, then the corresponding ζ n is identical zero.

Definition 2.1
Let G be a domain in R d , θ and γ, real numbers, and p ∈ [1, +∞). Take a collection {ζ n , n ∈ Z} as above. Then A detailed analysis of the spaces H γ p,θ (G) is given in [8]. In particular, it is shown that H γ p,θ (G) does not depend on the particular choice of the system {ζ n } and, for p > 1, is a reflexive Banach space.

Remark 2.2
If G is a bounded domain, then summation in (2.4) is carried out over n ≤ n 0 for some integer n 0 depending on the domain. In particular, if G is bounded, then for all x ∈ G and for every m = 0, 1, . . ..
Functions introduced in the above definition do exist; for example, (2.6) The conditions in the above definition imply that ρ is uniformly Lipschitz continuous in R d ; in particular, this is true for the function defined by (2.6). On the other hand, if G is a bounded domain and the boundary ∂G is of class C |γ|+2 , γ ∈ R, then, by Lemma 14.16 in [1], it is possible to choose the function ρ so that ρ ∈ C |γ|+2 (R d ).
For ν ≥ 0, define the space A ν (G) as follows: (2.10) 2. Given γ ∈ R define γ so that γ = 0 for integer γ and γ is any number from (0, 1) as long as |γ| + γ is not an integer for non-integer γ. Then, for every u ∈ H γ p,θ (G) and These and other properties of the spaces H γ p,θ (G) and A γ (G) can be found in [8].
The following Banach spaces were introduced in [4] to study stochastic parabolic equations on R d : 3. H γ p (τ ): the collection of processes from H γ+1 p (τ ) that can be written, in the sense of distributions, as (2.14) For a positive real number T > 0, a stopping time τ ≤ T , a real number δ ∈ (0, 1], and a (Banach space) X -valued process u, we will use the following notation: and if in addition 1/p < α < β < 1/2, then Next, we define the similar spaces on G.

Main result
Take a function ρ from Definition 2.3 and consider the following equation: with initial condition u(0, x) = u 0 (x). Summation over the repeated indices is assumed, and the Ito stochastic differential is used.

Definition 3.1 A process u ∈ H γ p,θ (τ, G) is a solution of (3.1) if and only if the equality
holds in H γ p,θ (τ, G).
Proof. The arguments are very similar to the proof of Theorem 3.2 in [7]. A more detailed description of the method can be found in [3,Sections 6.4,6.5].
Note also that, for every u ∈ H γ p,θ (T, G), S n ((|A, B|)(ζ −n u)) = ζ −n u. (3.13) Assume first that f and g depend only on t and x. Also, with no loss of generality, assume that n ζ 2 n (x) = 1 for all x ∈ G. If (|A, B|)u = (f, g, u 0 ), then it follows from (3.13) that (3.14) where Conversely, for every (f, g) ∈ F γ p,θ (T, G) and every u 0 ∈ U γ p,θ (G), equation (3.14) has a unique solution u ∈ H γ p,θ (T, G) so that u satisfies (3.8) and (3.1) (recall that so far we assume that f, g do not depend on u). Indeed, by inequalities (3.11) and (2.21), a sufficiently high power of the operator u → n ζ −n S n (A n u, B n u, 0) is a contraction in H γ p,θ (T, G), which implies the existence of a unique solution of (3.14). This solution satisfies . for almost all ω ∈ Ω, u is a continuous function of (t, x);

for almost all ω ∈ Ω and all t ∈ [0, T ], u is m times continuously differentiable in
G as a function of x; 3. for almost all ω ∈ Ω and all t ∈ [0, T ], u(t, x) and its spatial partial derivatives of order less than or equal to m vanish on the boundary of G.
Proof. By assumption, u 0 ∈ H 2−2/p p,θ (G) for every θ ∈ R, because compactness of support of u 0 means that the corresponding sum in (2.4) contains finitely many non-zero terms. Consequently, by Theorem 3.2, u ∈ H γ p,θ (G, T ) for all θ ∈ R. It remains to use (2.22) with β sufficiently close to 1/p, and then apply Theorem 2.4(1).

Application to nonlinear filtering of diffusion processes
The classical problem of nonlinear filtering is considered for a pair of diffusion processes with some initial conditions X 0 , Y 0 . It is assumed that W t is a d 1 -dimensional Wiener process on a complete probability space (Ω, F , P ), X t is d-dimensional state process, Nonetheless, for most applications, (4.1) is only an approximation. There are two main reasons for that. First, the actual process X t usually evolves in a bounded region, for example, because of mechanical restrictions, and therefore equations (4.1) are a suitable model only when the process X t is away from the boundary. Second, even if X t evolves in the whole space, the corresponding filtering equations, when solved numerically, are considered in a bounded domain, which effectively restricts the range of X t . Therefore, it seems natural to start with a filtering model in which the state process evolves in a bounded region. The model presented below is just one possible way to address the issue of replacing the whole of R d with a bounded domain. The model can be easily analysed using the theory developed in the previous sections of the paper, but it is certainly not the most general filtering model in a bounded domain.
Let G be a bounded domain and ρ, a scalar function as in Definition 2.3. Consider the following modification of the classical filtering model: with some initial conditions x 0 , y 0 . Since ρ is Lipschitz continuous in R n , by uniqueness of the solution of (4.2), the process x t can never cross the boundary of G. Note that G is any bounded domain. It is shown below (Lemma 4.1) that, if the domain G is sufficiently large, x 0 ∈ G, and the function ρ is chosen in a special way, then (x t , y t ) are close to (X t , Y t ).
For a matrix M denote by M * its transpose. We make the following assumptions. For the discussion of these assumptions see Section 8 in [4].

Assumption 4.3
There exists δ > 0 so that, for all (x, y)

Assumption 4.4
The initial condition (x 0 , y 0 ) is independent of W t , the conditional distribution of x 0 given y 0 has a density Π 0 , and Π 0 is compactly supported in G.
Define the processes x = x K , y = y K according to (4.2) with ρ = ρ K , x K 0 = X 0 I(X 0 ∈ B K ), y K 0 = Y 0 , and assume that E(|X 0 | p + |Y 0 | p ) < ∞ for some p > 0. Then, as K → ∞, sup 0≤t≤T |X t − x K t | + sup 0≤t≤T |Y t − y K t | converges to zero with probability one and in every L q (Ω), q < p.
Introduce the following notations: The filtering problem for (4.2) is to find conditional distribution of x t given the observations {y s , 0 < s ≤ t}. Since the function ρ is zero outside of G, equations (4.2) can be considered in the whole R d . Then, by Theorem 8.1 in [4], the conditional expectation of f (x t ) given {y s , 0 < s ≤ t}, for every bounded measurable function f = f (x), can be written as R d f (x)Π(t, x)dx. By the same theorem, the function Π = Π(t, x) belongs to H 1 p (T ), for very T > 0, and is the solution of a nonlinear equation with initial condition Π 0 , where V k is the kth row of the matrix V andh t =