A QUANTITATIVE INTERNAL UNIQUE CONTINUATION FOR STOCHASTIC PARABOLIC EQUATIONS

. This paper is addressed to a quantitative internal unique continuation property for stochastic parabolic equations, i.e., we show that each of their solutions can be determined by the observation on any nonempty open subset of the whole region in which the equations evolve. The proof is based on a global Carleman estimate.

Let (Ω, F, {F t } t∈[0,T ] , P ) be a complete filtered probability space (satisfying the usual conditions), on which a standard one-dimensional Brownian motion {B(t)} t≥0 is defined, and write F = {F t } t∈[0,T ] . Assume that H is a Fréchet space. Denote L 2 F (0, T ; H) by the Fréchet space consisting of all H-valued F-adapted processes X(·) such that E X(·) Assume that a ij ∈ W 1,∞ (0, T ; W 2,∞ loc (G)), i, j = 1, 2, · · · , n, and for any subset G 1 of G, there exists a constant Λ = Λ(G 1 ) > 0 such that n i,j=1 a ij ξ i ξ j ≥ Λ|ξ| 2 , ∀(t, x, ξ) ≡ (t, x, ξ 1 , ξ 2 , · · · , ξ n ) ∈ (0, T ) × G 1 × R n . (1) For simplicity, we use the symbol y i = y i (x) to stand for ∂y ∂xi , the partial derivative of y in the x i -direction, where x i is the i-th entry of a generic point x = (x 1 , x 2 , · · · , x n ) ∈ R n . In a similar way, we use the notations y ii , y ij , v i , u i , etc. By a · b we denote the scalar product of two vectors a and b in R n .
Consider the following stochastic parabolic equation:

ZHONGQI YIN
Here, a 1 ∈ L ∞ F (0, T ; L ∞ loc (G; R n )), a 2 ∈ L ∞ F (0, T ; L ∞ loc (G)) and a 3 ∈ L ∞ F (0, T ; W 1,∞ loc (G)). We call y ∈ L 2 F (Ω; C([0, T ]; L 2 loc (G))) ∩ L 2 F (0, T ; H 1 loc (G)) a solution to the equation (2) if (1) For any nonempty open set G ⊂⊂ G, The aim of this paper is to derive a quantitative internal unique continuation result for the equation (1.2), i.e., to show that the value of each solution to (1.2) can be determined by an observation on any nonempty open subset of the whole region G. The main result in this work is stated as follows: , there exists a subdomain G ⊂⊂ G with G ⊂⊂ G, such that for any solution y to the equation (2), it holds that for some positive constant C.
The study of unique continuation property for partial differential equations may date back to the classical results due to Holmgren and Carleman at the beginning of the last century. After that, there were many authors working on this topic. Many of the existing results in this respect were successfully applied to control and inverse problems for partial differential equations(see [3,14], etc.). Until now, there are numerous references on the unique continuation for deterministic parabolic equations (see [1,2,5,8,9,10,12,15], etc.). It is interesting and meaningful to extend the study of unique continuation properties from the deterministic case to its stochastic counterpart. However, there are only a very few works addressed to the unique continuation for stochastic partial differential equations. In [4], the authors obtained the boundary unique continuation for stochastic parabolic equations. The unique continuation property for stochastic Schrödinger and stochastic hyperbolic equations were established in [6] and [7], respectively. On the other hand, in [13], the author obtained a unique continuation property for (2) without giving the quantitative estimate (4).
There are two well-known tools to deal with the unique continuation property for deterministic partial differential equations, i.e., Carleman estimate and Holmgrentype uniqueness theorem. In general a stochastic equation does not admit a solution which is analytic in time even if all coefficients of the equation are constants. Thus, it cannot be expected to obtain a Holmgren-type unique continuation result for stochastic equations, except for some very special cases. We employ Carleman estimate as a main tool to prove the desired unique continuation result in the present work.
2. Proof of the main result. In order to prove our main result, Theorem 1.1, we need the following known result, which can be found in [11,Theorem 3.1].
Assume that u is an H 2 (R n )valued continuous semi-martingale. Set θ = e and v = θu. Then for a.e. x ∈ R n and P -a.s. ω ∈ Ω, it holds that Now, we are in a position to prove Theorem 1.1.
Proof of Theorem 1.1. We borrow some idea from [4]. Without loss of generality, we can assume that the boundary ∂G of G is smooth. Otherwise, we can choose a domain G contained in G such that G ⊂⊂ G ⊂⊂ G and the boundary ∂G of G is smooth and then study the unique continuation problem on G . It is well known that there exists a function ψ ∈ C 4 (G) such that Since G ⊂⊂ G, then by the choice of ψ we can fix an integer N large enough such that
Write z = ηy. It is easy to show that Here, f is given by Clearly, f is supported in Q 2 \ Q 3 .
Assume that λ and µ are two parameters with λ > 1 and µ > 1. Let = λα. Hereafter, we use the symbol O(1) to stand for a bounded quantity and O(µ k ) a function with the same order to µ k , where k is a positive integer.
Replace b ij by a ij and u by z in (6) and then integrate (6) over G × (0, T ). Noticing that z is supported in Q 1 , after taking mathematical expectation in both sides, we obtain the following inequality: According to the property of η, both the second and the third terms in the left hand side of (13) vanish. It is easy to verify that Therefore, Furthermore, by In what follows, we analyze the terms in the right hand side of (14) one by one. Recalling the definition of α and ψ, we see that Noticing that Ψ = 2 n i,j=1 a ij ij , we have that After some direct computations, it follows that n i,j=1 Thus, (19) Our next goal is to estimate the term B in (14). By (16) and (18), we find that Clearly, Aa ij ij .
Noticing that z = y in Q 3 and Q 3 ⊂ Q 1 , we see that the above inequality implies that Noticing Q 4 ⊂ Q 3 and by (11), it is easy to show that Further, from the definition of Q 2 and Q 3 , we find that E Q2\Q3 θ 2 (|∇y| 2 + y 2 )dxdt ≤ exp(2λβ 3 )E
Noticing that t 0 ∈ [ √ 2δ, T − √ 2δ], take As a direct result, we have that E