L 2 -Theory of Linear Degenerate SPDEs and L p ( p>0 ) Estimates for the Uniform Norm of Weak Solutions
aa r X i v : . [ m a t h . A P ] M a y L -Theory of Linear Degenerate SPDEs and L p ( p > Jinniao Qiu ∗ May 6, 2019
Abstract
In this paper, we are concerned with possibly degenerate stochastic partial differentialequations (SPDEs). An L -theory is introduced, from which we derive a H¨ormander-typetheorem with an analytical approach. With the method of De Giorgi iteration, we obtainthe maximum principle which states the L p ( p >
0) estimates for the time-space uniformnorm of weak solutions.
Mathematics Subject Classification (2010):
Keywords: stochastic partial differential equation, L -theory, H¨ormander theorem, maxi-mum principle, De Giorgi iteration Let (Ω , F , { F t } t ≥ , P ) be a complete filtered probability space, on which a d -dimensionalWiener process W = ( W t ) t ≥ is well defined. We consider SPDE of the form du ( t, x ) = (cid:20)
12 ( L k + M k ) u + b j D j u + cu + f + L ′ k g k + M ′ k h k (cid:21) ( t, x ) dt + h M k u + β k u + h k i ( t, x ) dW kt , ( t, x ) ∈ Q := [0 , T ] × R d ; u (0 , x ) = u ( x ) , x ∈ R d . (1.1)Here and throughout this paper, the summation over repeated indices is enforced unless statedotherwise, T ∈ (0 , ∞ ), D = ( D , . . . , D d ) is the gradient operator, and L k = σ jk D j , M k = θ jk D j , L ′ k = D j ( σ jk · ), M ′ k = D j ( θ jk · ), for k = 1 , . . . , d . SPDE (1.1) is said to be degenerate when itfails to satisfy the super-parabolicity ( SP ): There exists λ ∈ (0 , ∞ ) such that σ ik σ jk ( t, x ) ξ i ξ j ≥ λ | ξ | a.s., ∀ ( t, x, ξ ) ∈ [0 , T ] × R d × R d . We first investigate the solvability of linear, possibly degenerate SPDEs in L -spaces. An L -theory on linear degenerate SPDEs was initiated by Krylov and Rozovskii [20, 18], and itwas developed recently by [2, 10, 16, 21]. Along this line, obtaining a solution of SPDE (1.1) Department of Mathematics & Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB T2N1N4, Canada.
E-mail : [email protected] .The author is partially supported by the National Science and Engineering Research Council of Canada(NSERC) and by the start-up funds from the University of Calgary.
1n space L (Ω; C ([0 , T ]; H m )) not only requires that f + L ′ k g k + M ′ k h k is H m -valued but alsoassumes that h k is H m +1 -valued, while in this work, f, g and h are allowed to be just H m -valued. Moreover, we get the estimate L k u ∈ L (Ω × [0 , T ]; H m ), and under a H¨ormander-typecondition, we further have u ∈ L (Ω × [0 , T ]; H m + η ) for some η ∈ (0 , super-parabolic SPDEs in line with theapplications of pseudo-differential operator theory. As a byproduct, a H¨ormander-type theoremfor SPDE (1.1) is derived from the established L -theory and an estimate on the Lie bracket(Lemma 3.4).Most importantly, we prove the maximum principle for the weak solution of SPDE (1.1).More precisely, we obtain the L p ( p >
0) estimates for the time-space uniform norm of weaksolutions, i.e., under suitable integrability assumptions on u , f, g and h , we have Theorem 1.1.
Let the H¨ormander-type condition ( H ) hold. For the weak solution u of SPDE (1.1) , we have for any p ∈ (0 , ∞ ) E k u ∓ k pL ∞ ( Q ) ≤ C Ξ( u ∓ , f ∓ , g, h ) , where Ξ( u ∓ , f ∓ , g, h ) is expressed in terms of certain norms of ( u , f ∓ , g, h ) , and the constant C depends on d, p, T and the quantities related to the structure coefficients of SPDE (1.1) . The novelty of our result is that it does not require the super-parabolic condition ( SP ),which, to the best of our knowledge, is always assumed in the existing literature on such kindof maximum principles for SPDEs.For the super-parabolic SPDEs, Krylov [14] established the L p -theory ( p ≥ L p estimates of time-spaceuniform norm for the strong solutions that require stronger smoothness assumptions on thecoefficients. For the weak solutions of super-parabolic SPDEs in bounded domains, the maximumprinciple was obtained by Denis, Matoussi and Stoica [7] and further by [3, 6], but with p ∈ [2 , ∞ ). Their method relied on Moser’s iteration. Such method was also used by Denis, Matoussiand Zhang [8] to derive the maximum principle for weak solutions of super-parabolic SPDEswith obstacle. In comparison, we adopt a stochastic version of De Giorgi iteration scheme inthis paper. We would also note that our method is inspired by the other two different versionsof De Giorgi iteration used by Hsu, Wang and Wang [13] to investigate the regularity of strong solutions for super-parabolic SPDEs and by Qiu and Tang [23] to study the maximum principlesof weak solutions for quasilinear backward
SPDEs. For some more recent works on supremumbounds for solutions of SPDEs with iteration methods, we refer to [4, 5, 9, 25]; especially,[5] deals with some classes of degenerate nonlinear SPDEs under strong uniform boundednessassumptions on the external force terms.The remainder of this paper is organized as follows. In Section 2, we set some notations andstate our main result. The L -theory and the H¨ormander-type theorem are addressed in Section3. Finally, we prove the maximum principle in section 4. Let L ( R d ) ( L for short) be the usual Lebesgue integrable space with usual scalar product h· , ·i and norm k · k . For n ∈ ( −∞ , ∞ ), we denote by H n the space of Bessel potentials, that is2 n := (1 − ∆) − n L with the norm k φ k n := k (1 − ∆) n φ k , φ ∈ H n . For each l ∈ N + and domain Π ⊂ R l , denote by C ∞ c (Π) the space of infinitely differentiablefunctions with compact supports in Π. For convenience, we shall use h· , ·i to denote the dualitybetween ( H n ) k and ( H − n ) k ( k ∈ N + , n ∈ R ) as well as that between the Schwartz functionspace D and C ∞ c ( R d ). Moreover, we always omit the superscript associated to the dimensionwhen there is no confusion.For Banach space ( B , k · k B ) and p ∈ [1 , ∞ ], S p ( B ) is the set of all the B -valued, ( F t )-adaptedand continuous processes ( X t ) t ∈ [0 ,T ] such that k X k S p ( B ) := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ [0 ,T ] k X t k B (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) < ∞ . Denote by L p ( B ) the totality of all the B -valued, ( F t )-adapted processes ( X t ) t ∈ [0 ,T ] such that k X k L p ( B ) := kk X t k B k L p (Ω × [0 ,T ]) < ∞ . Obviously, both ( S p ( B ) , k · k S p ( B ) ) and ( L p ( B ) , k · k L p ( B ) ) are Banach spaces. In addition, for p ∈ (0 , L p (Ω; B ) the B -valued F -measurable functions f such that k f k p B ∈ L (Ω; R )with k f k L p (Ω; B ) := (cid:13)(cid:13) k f k p B (cid:13)(cid:13) /pL (Ω; R ) .By C ∞ b , we denote the set of infinitely differentiable functions with bounded derivatives of anyorder. Denote by L ∞ ( C ∞ b ) the set of functions h on Ω × [0 , T ] × R d such that h ( t, x ) is infinitelydifferentiable with respect to x and all the derivatives of any order belong to L ∞ ( L ∞ ( R d )).Throughout this paper, we denote I n = (1 − ∆) n for n ∈ R . Then I n belongs to Ψ n that isthe class of pseudo-differential operators of order n . By the pseudo-differential operator theory(see [12] for instance), the m -th order differential operator belongs to Ψ m for m ∈ N + , themultiplication by elements of C ∞ b lies in Ψ , and for the reader’s convenience, two basic resultsare collected below. Lemma 2.1. (i). If J ∈ Ψ n and J ∈ Ψ n with n , n ∈ R , then J J ∈ Ψ n + n and the Liebracket [ J , J ] := J J − J J ∈ Ψ n + n − .(ii). For m ∈ (0 , ∞ ) , let ζ belong to C mb which is defined as usual. Then for any n ∈ ( − m, m ) there exists constant C such that k ζφ k n ≤ C k ζ k C m k φ k n , ∀ φ ∈ H n . We introduce the definition for solution of SPDE (1.1).
Definition 2.1.
A process u is called a solution to SPDE (1.1) if u ∈ S ( H m ) for some m ∈ R and SPDE (1.1) holds in the distributional sense, i.e., for any ζ ∈ C ∞ c ( R ) ⊗ C ∞ c ( R d ) there holdsalmost surely h ζ ( t ) , u ( t ) i − Z t h ∂ s ζ ( s ) , u ( s ) i ds − Z t h ζ ( s ) , ( M k u + β k u + h k )( s ) i dW ks = h ζ (0) , u i + Z t (cid:28) ζ,
12 ( L k + M k ) u + b j D j u + cu + f + L ′ k g k + M ′ k h k (cid:29) ( s ) ds, ∀ t ∈ [0 , T ] . In particular, if u ∈ S ( L ), it is said to be a weak solution.3et V = { L , . . . , L d } and V n +1 = V n ∪ { [ L k , V ] : V ∈ V n , k = 1 , . . . , d } . Denote by L n the set of linear combinations of elements of V n with coefficients of L ∞ ( C ∞ b ). Weintroduce the following H¨ormander-type condition.( H ) There exists n ∈ N such that { D i : i = 1 , . . . , d } ⊂ L n . (Throughout this paper, n isalways chosen to be the smallest one.) Remark 2.1.
It is obvious that the super-parabolicity ( SP ) corresponds to the trivial case n = 0. A nontrivial example is the 2-dimensional case with d = d = 2: L = D and L = cos ((1 + α t ) x ) D where ( α t ) t ≥ can be any nonnegative bounded F t -adapted process.Then one has D / ∈ L , but { D , D } ⊂ L since [ L , L ] = − (1 + α t ) sin ((1 + α t ) x ) D . Hence,we have n = 1.We also make the following assumptions.( A σ ik , θ ik , b i , β, c ∈ L ∞ ( C ∞ b ) , for i = 1 , . . . , d , k = 1 , . . . , d ; ( A c ≥ , u ∈ L ∞ (Ω × R d ) ∩ ∩ q> L q (Ω , F ; L ) , f, g k , h k ∈ L ( L ) ∩ ∩ q> L q (Ω; L ( Q )) , for k = 1 , . . . , d , and moreover, for some ¯ p > d + 2 η ( f, g, h ) ∈ L ∞ (Ω; L ¯ p ( d +2 η )(¯ p + d +2 η ) η ( Q )) × L ∞ (Ω; L ¯ pη ( Q )) × (cid:18) L ∞ (Ω; L p ( d +2 η )(¯ p + d +2 η ) η ( Q )) ∩ L ∞ (Ω; L ¯ pη ( Q )) (cid:19) , where and in the following, we set η = 2 − n . Throughout this paper, we denote Λ ∓ ¯ p, ∞ = k u ∓ k L ∞ (Ω × R d ) + esssup ω ∈ Ω k f ∓ ( ω, · , · ) k L ¯ p ( d +2 η )(¯ p + d +2 η ) η ( Q ) + esssup ω ∈ Ω k ( g, h )( ω, · , · ) k L ¯ pη ( Q ) + esssup ω ∈ Ω k h ( ω, · , · ) k L p ( d +2 η )(¯ p + d +2 η ) η ( Q ) , Λ ∓ p = k u ∓ k L p (Ω; L ) + k ( f ∓ , g, h ) k L p (Ω; L ( Q )) , p ∈ (0 , ∞ ) . We now state our main results.
Theorem 2.2.
Let assumption ( A hold. Given f ∈ L ( H m ) , g, h ∈ L (( H m ) d ) and u ∈ L (Ω , F ; H m ) with some m ∈ R , the following three assertions hold:(i) SPDE (1.1) admits a unique solution u ∈ S ( H m ) with L k u ∈ L ( H m ) , k = 1 , . . . , d ,and E sup t ∈ [0 ,T ] k u ( t ) k m + d X k =1 E Z T k L k u ( t ) k m dt ≤ C (cid:26) E k u k m + E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m (cid:1) ds (cid:27) , with the constant C depending on T, m, θ, σ, b, c and β . In particular, if condition ( H ) holds, wehave further E Z T k u ( t ) k m + η dt ≤ C (cid:26) E k u k m + E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m (cid:1) ds (cid:27) , ith C depending on T, m, n , θ, σ, b, c and β .(ii) Assume further ( H ) and f ∈ ∩ n ∈ R L ( H n ) , g, h ∈ ∩ n ∈ R L (( H n ) d ) . For any ε ∈ (0 , T ) ,one has u ∈ ∩ n ∈ R L (Ω; C ([ ε, T ]; H n )) , and for each n ∈ R , E sup t ∈ [ ε,T ] k u ( t ) k n + E Z Tε k u ( t ) k n + η dt ≤ C (cid:26) E k u k m + E Z T (cid:0) k f ( s ) k n + k g ( s ) k n + k h ( s ) k n (cid:1) ds (cid:27) , (2.1) with the constant C depending on ε, n, T, m, n , σ, θ, γ, b and c . In particular, the random field u ( t, x ) is almost surely infinitely differentiable with respect to x on (0 , T ] × R d and each derivativeis a continuous function on (0 , T ] × R d .(iii) Let assumption ( A and condition ( H ) hold. For the weak solution u of SPDE (1.1) ,there exists θ ∈ (0 , such that for any p > , E k u ∓ k pL ∞ ( Q ) ≤ C (cid:18) Λ ∓ ¯ p, ∞ + Λ ∓ pθ (cid:19) p , with the constant C depending on d, p, n , T and the quantities related to the coefficients σ, θ, b, c and β . Remark 2.2.
Assertion (i) is a summary of Theorem 3.3 and Corollary 3.5, in which an L -theory is presented for the linear, possibly degenerate SPDEs. Assertion (ii) is from Theorem 3.6,which is a H¨ormander-type theorem. The most important result of this paper is the maximumprinciple of assertion (iii), which corresponds to Theorem 4.1 below and states the L p ( p > degenerate SPDE (1.1)in the whole space. L -theory and H¨ormander-type theorem for SPDEs L -theory of SPDEs We consider the following SPDE du ( t, x ) = (cid:20) δ ∆ u + 12 ( L k + M k ) u + b j D j u + cu + f + L ′ k g k + M ′ k h k (cid:21) ( t, x ) dt + h M k u + β k u + h k i ( t, x ) dW kt , ( t, x ) ∈ Q ; u (0 , x ) = u ( x ) , x ∈ R d , (3.1)with δ ∈ [0 , ∞ ).We first give an a priori estimate for the solution of SPDE (3.1). Proposition 3.1.
Let assumption ( A hold. Assume u ∈ L (Ω , F ; H m ) and f, g k , h k ∈L ( H m ) with m ∈ R , for k = 1 , . . . , d . If u ∈ S ( H m +1 ) ∩ L ( H m +2 ) is a solution of SPDE (3.1) , one has E sup t ∈ [0 ,T ] k u ( t ) k m + E Z T δ k Du ( t ) k m + d X k =1 k L k u ( t ) k m ! dt C (cid:26) E k u k m + E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m (cid:1) ds (cid:27) , (3.2) with C being independent of δ .Proof. We have decompositions L k = L ′ k + c k and M k = M ′ k + α k with c k = − ( D i σ ik ) · and α k = − ( D i θ ik ) · , for k = 1 , . . . , d . Applying Itˆo formula for the square norm (see e.g. [19,Theorem 3.1]), one has almost surely for t ∈ [0 , T ], k I m u ( t ) k + Z t δ k I m Du ( s ) k ds − Z t h I m u ( s ) , I m (( Du ) θ + βu + h )( s ) dW s i = k I m u k + Z t D I m u, I m (cid:16) ( L k + M k ) u + 2 M ′ k h k + 2 L ′ k g k (cid:17)E ( s ) ds + Z t (cid:10) I m u, I m (cid:0) b j D j u + cu + f (cid:1)(cid:11) ( s ) ds + Z t k I m (( Du ) θ + βu + h )( s ) k ds. (3.3)First, basic calculations yield h I m u, I m ( L k u + 2 L ′ k g k + 2 cu + 2 f ) i = h I m u, I m ( L ′ k + c k ) L k u i + 2 h I m u, I m L ′ k g k i + 2 h I m u, I m ( cu + f ) i = −k I m L k u k + h [ I m , L k ] u, I m L k u i + h I m u, [ I m , L ′ k ] L k u + I m c k L k u i− h I m L k u, I m g k i + 2 h [ I m , L k ] u, I m g k i + 2 h I m u, [ I m , L ′ k ] g k i + 2 h I m u, I m ( cu + f ) i≤ − (1 − ε ) k I m L k u k + C ε (cid:16) k I m u k + k I m g k k + k I m f k (cid:17) , ε ∈ (0 , , (3.4)and h I m u, I m ( M k u + 2 M ′ k h k ) i = −k I m M k u k + h [ I m , M k ] u, I m M k u i + h I m u, [ I m , M ′ k ] M k u + I m α k M k u i− h I m M k u, I m h k i + 2 h [ I m , M k ] u, I m h k i + 2 h I m u, [ I m , M ′ k ] h k i≤ −k I m M k u k − h I m M k u, I m h k i + h [ I m , M k ] u, M k I m u i + h I m u, [ I m , M k ] M k u + α k M k I m u i + C (cid:16) k I m u k + k I m h k k (cid:17) ≤ −k I m M k u k − h I m M k u, I m h k i + h I m u, [[ I m , M k ] , M k ] u + α k M k I m u i + C (cid:16) k I m u k + k I m h k k (cid:17) ≤ −k I m M k u k − h I m M k u, I m h k i + C (cid:16) k I m u k + k I m h k k (cid:17) , (3.5)where we have used the relation h I m u, α k M k I m u i = − h I m u, D i ( α k θ ik ) I m u i . (3.6)Notice that for i = 1 , . . . , d , k = 1 , . . . , d , k I m ( h k + β k u + M k u ) k = k I m h k k + 2 h I m h k , I m M k u i + k I m M k u k + 2 h I m ( h k + M k u ) , I m ( β k u ) i + k I m ( β k u ) k , (3.7) h I m u, I m ( b i D i u ) i = − h I m u, D i b i I m u + 2[ b i D i , I m ] u i , I m M k u, I m ( β k u ) i ≤ h M k I m u, β k I m u i + C k I m u k = − h I m u, D i ( β k θ ik ) I m u i + C k I m u k . Putting (3.3), (3.4) and (3.5) together, and taking expectations on both sides of (3.3), one getsby Gronwall inequalitysup t ∈ [0 ,T ] E k u ( t ) k m + E Z T δ k Du ( t ) k m + d X k =1 k L k u ( t ) k m ! dt ≤ C (cid:26) E k u k m + E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m (cid:1) ds (cid:27) . (3.8)On the other hand, one has for each t ∈ [0 , T ), E sup τ ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12) Z τ h I m u ( s ) , I m ( h + βu + ( Du ) θ )( s ) dW s i (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) E d X k =1 Z t (cid:16) |h I m u ( s ) , I m ( h k + β k u )( s ) i| + |h I m u ( s ) , ( M k I m + [ I m , M k ]) u ( s ) i| (cid:17) ds (cid:19) / ≤ C (cid:18) E Z t (cid:0) k I m u ( s ) k k I m h ( s ) k + k I m u ( s ) k (cid:1) ds (cid:19) / ≤ εE sup s ∈ [0 ,t ] k I m u ( s ) k + C ε E Z t (cid:0) k I m h ( s ) k + k I m u ( s ) k (cid:1) ds, ε ∈ (0 , . Together with (3.3), (3.4), (3.5) and (3.8), the above estimate implies (3.2).
Remark 3.1.
The estimate (3.2) plays an important role in our L -theory for SPDEs, for whichsome unusual techniques are applied in the calculations of (3.4), (3.5) and (3.7). Especially, wetreat the term 2 h I m M k u, I m h k i as a unity and it allows us to weaken the assumptions on h inthe L -theory.An immediate consequence of Proposition 3.1 is the following Corollary 3.2.
Let assumption ( A hold. Given u ∈ L (Ω , F ; H m ) and f, g k , h k ∈ L ( H m ) with m ∈ R , for k = 1 , . . . , d , the solution of SPDE (3.1) is unique. Theorem 3.3.
Let assumption ( A hold. Assume u ∈ L (Ω , F ; H m ) and f, g k , h k ∈ L ( H m ) with m ∈ R , for k = 1 , . . . , d . SPDE (3.1) with δ = 0 (equivalently, SPDE (1.1) ) admits aunique solution u ∈ S ( H m ) with L k u ∈ L ( H m ) , k = 1 , . . . , d , and E sup t ∈ [0 ,T ] k u ( t ) k m + d X k =1 E Z T k L k u ( t ) k m dt ≤ C (cid:26) E k u k m + E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m (cid:1) ds (cid:27) , (3.9) with C depending on T, m, σ, θ, b, c and β . roof. Choose { δ l } l ∈ N + ⊂ (0 , { u n } n ∈ N + ⊂ L (Ω , F ; H m +5 ) and { f n , g kn , h kn } n ∈ N + ⊂ L ( H m +5 ),for k = 1 , . . . , d , such that δ l converges down to 0 andlim n →∞ k u n − u k L (Ω; H m ) + k ( f n − f, g n − g, h n − h ) k L ( H m ) = 0 . By L p -theory for SPDEs (see [14] for instance), SPDE (3.1) admits a unique solution u l,n ∈S ( H m +5 ) ∩ L ( H m +6 ) associated with ( δ l , f n , g n , h n , u n ).Fixing n , one deduces from Proposition 3.1 that { ( u l,n , L k u l,n ) } l ∈ N + is bounded in S ( H m +4 ) ×L ( H m +4 ), k = 1 , . . . , d . Observe that δ l ∆ u l,n tends to zero in L ( H m +2 ) as l goes to infin-ity. Therefore, letting l tend to infinity, we derive from Proposition 3.1 and Corollary 3.2the unique solution u n for SPDE (3.1) associated with ( f n , g n , h n , u n ) and δ = 0 such that( u n , L k u n ) ∈ S ( H m +2 ) × L ( H m +2 ), for k = 1 , . . . , d .Furthermore, letting n go to infinity, again by Proposition 3.1 and Corollary 3.2, one obtainsthe unique solution u and associated estimates. This completes the proof.Here, we would note that the above proof is based on methods of strong convergence which isdifferent from the weak convergence developed in [20]. This is basically because of the linearityof the concerned equations and the smoothness assumptions on coefficients in ( A Remark 3.2.
Consider the particular case m = 0 in Theorem 3.3. In view of the approximationsin the above proof, through similar calculations as in the proof of Proposition 3.1, we can getthe following estimate k u ( t ) k − Z t D u ( s ) , ( − D i θ ik u + 2 β k u + 2 h k )( s ) dW ks E ≤ k u k − (1 − ε ) Z t d X k =1 k L k u ( s ) k ds + C ε Z t k u ( s ) k ds + Z t (cid:16) k h ( s ) k + 2 D u ( s ) , ( L ′ k g k + cu + f )( s ) E(cid:17) ds a.s. , ∀ ε ∈ (0 , . (3.10)Assume further c ≥
0. Put u λ = ( u − λ ) + := max { u − λ, } for λ ∈ [0 , ∞ ). If we start from theItˆo formula for the square norm of the positive part of solution (see [24, Corollary 3.11]), in asimilar way to the above estimate, we have k u λ ( t ) k − Z t h u λ ( s ) , ( − D i θ ik u λ + 2 β k u λ + 2 h k )( s ) dW ks i≤ k u λ (0) k − (1 − ε ) Z t d X k =1 k L k u λ ( s ) k ds + C ε Z t (cid:0) k u λ ( s ) k + h| u λ | , λ { u λ > } i ( s ) (cid:1) ds + Z t (cid:16) k h ( s )1 { u λ > } k + 2 D u λ ( s ) , ( L ′ k g k + f )( s ) E(cid:17) ds a.s. , ∀ ε ∈ (0 , . (3.11)where we note that u ≤ u λ + λ { u λ > } .Note that we do not assume the H¨ormander-type condition ( H ) in Theorem 3.3. In fact, wemay get more regularity properties of solutions of SPDE (1.1) under condition ( H ), for whichwe first recall an estimate on the Lie bracket. 8 emma 3.4. ([22, Lemma 4.1]). For { J, L } ⊂ ∪ l ≥ V l , m ∈ R and ε ∈ [0 , , there exists apositive constant C such that almost surely for any φ ∈ H m with J φ ∈ H m − ε and Lφ ∈ H m ,it holds that k [ J, L ] φ k m − ε ≤ C ( k J φ k m − ε + k Lφ k m + k φ k m ) . The above lemma basically generalizes [17, Lemma 4.2] from the deterministic case when m = 0 to the stochastic case for any m ∈ R . Starting from estimate (3.9) of Theorem 3.3 andapplying Lemma 3.4 iteratively to elements of V , . . . , V n , we have Corollary 3.5.
Assume the same hypothesis of Theorem 3.3. Let condition ( H ) hold. For theunique solution u of SPDE (1.1) , one has further u ∈ L ( H m + η ) with E Z T k u ( s ) k m + η ds ≤ C (cid:26) E k u k m + E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m (cid:1) ds (cid:27) , where the constant C depends on T, m, n , σ, θ, b, c and β . The estimate on solution of SPDE (1.1) for the case m = 0 in Corollary 3.5 plays animportant role in Section 4 for the maximum principle of weak solutions. Therefore, for thereader’s convenience, we would provide a sketched proof of Lemma 3.4 from which Corollary 3.5follows immediately. Proof of Lemma 3.4.
Assume first φ ∈ H m +1 . Setting A n = I n − [ J, L ], we have A n ∈ Ψ n almostsurely for each n ∈ R . As the adjoint operator of J and L , J ∗ = − J + ˜ c and L ∗ = − L + ¯ c with˜ c, ¯ c ∈ L ∞ ( C ∞ b ). By Lemma 2.1, we have h J Lφ, I m A m − ε φ i = h Lφ, ( I m J ∗ + [ J ∗ , I m ]) A m − ε φ i = h I m Lφ, ( A m − ε J ∗ + [ J ∗ , A m − ε ]) φ i + h [ I m , J ] Lφ, A m − ε φ i≤ C (cid:0) k Lφ k m + k J φ k m − ε + k φ k m (cid:1) and h LJ φ, I m A m − ε φ i = h J φ, ( I m − ε L ∗ + [ L ∗ , I m − ε ]) A m φ i = h I m − ε J φ, ( A m L ∗ + [ L ∗ , A m ]) φ i + h I m − ε J φ, I − ( m − ε ) [ L ∗ , I m − ε ] A m φ i≤ C (cid:0) k J φ k m − ε + k Lφ k m + k φ k m (cid:1) . Hence, k [ J, L ] φ k m − ε = h [ J, L ] φ, I m A m − ε φ i ≤ C ( k J φ k m − ε + k Lφ k m + k φ k m ) . Through standard density arguments, one verifies that the above estimate also holds for any φ ∈ H m with J φ ∈ H m − ε and Lφ ∈ H m . 9 .2 H¨ormander-type theorem for SPDEs Inspired by the filtering theory of partially observable diffusion processes, Krylov [16, 15] hasjust obtained the H¨ormander-type theorem for SPDEs, which states the spatial smoothness ofsolutions. The method therein relies on the generalized Itˆo-Wentzell formula and associatedresults on deterministic PDEs. Next to the above established L -theory, we intend to derive thefollowing H¨ormander-type theorem for SPDE (1.1) under the condition ( H ) with an analyticalapproach. Theorem 3.6.
Let assumptions ( H ) and ( A hold. If f ∈ ∩ n ∈ R L ( H n ) , g, h ∈ ∩ n ∈ R L (( H n ) d ) ,and u ∈ L (Ω; H m ) for some m ∈ R , then for the unique solution u of SPDE (1.1) in Theorem3.3, one has for any ε ∈ (0 , T ) , u ∈ ∩ n ∈ R L (Ω; C ([ ε, T ]; H n )) , and for any n ∈ R , E sup t ∈ [ ε,T ] k u ( t ) k n + E Z Tε k u ( t ) k n + η dt ≤ C (cid:26) E k u k m + E Z T (cid:0) k f ( s ) k n + k g ( s ) k n + k h ( s ) k n (cid:1) ds (cid:27) , (3.12) with the constant C depending on ε, n, T, m, n , σ, θ, γ, b and c . In particular, the random field u ( t, x ) is almost surely infinitely differentiable with respect to x on (0 , T ] × R d and each derivativeis a continuous function on (0 , T ] × R d .Proof. By Theorem 3.3, SPDE (1.1) admits a unique solution u ∈ S ( H m ) and the random field¯ u ( t, x ) := tu ( t, x ) is the unique solution of SPDE d ¯ u ( t, x ) = (cid:20)
12 ( L k + M k )¯ u + b j D j ¯ u + c ¯ u + u + t (cid:16) f + L ′ k g k + M ′ k h k (cid:17)(cid:21) ( t, x ) dt + h tM k ¯ u + tβ k ¯ u + th k i ( t, x ) dW kt , ( t, x ) ∈ Q ;¯ u (0 , x ) = 0 , x ∈ R d , (3.13)with E sup t ∈ [0 ,T ] k ¯ u ( t ) k m + d X k =1 E Z T k L k ¯ u ( t ) k m dt ≤ C (cid:0) T + 1 (cid:1) E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m + k u ( s ) k m (cid:1) ds. Starting from the above estimate, we apply Lemma 3.4 iteratively to elements of V , . . . , V n .Under condition ( H ), there arrives the estimate Z T k D ¯ u k m − η ds ≤ C (cid:0) T + 1 (cid:1) E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m + k u ( s ) k m (cid:1) ds. (3.14)10ix any ε ∈ (0 , T ∧
1) and define ε l = P li =1 ε i for l ∈ N + . By interpolation and Theorem 3.3,we have E sup t ∈ [ ε ,T ] k u ( t ) k m + E Z Tε k u ( t ) k m +2 − n dt ≤ C T + 1) ε E Z T (cid:0) k f ( s ) k m + k g ( s ) k m + k h ( s ) k m + k u ( s ) k m (cid:1) ds. Since f ∈ ∩ n ∈ R L ( H n ) and g, h ∈ ∩ n ∈ R L (( H n ) d ), by iteration we obtain for any j ∈ N + , E sup t ∈ [ ε j ,T ] k u ( t ) k m +( j − η + E Z Tε j k u ( t ) k m + jη dt ≤ C j ( T + 1) ε E Z Tε j − (cid:16) k f ( s ) k m +( j − η + k g ( s ) k m +( j − η + k h ( s ) k m +( j − η + k u ( s ) k m +( j − η (cid:17) ds, which together with estimate (3.9), implies by iteration that E sup t ∈ [ ε j ,T ] k u ( t ) k m +( j − η + E Z Tε j k u ( t ) k m + jη dt ≤ C ( j ) (cid:26) E k u k m + E Z T (cid:16) k f ( s ) k m +( j − η + k g ( s ) k m +( j − η + k h ( s ) k m +( j − η (cid:17) ds (cid:27) . Hence, for any ε ∈ (0 , T ), one has u ∈ ∩ n ∈ R L (Ω; C ([ ε, T ]; H n )) and the estimate (3.12) holds.In particular, by Sobolev embedding theorem, u ( t, x ) is almost surely infinitely differentiablewith respect to x and each derivative is a continuous function on (0 , T ] × R d . Remark 3.3.
By Theorem 3.6, we have the global spatial smoothness of the solution in timeinterval (0 , T ]. A similar result exists in Krylov’s recent work [15, 16], which states a local spatialsmoothness of solution under a H¨ormander-type condition of local type; roughly speaking, asclaimed in [15], if a H¨ormander-type condition and all the assumptions on coefficients just holdon a measurable subset Ω × ( t , t ) × B ⊂ Ω × [0 , ∞ ) × R d where Ω ∈ F , and B is a ball in R d , then any solution u ( ω, t, x ) satisfying the concerned SPDE on Ω × ( t , t ) × B admits aversion that is, for almost all ( ω, t ) ∈ Ω × ( t , t ), infinitely differentiable with respect to x on B . However, the method therein relies on the generalized Itˆo-Wentzell formula and associatedresults on deterministic PDEs, while herein, we use an analytical approach on the basis of our L -theory and an estimate on the Lie bracket (Lemma 3.4). In fact, our method has the potentialto derive the associated local results, but we would not seek such a generality in the presentpaper. In addition, we would mention that, to the best of our knowledge, the hypoellipticity forSPDEs was first considered by Chaleyat-Maurel and Michel [1], where the coefficients dependon ( t, ω ) only through a substituted Wiener process. L p estimates for the uniform norm of solutions In this section, let assumptions ( A A
2) and ( H ) hold. By Theorem 3.3, SPDE (1.1) has aunique weak solutoin. In this section, we shall prove the L p -estimates for the time-space uniformnorm of the weak solution. 11 heorem 4.1. For the weak solution u of SPDE (1.1) , there exists θ ∈ (0 , such that for any p ∈ (0 , ∞ ) , E k u ∓ k pL ∞ ( Q ) ≤ C (cid:18) Λ ∓ ¯ p, ∞ + Λ ∓ pθ (cid:19) p , with the constant C depending on d, p, n , T and the quantities related to the coefficients σ, θ, b, c and β . An immediate consequence is the following comparison principle.
Corollary 4.2.
Suppose that random field u is the weak solution of SPDE (1.1) . Let ˜ u be thesolution of SPDE (1.1) with the initial value u and external force f being replaced by ˜ u and ˜ f respectively. Suppose further that f ≤ ˜ f , P ⊗ dt ⊗ dx -a.e. and u ≤ ˜ u , P ⊗ dx -a.e.Then, there holds u ≤ ˜ u , P ⊗ dt ⊗ dx -a.e. Before proving Theorem 4.1, we give the following embedding lemma that will be usedfrequently in what follows.
Lemma 4.3.
For ψ ∈ L (0 , T ; H η ) ∩ C ([0 , T ]; L ) , one has ψ ∈ L d +2 η ) d ( Q ) and k ψ k L d +2 η ) d ( Q ) ≤ k ψ k dd +2 η L (0 ,T ; H η ) k ψ k ηd +2 η C ([0 ,T ]; L ) (4.1) with the positive constant C depending on d and η .Proof. By the fractional Gagliard-Nirenberg inequality (see [11, Corollary 2.3] for instance), wehave k ψ ( s, · ) k qL q ≤ C k ψ ( s, · ) k αqη k ψ ( s, · ) k q (1 − α ) , a.e. s ∈ [0 , T ] , where α = d/ ( d + 2 η ) and q = 2( d + 2 η ) /d . Integrating on [0 , T ], we obtain Z Q | ψ ( s, x ) | q dxds ≤ C k ψ k η max s ∈ [0 ,T ] k ψ ( s, · ) k (1 − α ) q . Therefore, ψ ∈ L d +2 η ) d ( Q ) and there holds (4.1).For λ > z ∈ N , set u z = ( u − λ (1 − − z )) + and U z = sup t ∈ [0 ,T ] k u z ( t ) k + Z T k u z ( t ) k η + d X k =1 k L k u z ( t ) k ! dt. Obviously, for each z ∈ N + , one has | D i u z − | ≥ | D i u z | for i = 1 , . . . , d , u z − ≥ u z , u { u z > } = u z + λ (1 − − z )1 { u z > } and 1 { u z > } ≤ (cid:18) z u z − λ (cid:19) q , ∀ q > . (4.2)As an immediate consequence of Lemma 4.3, there follows12 orollary 4.4. k u z k L d +2 η ) d ( Q ) ≤ C U z , a.s. with the constant C depending on d and η . In view of Remark 3.2, the weak solution u of SPDE (1.1) satisfies k u z ( t ) k − Z t h u z ( s ) , ( − D i θ ik u z + 2 β k u z + 2 h k )( s ) dW ks i≤ k u z (0) k − (1 − ε ) Z t d X k =1 k L k u z ( s ) k ds + C ε Z t (cid:0) k u z ( s ) k + (cid:10) | u z | , λ (1 − − z )1 { u z > } (cid:11) ( s ) (cid:1) ds + Z t (cid:16) k h ( s )1 { u z > } k + 2 D u z ( s ) , ( L ′ k g k + f )( s ) E(cid:17) ds, a.s., ∀ ε ∈ (0 , . (4.3)Taking ε = 1 /
2, we have by Gronwall inequalitysup s ∈ [0 ,t ] k u z ( s ) k + Z t d X k =1 k L k u z ( s ) k ds ≤ C (cid:26) λ (1 − − z ) Z t h| u z | , { u z > } i ( s ) ds + sup τ ∈ [0 ,t ] Z τ D u z ( s ) , ( − D i θ ik u z + 2 β k u z + 2 h k )( s ) dW ks E + Z t (cid:16) k h ( s )1 { u z > } k + 2 (cid:12)(cid:12)(cid:12)D u z ( s ) , ( L ′ k g k + f + )( s ) E(cid:12)(cid:12)(cid:12)(cid:17) ds + k u z (0) k (cid:27) , a.s.Under condition ( H ), starting from the above estimate and applying Lemma 3.4 iteratively toelements of V , . . . , V n , we getsup s ∈ [0 ,t ] k u z ( s ) k + Z t k u z ( s ) k η + d X k =1 k L k u z ( s ) k ! ds ≤ C (cid:26) λ (1 − − z ) Z t h| u z | , { u z > } i ( s ) ds + sup τ ∈ [0 ,t ] Z τ D u z ( s ) , ( − D i θ ik u z + 2 β k u z + 2 h k )( s ) dW ks E + Z t (cid:16) k h ( s )1 { u z > } k + 2 (cid:12)(cid:12)(cid:12)D u z ( s ) , ( L ′ k g k + f + )( s ) E(cid:12)(cid:12)(cid:12)(cid:17) ds + k u z (0) k (cid:27) , a.s. (4.4)Set M z ( t ) = Z t D u z ( s ) , ( − D i θ ik u z + 2 β k u z + 2 h k )( s ) dW ks E , t ∈ [0 , T ] . The proof of Theorem 4.1 is started from the iteration inequality of the following lemma.
Lemma 4.5.
Assume λ ≥ +¯ p, ∞ > . For the solution of SPDE (1.1) , there exists a positiveconstant N such that for any z ∈ N + , U z ≤ N z λ α ( U z − ) α + N sup t ∈ [0 ,T ] M z ( t ) , a.s. (4.5) where < α := (¯ p − η )( d + 2 η )2¯ pd − . roof. We estimate each item involved in relation (4.4). Since ¯ p > d + 2 η , basic calculationsyield that 2 < α < d +2 η ) d . Then, it holds that λ (1 − − z ) Z T h| u z | , { u z > } i ( s ) ds ≤ λ (1 − − z ) Z T * | u z − | , (cid:18) z u z − λ (cid:19) α + ( s ) ds = (1 − − z )2 (1+2 α ) z λ α k u z − k α L α ( Q ) ≤ (1 − − z )2 (1+2 α ) z λ α k u z − k (2+2 α ) εL d +2 η ) d ( Q ) k u z − k (2+2 α )(1 − ε ) L ( Q ) ≤ C (1 − − z )2 (1+2 α ) z λ α ( U k − ) α , a.s.where by Lyapunov’s inequality, ε ∈ (0 ,
1) is chosen to satisfy12 + 2 α = dε d + 2 η ) + 1 − ε . Furthermore, we have Z T h u z , f + i ( s ) ds ≤ k u z k L d +2 η ) d ( Q ) k f + k L ¯ p ( d +2 η )(¯ p + d +2 η ) η ( Q ) (cid:18)Z Q { u z > } dxds (cid:19) − η ¯ p ≤ k u z k L d +2 η ) d ( Q ) k f + k L ¯ p ( d +2 η )(¯ p + d +2 η ) η ( Q ) Z Q (cid:12)(cid:12)(cid:12)(cid:12) z u z − λ (cid:12)(cid:12)(cid:12)(cid:12) d +2 η ) d dxds − η ¯ p ≤ (cid:18) z λ (cid:19) α k f + k L ¯ p ( d +2 η )(¯ p + d +2 η ) η ( Q ) k u z − k α L d +2 η ) d ( Q ) ≤ C (cid:18) z λ (cid:19) α k f + k L ¯ p ( d +2 η )(¯ p + d +2 η ) η ( Q ) ( U z − ) α , a.s.and Z T |h u z , L ′ k g k i ( s ) | ds = Z T |h L k u z , g k i ( s ) | ds ≤ k L k u z k L ( Q ) (cid:18)Z Q g { u z > } dxds (cid:19) ≤ k L k u z k L ( Q ) k g k L ¯ pη ( Q ) (cid:18)Z Q { u z > } dxds (cid:19) − η ¯ p ≤ (cid:18) z λ (cid:19) α k g k L ¯ pη ( Q ) k L k u z − k L ( Q ) k u z − k α L d +2 η ) d ( Q ) ≤ C (cid:18) z λ (cid:19) α k g k L ¯ pη ( Q ) ( U z − ) α , a.s.14et q = ¯ p ( d +2 η ) η (¯ p + d +2 η ) and ˜ q = qq − . There follows d +2 η ) d ˜ q = 2 + 2 α and thus Z T k h ( s )1 { u z > } k ds ≤ k h k L q ( Q ) (cid:18)Z Q { u z > } dxds (cid:19) q ≤ k h k L q ( Q ) Z Q (cid:18) z u z − λ (cid:19) d +2 η ) d dxds q = (cid:18) z λ (cid:19) α k h k L q ( Q ) k u z − k α L d +2 η ) d ( Q ) ≤ C (2+2 α ) z λ α k h k L q ( Q ) ( U z − ) α , a.s.Since λ ≥ +¯ p, ∞ , it follows that u z (0) ≡ z ∈ N + . Choosing N to be big enough,we have by relation (4.4), U z ≤ N z λ α ( U z − ) α + N sup t ∈ [0 ,T ] M z ( t ) , a.s.Next, let us deal with the martingale part M z ( · ) in the iteration inequality (4.5). We shallprove that M z ( · ) is comparable with ( U z − ) α , and the techniques are generalized from [13]for the superparabolic cases. Lemma 4.6.
Let λ ≥ Λ +¯ p, ∞ . There exists N ∈ (1 , ∞ ) such that for any κ, ζ ∈ (0 , ∞ ) , P ( sup t ∈ [0 ,T ] M z ( t ) ≥ κζ, ( U z − ) α ≤ ζ )! ≤ exp (cid:26) − κ λ α N z (cid:27) , ∀ z ∈ N + . Proof.
First, we have h M z i T = d X k =1 Z T (cid:12)(cid:12)(cid:12)D u z , ( − D i θ ik u z + 2 β k u z + 2 h k )( s ) E(cid:12)(cid:12)(cid:12) ds ≤ C Z T (cid:0) k u z k + k u z k k h { u z > } k (cid:1) ds, a.s.with the constant C being independent of z . On the other hand, we have Z T k u z k k h { u z > } k ds ≤ sup s ∈ [0 ,T ] k u z ( s ) k Z T k h { u z > } k ds ≤ sup s ∈ [0 ,T ] k u z ( s ) k k h k L ¯ pη ( Q ) (cid:18)Z Q { u z > } dxds (cid:19) − η ¯ p k h k L ¯ pη ( Q ) sup s ∈ [0 ,T ] k u z ( s ) k Z Q (cid:12)(cid:12)(cid:12)(cid:12) z u z − λ (cid:12)(cid:12)(cid:12)(cid:12) d +2 η ) d dxds − η ¯ p = (cid:18) z λ (cid:19) α k h k L ¯ pη ( Q ) sup s ∈ [0 ,T ] k u z ( s ) k k u z − k α L d +2 η ) d ( Q ) ≤ C (cid:18) z λ (cid:19) α k h k L ¯ pη ( Q ) ( U z − ) α , a.s.and Z T k u z ( s ) k ds ≤ sup s ∈ [0 ,T ] k u z ( s ) k Z T k u z k ds ≤ sup s ∈ [0 ,T ] k u z ( s ) k Z Q | u z | (cid:12)(cid:12)(cid:12)(cid:12) z u z − λ (cid:12)(cid:12)(cid:12)(cid:12) α dxds ≤ (cid:18) z λ (cid:19) α sup s ∈ [0 ,T ] k u z ( s ) k k u z − k α α ≤ C (cid:18) z λ (cid:19) α ( U z − ) α , a.s.Therefore, there exists N ∈ (1 , ∞ ) such that for any z ∈ N + , h M z i T ≤ C ((cid:18) z λ (cid:19) α + (cid:18) z λ (cid:19) α k h k L ¯ pη ( Q ) ) ( U z − ) α ≤ N z λ α ( U z − ) α , a.s. (4.6)with the constant C being independent of z .In view of relation (4.6), ( U z − ) α ≤ ζ implies that h M z i T ≤ γ := N z ζ λ α . Note that thereexists a Brownian motion B such that M t = B h M i t . Hence, P ( sup t ∈ [0 ,T ] M z ( t ) ≥ κζ, ( U z − ) α ≤ ζ )! ≤ P ( sup t ∈ [0 ,T ] M z ( t ) ≥ κζ, h M z i T ≤ γ )! ≤ P ( sup t ∈ [0 ,γ ] B t ≥ κζ )! (by the reflection principle) = 2 P ( { B γ ≥ κζ } ) ≤ exp (cid:26) − κ ζ γ (cid:27) = exp (cid:26) − κ λ α N z (cid:27) , which completes the proof.Combining the iteration inequality (4.5) and the estimate on martingale part M z ( · ), we shallestimate the tail probability of k u + k L ∞ ( Q ) . Proposition 4.7.
There exist θ ∈ (0 , and λ ∈ (1 , ∞ ) such that for any λ ≥ λ , P (cid:16)n k u + k L ∞ ( Q ) > λ, U ≤ λ θ o(cid:17) ≤ (cid:8) − λ α (cid:9) . (4.7)16 roof. For z ∈ N , set A z = (cid:26) U z ≤ λ θ ν z (cid:27) , with the parameter ν > n k u + k L ∞ ( Q ) > λ, U ≤ λ θ o ⊂ ∪ z ∈ N ( A z ) c ∩ A ⊂ ∪ z ∈ N + ( A z ) c ∩ A z − which implies that P (cid:16)n k u + k L ∞ ( Q ) > λ, U ≤ λ θ o(cid:17) ≤ X z ∈ N + P (( A z ) c ∩ A z − ) . (4.8)In view of Lemma 4.5, the event in ( A z ) c ∩ A z − implies thatsup t ∈ [0 ,T ] M z ( t ) ≥ λ θ N ν z − N z − λ α − θ (1+ α ) ν ( z − α ) = λ θ (1+ α ) ν ( z − α ) (cid:20) ν α z − − α N λ α θ − N z − λ α (cid:21) . Put ζ z = λ θ (1+ α ) ν ( z − α ) and κ z = ν α z − − α N λ α θ − N z − λ α , and take θ = 14 and ν = (2 N + 1) α . There exists λ ∈ (1 , ∞ ) such that for any λ ≥ λ , one has κ z ≥ (2 N + 1) z λ α , ∀ z ∈ N + . By Lemma 4.6, it follows that for any z ∈ N + , P (( A z ) c ∩ A z − ) ≤ P ( sup t ∈ [0 ,T ] M z ( t ) ≥ κ z ζ z , ( U z − ) α ≤ ζ z )! ≤ exp (cid:26) − κ z λ α N z (cid:27) ≤ exp (cid:26) − (2 N + 1) z λ α N z (cid:27) ≤ exp (cid:8) − z λ α (cid:9) ≤ exp (cid:8) − zλ α (cid:9) , which together with relation (4.8) implies estimate (4.7).Finally, equipped with the above estimate on the tail probability, we are now at a positionto prove the L p -estimates for the time-space uniform norm of weak solutions. Proof of Theorem 4.1.
Taking z = 0 in relation (4.4) and applying H¨older inequality, we havefor 0 ≤ τ ≤ T ,sup t ∈ [0 ,τ ] k u + ( t ) k + Z τ k u + ( s ) k η + d X k =1 k L k u + ( s ) k ! ds ≤ C (cid:26) sup t ∈ [0 ,τ ] Z t D u + ( s ) , ( − D i θ ik u + + 2 β k u + + 2 h k )( s ) dW ks E Z τ (cid:16) k h ( s )1 { u> } k + 2 (cid:12)(cid:12)(cid:12)D u + ( s ) , ( L ′ k g k + f + )( s ) E(cid:12)(cid:12)(cid:12)(cid:17) ds + k u +0 k (cid:27) ≤ C (cid:26) sup t ∈ [0 ,τ ] Z t D u + ( s ) , ( − D i θ ik u + + 2 β k u + + 2 h k )( s ) dW ks E + Z τ (cid:0) k h ( s )1 { u> } k + k g ( s )1 { u> } k + k f + ( s )1 { u> } k (cid:1) ds + k u +0 k (cid:27) + 12 Z τ k u + k + d X k =1 k L k u + ( s ) k ! ds, a.s.,which implies thatsup t ∈ [0 ,τ ] k u + ( t ) k + Z τ k u + ( s ) k η + d X k =1 k L k u + ( s ) k ! ds ≤ C ( sup t ∈ [0 ,τ ] ˜ M t + k ( f + , g, h )1 { u> } k L ([0 ,τ ] × R d ) + k u +0 k ) , a.s., (4.9)with ˜ M t := Z t D u + ( s ) , ( − D i θ ik u + + 2 β k u + + 2 h k )( s ) dW ks E , t ∈ [0 , T ] . Observe that for any t ∈ [0 , T ] and q > h ˜ M i q t = d X k =1 Z t (cid:12)(cid:12)(cid:12)D u + ( s ) , ( − D i θ ik u + + 2 β k u + + 2 h k )( s ) E(cid:12)(cid:12)(cid:12) ds ! q ≤ C (cid:18)Z t (cid:0) k u + k + k u + k k h { u> } k (cid:1) ( s ) ds (cid:19) q ≤ (cid:16) ε + Cτ q (cid:17) sup s ∈ [0 ,t ] k u + ( s ) k q + C ε (cid:18)Z t k h { u> } k ds (cid:19) q . Take ε = 14 and τ = T ∧ (cid:18) C (cid:19) q . By relation (4.9) and the Burkholder-Davis-Gundy inequality, we have for q > E sup t ∈ [0 ,τ ] k u + ( t ) k q + E "Z τ k u + ( s ) k η + d X k =1 k L k u + ( s ) k ! ds q ≤ E sup s ∈ [0 ,τ ] k u + ( s ) k q + CE h(cid:16) k ( f + , g, h )1 { u> } k L ([0 ,τ ] × R d ) + k u +0 k (cid:17) q i . Starting from the interval [0 , τ ], within (cid:6) Tτ (cid:7) steps we arrive at E ( U ) q ≤ CE h k ( f + , g, h )1 { u> } k qL ( Q ) + k u +0 k q i . Taking q = p θ in the above inequality, we have by Proposition 4.7, E k u + k pL ∞ ( Q ) p Z ∞ P (cid:0)(cid:8) k u + k L ∞ ( Q ) > λ (cid:9)(cid:1) λ p − dλ ≤ λ p + Z ∞ λ P (cid:16)n U > λ θ o(cid:17) λ p − dλ + Z ∞ λ P (cid:16)n k u + k L ∞ ( Q ) > λ, U ≤ λ θ o(cid:17) λ p − dλ ≤ λ p + 12 θ E | U | p θ + Z ∞ λ (cid:8) − λ α (cid:9) λ p − dλ< ∞ . Hence, in view of Lemmas 4.5 and 4.7, we have by scaling E k u + k pL ∞ ( Q ) ≤ C (cid:18) Λ +¯ p, ∞ + Λ + pθ (cid:19) p , with the constant C depending on d, p, n , T and the quantities related to the coefficients σ, θ, b, c and β . The estimate on u − follows in a similar way. We complete the proof. Remark 4.1.
Theorem 4.1 addresses the L p ( p >
0) estimates for the time-space uniform normof weak solutions for possibly degenerate
SPDE (1.1) in the whole space. It seems to be new,even for the super-parabolic case (that is n = 0 in ( H )), as the existing results on such kind ofestimates for weak solutions of super-parabolic SPDEs are restricted in bounded domains (see[3, 6, 7]) with p ∈ [2 , ∞ ). In fact, our method of De Giorgi iteration in this section is applicableto the local maximum principle for weak solutions of SPDEs in either bounded or unboundeddomains, by using the techniques of cut-off functions (see [23] for instance). On the other hand,in Theorem 4.1 as well as in assertion (i) of Theorem 2.2, we assume ( A
1) which requires thespatial smoothness of coefficients σ, θ, b, c and β ; in fact, such assumption is made for the sakeof simplicity and it can be relaxed in a standard way due to the properties of multipliers in(ii) of Lemma 2.1. However, we would postpone such generalizations in domains with relaxedassumption ( A
1) to a future work.
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