On divergence form SPDEs with growing coefficients in W 1 2 spaces without weights
aa r X i v : . [ m a t h . P R ] A ug ON DIVERGENCE FORM SPDES WITH GROWINGCOEFFICIENTS IN W SPACES WITHOUT WEIGHTS
N.V. KRYLOV
Abstract.
We consider divergence form uniformly parabolic SPDEswith bounded and measurable leading coefficients and possibly growinglower-order coefficients in the deterministic part of the equations. Welook for solutions which are summable to the second power with respectto the usual Lebesgue measure along with their first derivatives withrespect to the spatial variable. Introduction
We consider divergence form uniformly parabolic SPDEs with boundedand measurable leading coefficients and possibly growing lower-order coeffi-cients in the deterministic part of the equation. We look for solutions whichare summable to the second power with respect to the usual Lebesgue mea-sure along with their first derivatives with respect to the spatial variable.To the best of our knowledge our results are new even for deterministicPDEs when one deletes all stochastic terms in the results below. If there areno stochastic terms and the coefficients are nonrandom and time indepen-dent, our results allow one to obtain the corresponding results for ellipticdivergence-form equations which also seem to be new. A sample result inthis case is the following. Consider the equation D i (cid:0) a ij ( x ) D j u ( x ) + b i ( x ) u ( x ) (cid:1) + b i ( x ) D i u ( x ) − ( c ( x ) + λ ) u ( x ) = D i f i ( x ) + f ( x ) (1.1)in R d which is the Euclidean space of points x = ( x , ..., x d ). Here and belowthe summation convention is enforced and D i = ∂∂x i . Assume that (1.1) is uniformly elliptic, a ij are bounded, and c ≥
0. Alsoassume that f j ∈ L = L ( R d ), j = 0 , ..., d , andsup | x − y |≤ ( | b ( x ) − b ( y ) | + | b ( x ) − b ( y ) | + | c ( x ) − c ( y ) | ) < ∞ Mathematics Subject Classification.
Key words and phrases.
Stochastic partial differential equations, Sobolev-Hilbertspaces without weights, growing coefficients, divergence type equations.The work was partially supported by NSF grant DMS-0653121. and that the constant λ > u ∈ W = W ( R d ). Notice that theabove condition on b , b , and c allow them to grow linearly as | x | → ∞ .As in [3] one of the main motivations for studying SPDEs with growingfirst-order coefficients is filtering theory for partially observable diffusionprocesses.It is generally believed that introducing weights is the most natural settingfor equations with growing coefficients. When the coefficients grow it is quitenatural to consider the equations in function spaces with weights that wouldrestrict the set of solutions in such a way that all terms in the equation willbe from the same space as the free terms. The present paper seems to bethe first one treating the unique solvability of these equations with growinglower-order coefficients in the usual Sobolev spaces W without weightsand without imposing any special conditions on the relations between thecoefficients or on their derivatives .The theory of SPDEs in Sobolev-Hilbert spaces with weights attractedsome attention in the past. We do not use weights and only mention a fewpapers about stochastic PDEs in L p -spaces with weights in which one canfind further references: [1] (mild solutions, general p ), [3], [8], [9], [10] ( p = 2in the four last articles).Many more papers are devoted to the theory of deterministic PDEs withgrowing coefficients in Sobolev spaces with weights. We cite only a few ofthem sending the reader to the references therein again because neither dowe deal with weights nor use the results of these papers. It is also worthsaying that our results do not generalize the results of the above cited papers.In most of these papers the coefficients are time independent, see [2], [4],[7], [20], [22], part of the result of which are extended in [6] to time-dependentOrnstein-Uhlenbeck operators.It is worth noting that many issues for deterministic divergence-type equa-tions with time independent growing coefficients in L p spaces with arbitrary p ∈ (1 , ∞ ) without weights were also treated previously in the literature.This was done mostly by using the semigroup approach which excludes timedependent coefficients and makes it almost impossible to use the results inthe more or less general filtering theory. We briefly mention only a fewrecent papers sending the reader to them for additional information.In [21] a strongly continuous in L p semigroup is constructed correspond-ing to elliptic operators with measurable leading coefficients and Lipschitzcontinuous drift coefficients. In [23] it is assumed that if, for | x | → ∞ , thedrift coefficients grow, then the zeroth-order coefficient should grow, basi-cally, as the square of the drift. There is also a condition on the divergenceof the drift coefficient. In [24] there is no zeroth-order term and the semi-group is constructed under some assumptions one of which translates intothe monotonicity of ± b ( x ) − Kx , for a constant K , if the leading term isthe Laplacian. In [5] the drift coefficient is assumed to be globally Lipschitzcontinuous if the zeroth-order coefficient is constant. PDES WITH GROWING COEFFICIENTS 3
Some conclusions in the above cited papers are quite similar to ours butthe corresponding assumptions are not as general in what concerns the reg-ularity of the coefficients. However, these papers contain a lot of additionalimportant information not touched upon in the present paper (in particular,it is shown in [21] that the corresponding semigroup is not analytic).The technique, we apply, originated from [18] and [13] and uses specialcut-off functions whose support evolves in time in a manner adapted to thedrift. We do not make any regularity assumptions on the coefficients and arerestricted to only treat equations in W . Similar, techniques could be usedto consider equations in the spaces W p with any p ≥
2. This time one canuse the results of [11] and [14] where some regularity on the coefficients in x variable is needed like, say, the condition that the second order coefficientsbe in VMO uniformly with respect to the time variable (see [14]). However,for the sake of brevity and clarity we concentrate only on p = 2. The mainemphasis here is that we allow the first-order coefficients to grow as | x | → ∞ and still measure the size of the derivatives with respect to Lebesgue measurethus avoiding using weights.It is worth noting that considering divergence form equations in L p -spacesis quite useful in the treatment of filtering problems (see, for instance, [17])especially when the power of summability is taken large and we intend totreat this issue in a subsequent paper.The article is organized as follows. In Section 2 we describe the problem,Section 3 contains the statements of two main results, Theorem 3.1 on anapriori estimate providing, in particular, uniqueness of solutions and Theo-rem 3.2 about the existence of solutions. Theorem 3.1 is proved in Section 5after we prepare the necessary tools in Section 4. Theorem 3.2 is proved inthe last Section 6.As usual when we speak of “a constant” we always mean “a finite con-stant”. 2. Setting of the problem
Let (Ω , F , P ) be a complete probability space with an increasing filtration {F t , t ≥ } of complete with respect to ( F , P ) σ -fields F t ⊂ F . Denoteby P the predictable σ -field in Ω × (0 , ∞ ) associated with {F t } . Let w kt , k = 1 , , ... , be independent one-dimensional Wiener processes with respectto {F t } . Finally, let τ be a stopping time with respect to {F t } .We consider the second-order operator LL t u t ( x ) = D i (cid:0) a ijt ( x ) D j u t ( x )+ b it ( x ) u t ( x ) (cid:1) + b it ( x ) D i u t ( x ) − c t ( x ) u t ( x ) , (2.1)and the first-order operatorsΛ kt u t ( x ) = σ ikt ( x ) D i u t ( x ) + ν kt ( x ) u t ( x )acting on functions u t ( x ) defined on Ω × R d +1+ , where R d +1+ = [0 , ∞ ) × R d ,and given for k = 1 , , ... (the summation convention is enforced throughoutthe article). We set R + = [0 , ∞ ). N.V. KRYLOV
Our main concern is proving the unique solvability of the equation du t = ( L t u t − λu t + D i f it + f t ) dt + (Λ kt u t + g kt ) dw kt , t ≤ τ, (2.2)with an appropriate initial condition at t = 0, where λ > C ∞ = C ∞ ( R d ), L = L ( R d ), and let W = W ( R d ) be theSobolev space of functions u of class L , such that Du ∈ L , where Du isthe gradient of u . Introduce L ( τ ) = L ( | (0 , τ ]] , ¯ P , L ) , W ( τ ) = L ( | (0 , τ ]] , ¯ P , W ) , where ¯ P is the completion of P with respect to the product measure. Re-member that the elements of L ( τ ) need only belong to L on a predictablesubset of | (0 , τ ]] of full measure. For the sake of convenience we will alwaysassume that they are defined everywhere on | (0 , τ ]] at least as generalizedfunctions. Similar situation occurs in the case of W ( τ ). We also use thesame notation L ( τ ) for ℓ -valued functions like g t = ( g kt ). For such a func-tion, naturally, k g k L = k | g | ℓ k L = (cid:13)(cid:13)(cid:0) ∞ X k =1 ( g k ) (cid:1) / k L = (cid:0) ∞ X k =1 Z R d | g k | dx (cid:1) / . The following definition turns out to be useful if the coefficients of L andΛ k are bounded. Definition 2.1.
We introduce the space W ( τ ), which is the space of func-tions u t = u t ( ω, · ) on { ( ω, t ) : 0 ≤ t ≤ τ, t < ∞} with values in the space ofgeneralized functions on R d and having the following properties:(i) We have u ∈ L (Ω , F , L );(ii) We have u ∈ W ( τ );(iii) There exist f i ∈ L ( τ ), i = 0 , ..., d , and g = ( g , g , ... ) ∈ L ( τ ) suchthat for any φ ∈ C ∞ with probability 1 for all t ∈ R + we have( u t ∧ τ , φ ) = ( u , φ ) + ∞ X k =1 Z t I s ≤ τ ( g ks , φ ) dw ks + Z t I s ≤ τ (cid:0) ( f s , φ ) − ( f is , D i φ ) (cid:1) ds. (2.3)In particular, for any φ ∈ C ∞ , the process ( u t ∧ τ , φ ) is F t -adapted and (a.s.)continuous. In case that property (iii) holds, we write du t = ( D i f it + f t ) dt + g kt dw kt , t ≤ τ. It is a standard fact that for g ∈ L ( τ ) and any φ ∈ C ∞ the series in (2.3)converges uniformly on R + in probability.Similarly to this definition we understand equation (2.2) in the generalcase as the requirement that for any φ ∈ C ∞ with probability one the PDES WITH GROWING COEFFICIENTS 5 relation( u t ∧ τ , φ ) = ( u , φ ) + ∞ X k =1 Z t I s ≤ τ ( σ iks D i u s + ν ks u s + g ks , φ ) dw ks + Z t I s ≤ τ (cid:2) ( b is D i u s − ( c s + λ ) u s + f s , φ ) − ( a ijs D j u s + b is u s + f is , D i φ ) (cid:3) ds (2.4)hold for all t ∈ R + .Observe that at this moment it is not clear that the right-hand side makessense. Also notice that, if the coefficients of L and Λ k are bounded, thenany u ∈ W ( τ ) is a solution of (2.2) with appropriate free terms since if(2.3) holds, then (2.2) holds as well with f it − a ijt D j u t − b i u t , i = 1 , ..., d, f t + ( c t + λ ) u t − b it D i u t ,g kt − σ ik D i u t − ν kt u t in place of f it , i = 1 , ..., d , f t , and g kt , respectively.3. Main results
For ρ > B ρ ( x ) = { y ∈ R d : | x − y | < ρ } , B ρ = B ρ (0). Assumption 3.1. (i) The functions a ijt ( x ), b it ( x ), b it ( x ), c t ( x ), σ ikt ( x ), ν kt ( x )are real valued, measurable with respect to F ⊗ B ( R d +1+ ), F t -adapted for any x , and c ≥ K, δ > ξ ∈ R d ( a ij − α ij ) ξ i ξ j ≥ δ | ξ | , | a ij | ≤ δ − , | ν | ℓ ≤ K, where α ij = (1 / σ i · , σ j · ) ℓ . Also, the constant λ > x ∈ R d (and ω ) the function Z B ( | b t ( x + y ) | + | b t ( x + y ) | + c t ( x + y )) dy is locally square integrable on R + = [0 , ∞ ).Notice that the matrix a = ( a ij ) need not be symmetric. Also noticethat in Assumption 3.1 (iii) the ball B can be replaced with any other ballwithout changing the set of admissible coefficients b , b, c .We take some f j , g ∈ L ( τ ) and before we give the definition of solu-tion of (2.2) we remind the reader that, if u ∈ W ( τ ), then owing to theboundedness of ν and σ and the fact that Du, u, g ∈ L ( τ ), the first serieson the right in (2.4) converges uniformly in probability and the series is acontinuous local martingale. Definition 3.1.
By a solution of (2.2) for t ≤ τ with initial condition u ∈ L (Ω , F , L ) we mean a function u ∈ W ( τ ) (not W ( τ )) such that(i) For any φ ∈ C ∞ with probability one the integral with respect to ds in (2.4) is well defined and is finite for all t ∈ R + ; N.V. KRYLOV (ii) For any φ ∈ C ∞ with probability one equation (2.4) holds for all t ∈ R + .For d = 2 define q = d ∨ , and if d = 2 let q be a fixed number such that q >
2. The followingassumption contains a parameter γ ∈ (0 , Assumption 3.2 ( γ ) . There exists a ρ ∈ (0 ,
1] such that, for any ω ∈ Ωand b := ( b , ..., b d ) and b := ( b , ..., b d ) and ( t, x ) ∈ R d +1+ we have ρ − d Z B ρ Z B ρ | b t ( x + y ) − b t ( x + z ) | q dydz ≤ γ,ρ − d Z B ρ Z B ρ | b t ( x + y ) − b t ( x + z ) | q dydz ≤ γ,ρ − d Z B ρ Z B ρ | c t ( x + y ) − c t ( x + z ) | q dydz ≤ γ. Obviously, Assumption 3.2 is satisfied with any γ ∈ (0 ,
1] if b , b , and c are independent of x . Also notice that Assumption 3.2 allows b , b , and c growing linearly in x . Theorem 3.1.
There exist γ = γ ( d, δ, K ) ∈ (0 , ,N = N ( d, δ, K ) , λ = λ ( d, δ, K, ρ ) ≥ such that, if the above assumptions are satisfied and λ ≥ λ and u is asolution of (2.2) with initial condition u and some f j , g ∈ L ( τ ) , then k u √ λ + c k L ( τ ) + k Du k L ( τ ) ≤ N (cid:0) d X i =1 k f i k L ( τ ) + k g k L ( τ ) + λ − k f k L ( τ ) + E k u k L (cid:1) . (3.1)This theorem provides an apriori estimate implying uniqueness of solu-tions u . Observe that the assumption that such a solution exists is quitenontrivial because if b t ( x ) ≡ x , it is not true that b u ∈ L ( τ ) for arbitrary u ∈ W ( τ ).To prove the existence we need stronger assumptions because, generally,under only the above assumptions the term D i ( b it u t ) + b it D i u t cannot be written even locally as D i ˆ f it + ˆ f t with ˆ f j ∈ L ( τ ) if we only knowthat u ∈ W ( τ ) even if b and b are independent of x . We can only prove ourcrucial Lemma 6.5 if such a representation is possible. PDES WITH GROWING COEFFICIENTS 7
Assumption 3.3.
For any
T, R ∈ R + , and ω ∈ Ω we havesup t ≤ T Z B R ( | b t ( x ) | + | b t ( x ) | + c t ( x )) dx < ∞ . Theorem 3.2.
Let the above assumptions be satisfied with γ taken fromTheorem 3.1. Take λ ≥ λ , where λ is defined in Theorem 3.1, and take u ∈ L (Ω , F , L ) . Then there exists a unique solution of (2.2) with initialcondition u .Remark . If the stopping time τ is bounded, then in the above theoremone can take λ = 0. To show this take a large λ > u t with v t e λt . This leads to an equation for v t with theadditional term − λv t dt and the free terms multiplied by e − λt . The existenceof v ∈ W ( τ ) will be then equivalent to the existence of u ∈ W ( τ ) if τ isbounded. 4. A version of the Itˆo-Wentzell formula
Let D be the space of generalized functions on R d . We remind a definitionand a result from [16]. Recall that for any v ∈ D and φ ∈ C ∞ the function( v, φ ( · − x )) is infinitely differentiable with respect to x , so that the sup in(4.1) below is predictable. Definition 4.1.
Denote by D the set of all D -valued functions u (writtenas u t ( x ) in a common abuse of notation) on Ω × R + such that, for any φ ∈ C ∞ := C ∞ ( R d ), the restriction of the function ( u t , φ ) on Ω × (0 , ∞ ) is P -measurable and ( u , φ ) is F -measurable. For p = 1 , D p thesubset of D consisting of u such that, for any φ ∈ C ∞ and T, R ∈ R + , wehave Z T sup | x |≤ R | ( u t , φ ( · − x )) | p dt < ∞ (a.s.) . (4.1)In the same way, considering ℓ -valued distributions g on C ∞ , that is linear ℓ -valued functionals such that ( g, φ ) is continuous as an ℓ -valued functionwith respect to the standard convergence of test functions, we define D ( ℓ )and D ( ℓ ) replacing | · | in (4.1) with p = 2 by | · | ℓ .Observe that if g ∈ D ( l ) then for any φ ∈ C ∞ , and T ∈ R + ∞ X k =1 Z T ( g kt , φ ) dt = Z T | ( g t , φ ) | ℓ dt < ∞ (a.s.) , which, by well known theorems about convergence of series of martingales,implies that the series in (4.3) below converges uniformly on [0 , T ] in prob-ability for any T ∈ R + . Definition 4.2.
Let f, u ∈ D , g ∈ D ( l ). We say that the equality du t ( x ) = f t ( x ) dt + g kt ( x ) dw kt , t ≤ τ, (4.2) N.V. KRYLOV holds in the sense of distributions if f I | (0 ,τ ]] ∈ D , gI | (0 ,τ ]] ∈ D ( l ), and forany φ ∈ C ∞ , with probability one we have for all t ∈ R + ( u t ∧ τ , φ ) = ( u , φ ) + Z t I s ≤ τ ( f s , φ ) ds + ∞ X k =1 Z t I s ≤ τ ( g ks , φ ) dw ks . (4.3)Let x t be an R d -valued stochastic process given by x it = Z t ˆ b is ds + ∞ X k =1 Z t ˆ σ iks dw ks , where ˆ b t = (ˆ b it ) , ˆ σ kt = (ˆ σ ikt ) are predictable R d -valued processes such that forall ω and s, T ∈ R + we have tr ˆ α s < ∞ and Z T ( | ˆ b t | + tr ˆ α t ) dt < ∞ , where ˆ α t = ( ˆ α ijt ) and 2 ˆ α ijt = (ˆ σ i · , ˆ σ j · ) ℓ . Finally, before stating the mainresult of [16] we remind the reader that for a generalized function v , andany φ ∈ C ∞ the function ( v, φ ( · − x )) is infinitely differentiable and for anyderivative operator D of order n with respect to x we have D ( v, φ ( · − x )) = ( − n ( v, ( Dφ )( · − x )) =: ( Dv, φ ( · − x )) =: (( Dv )( · + x ) , φ )(4.4)implying, in particular, that Du ∈ D if u ∈ D . Theorem 4.1.
Let f, u ∈ D , and g ∈ D ( l ) . Introduce v t ( x ) = u t ( x + x t ) and assume that (4.2) holds (in the sense of distributions). Then dv t ( x ) = [ f t ( x + x t ) + ˆ L t v t ( x ) + ( D i g t ( x + x t ) , ˆ σ i · t ) ℓ ] dt + [ g kt ( x + x t ) + D i v t ( x )ˆ σ ikt ] dw kt , t ≤ τ (4.5) (in the sense of distributions), where ˆ L t v t = ˆ α ijt D i D j v t ( x ) + ˆ b it D i v t ( x ) . Inparticular, the terms on the right in (4.5) belong to the right class of func-tions. We remind the reader that the summation convention over the repeatedindices i, j = 1 , ..., d (and k = 1 , , ... ) is enforced throughout the article. Inthe main part of this paper we are going to use Theorem 4.1 only for ˆ σ ≡ Corollary 4.2.
Under the assumptions of Theorem 4.1 for any η ∈ C ∞ wehave d [ u t ( x ) η ( x − x t )] = [ g kt ( x ) η ( x − x t ) − u t ( x )ˆ σ ikt ( D i η )( x − x t )] dw kt +[ f t ( x ) η ( x − x t )+ u t ( x )( ˆ L ∗ t η )( x − x t ) − ( g t ( x ) , ˆ σ i · ( D i η )( x − x t )) ℓ ] dt, t ≤ τ, where ˆ L ∗ t is the formal adjoint to ˆ L t . PDES WITH GROWING COEFFICIENTS 9
Indeed, what we claim is that for any φ ∈ C ∞ with probability one(( u t ∧ τ φ )( · + x t ∧ τ ) , η ) = ( u φ, η )+ Z t I s ≤ τ (cid:0)(cid:2) g ks φ + ˆ σ iks D i ( u s φ ) (cid:3) ( · + x s ) , η (cid:1) dw ks + Z t I s ≤ τ (cid:0)(cid:2) f s φ + ˆ L s ( u t φ ) + (ˆ σ i · s , D i ( g s φ )) ℓ (cid:3) ( · + x s ) , η (cid:1) ds for all t . However, to obtain this result it suffices to write down an obviousequation for u t φ , then use Theorem 4.1 and, finally, use Definition 4.2 tointerpret the result. 5. Proof of Theorem 3.1
Throughout this section we suppose that the assumptions of Theorem 3.1are satisfied and start with analyzing the second integral in (2.4). Recallthat q was introduced before Assumption 3.2. Lemma 5.1.
Let h ∈ L q , v ∈ L , and u ∈ W . Then there exist V j ∈ L , j = 0 , , ..., d , such that hv = D i V i + V , d X j =0 k V j k L ≤ N k h k L q k v k L , where N is independent of h and v . In particular, | ( hv, u ) | ≤ N k h k L q k v k L k u k W . (5.1) Furthermore, if a number ρ > , then for any ball B of radius ρ we have k I B hu k L ≤ N k h k L q (cid:0) ρ − d/q k I B Du k L + ρ − d/q k I B u k L (cid:1) , (5.2) where N is independent of h , u , ρ , and B . Proof. Observe that by H¨older’s inequality for r = 2 q/ (2 + q ) ( ∈ [1 , k hv k L r ≤ k h k L q k v k L . Next we use the classical theory and introduce V ∈ W r (note that r > d = 1 and r = 1 if d = 1) as a unique solution of∆ V − V = hv. We know that for a constant N = N ( d, r ) we have k V k W r ≤ N k hv k L r , k V k W ≤ N k V k W r , where the last inequality follows by embedding theorems (2 − d/r ≥ − d/ V i = D i V , i = 1 , ..., d , V = − V itholds that hv = D i V i + V .To prove the second assertion, first let q >
2. Observe that by H¨older’sinequality k I B hu k L ≤ k h k L q k I B u k L s , where s = 2 q/ ( q − d/s ≥ d/ − k I B u k L s ≤ N ( ρ − d/q k I B Du k L + ρ − d/q k I B u k L (cid:1) and the result follows. In the remaining case q = 2, which happens only if d = 1. In that case the above estimates remain true if we set s = ∞ . Thelemma is proved.Before we extract some consequences from the lemma we take a nonneg-ative ξ ∈ C ∞ ( B ρ ) with unit integral and define¯ b s ( x ) = Z B ρ ξ ( y ) b s ( x − y ) dy, ¯ b s ( x ) = Z B ρ ξ ( y ) b s ( x − y ) dy, ¯ c s ( x ) = Z B ρ ξ ( y ) c s ( x − y ) dy. (5.3)We may assume that | ξ | ≤ N ( d ) ρ − d .One obtains the first two assertions of the following corollary from (5.1)and (5.2) by performing estimates like k I B ρ ( x t ) ( b t − ¯ b t ( x t )) k q L q = Z B ρ ( x t ) | b t − ¯ b t ( x t ) | q dx = Z B ρ ( x t ) (cid:12)(cid:12) Z B ρ ( x t ) [ b t ( x ) − b t ( y )] ξ ( x t − y ) dy (cid:12)(cid:12) q dx ≤ N Z B ρ ( x t ) (cid:12)(cid:12) ρ − d Z B ρ ( x t ) | b t ( x ) − b t ( y ) | dy (cid:12)(cid:12) q dx ≤ N ρ − d Z B ρ ( x t ) Z B ρ ( x t ) | b t ( x ) − b t ( y ) | q dy dx ≤ N γ, (5.4)
Corollary 5.2.
Let u ∈ W ( τ ) , let x s be an R d -valued predictable process,and let η ∈ C ∞ ( B ρ ) . Set η s ( x ) = η ( x − x s ) . Then on | (0 , τ ]] (i) For any v ∈ W ( | b is − ¯ b is ( x s ) | I B ρ ( x s ) | D i u s | , | v | ) ≤ N ( d ) γ /q k I B ρ ( x s ) Du s k L k v k W ; (ii) We have k I B ρ ( x s ) | b s − ¯ b s ( x s ) | u s k L + k I B ρ ( x s ) | c s − ¯ c s ( x s ) | u s k L ≤ N ( d ) γ /q (cid:0) ρ − d/q k I B ρ ( x s ) Du s k L + ρ − d/q k I B ρ ( x s ) u s k L (cid:1) ; (iii) Almost everywhere on | (0 , τ ]] we have ( b is − ¯ b is ( x s )) η s D i u s = D i V is + V s , (5.5) d X j =0 k V js k L ≤ N ( d ) γ /q k I B ρ ( x s ) Du s k L sup B ρ | η | , (5.6) where V js , j = 0 , ..., d , are some predictable L -valued functions on | (0 , τ ]] . PDES WITH GROWING COEFFICIENTS 11
To prove (iii) observe that one can find a predictable set A ⊂ | (0 , τ ]] offull measure such that I A D i u , i = 1 , ..., d , are well defined as L -valuedpredictable functions. Then (5.5) with I A D i u in place of D i u and (5.6)follow from (5.4), the first assertion of Lemma 5.1, and the fact that the way V j are constructed uses bounded hence continuous operators and translatesthe measurability of the data to the measurability of the result. Since weare interested in (5.5) and (5.6) holding only almost everywhere on | (0 , τ ]],there is no actual need for the replacement. Corollary 5.3.
Let u ∈ W ( τ ) . Then for almost any ( ω, s ) the mappings φ → I s ≤ τ ( b is D i u s , φ ) , I s ≤ τ ( b is u s , D i φ ) , I s ≤ τ ( c s u s , φ ) (5.7) are generalized functions on R d . Furthermore, for any T ∈ R + almost surely Z T I s ≤ τ ( | ( b is D i u s , φ ) | + | ( b is u s , D i φ ) | + | ( c s u s , φ ) | ) ds < ∞ , (5.8) so that requirement (i) in Definition 3.1 can be dropped. Proof. By having in mind partitions of unity we convince ourselves thatit suffices to prove that the mappings (5.7) are generalized functions on anyball B of radius ρ and that (5.8) holds if φ ∈ C ∞ ( B ). Let x be the centerof B and set x s ≡ x . Then to prove the first assertion concerning the lasttwo functions in (5.7) it suffices to use the first assertion of Corollary 5.2along with the observation that, say,( b is u s , D i φ ) = (( b is − ¯ b is ( x )) u s , D i φ ) + ¯ b is ( x )( u s , D i φ ) . Similar transformation and Corollary 5.2 (i) prove that the first function in(5.7) is also a generalized function. Assumption 3.1 (iii) and the estimatesfrom Corollary 5.2 also easily imply (5.8) thus finishing the proof of thecorollary.Before we continue with the proof of Theorem 3.1, we notice that, if u ∈ W ( τ ), then as we know (see, for instance, Theorem 2.1 of [15]), thereexists an event Ω ′ of full probability such that u t ∧ τ I Ω ′ is a continuous L -valued F t -adapted process on R + . Substituting, u t ∧ τ I Ω ′ in place of u in ourassumptions and assertions does not change them. Furthermore, replacing τ with τ ∧ n and then sending n to infinity allows us to assume that τ isbounded. Therefore, without losing generality we assume that(H) If we are considering a u ∈ W ( τ ), the process u t ∧ τ is a continuous L -valued F t -adapted process on R + . The stopping time τ is bounded.Now we are ready to prove Theorem 3.1 in a particular case. Lemma 5.4.
Let ν k ≡ and let b i , b i , and c be independent of x . Assumethat u is a solution of (2.2) with some f j , g ∈ L ( τ ) and λ > . Then (3.1) holds with N = N ( d, δ, K ) . Proof. We want to use Theorem 4.1 to get rid of the first order terms.Observe that (2.2) reads as du t = ( σ ikt D i u t + g kt ) dw kt + (cid:0) D i ( a ijt D j u t + [ b it + b it ] u t + f it ) + f t − ( c t + λ ) u t (cid:1) dt, t ≤ τ. (5.9)One can find a predictable set A ⊂ | (0 , τ ]] of full measure such that I A f j , j = 0 , , ..., d , and I A D i u , i = 1 , ..., d , are well defined as L -valued pre-dictable functions satisfying Z ∞ I A (cid:0) d X j =0 k f jt k L + k Du t k L (cid:1) dt < ∞ . Replacing f j and D i u in (5.9) with I A f j and I A D i u , respectively, will notaffect (5.9). Similarly, one can handle the function g and the terms h t = I | (0 ,τ ]] [ b i + b i ] u, I | (0 ,τ ]] cu for which Z T k h t k L dt < ∞ (a.s.)for each T ∈ R d owing to Assumption 3.1 (iii) and the fact that u ∈ W ( τ ).After these replacements all terms in (5.9) will be of class D or D ( ℓ )as appropriate since a and σ are bounded. This allows us to apply Theorem4.1 and for B it = Z t ( b is + b is ) ds, ˆ u t ( x ) = u t ( x − B t )obtain that d ˆ u t = (cid:0) D i (ˆ a ijt D j ˆ u t ) − ( c t + λ )ˆ u t + D i ˆ f it + ˆ f t (cid:1) dt + (cid:0) ˆ σ ikt D i ˆ u t + ˆ g kt (cid:1) dw kt , t ≤ τ, (5.10)where (ˆ a ijt , ˆ σ ikt , ˆ f jt , ˆ g kt )( x ) = ( a ijt , σ ikt , f jt , g kt )( x − B t ) . Obviously, ˆ u is in W ( τ ) and its norm coincides with that of u . Moreover,having in mind that c t is independent of x and is locally (square) integrable,one can find stopping times τ n ↑ τ such that I τ n = τ ↓ ξ τ n ≤ n, ξ t := Z t c s ds ≤ n. Then it follows from from the equation d ( ξ t ˆ u t ) = (cid:0) D i ( ξ t ˆ a ijt D j ˆ u t ) − λξ t ˆ u t + D i ξ t ˆ f it + ξ t ˆ f t (cid:1) dt + (cid:0) ˆ σ ikt ξ t D i ˆ u t + ξ t ˆ g kt (cid:1) dw kt , t ≤ τ n that ξu ∈ W ( τ n ) and hence ξ t ∧ τ n u t ∧ τ n is a continuous L -valued functionand so are u t ∧ τ n and u t ∧ τ .Furthermore, since τ is bounded and u t ∧ τ is a continuous L -valued func-tion and c t is independent of x and is locally square integrable, we have Z τ k c t ˆ u t k L dt = Z τ c t k u t k L dt ≤ sup t ≤ τ k u t k L Z τ c t dt < ∞ (5.11) PDES WITH GROWING COEFFICIENTS 13 and there is a sequence of, perhaps, different from the above stopping times τ n ↑ τ such that for each nE Z τ n k c t ˆ u t k L dt < ∞ . (5.12)Then (5.10) implies that ˆ u ∈ W ( τ n ) for each n . Also observe that if wecan prove (3.1) with τ n in place of τ , then we can let n → ∞ and use themonotone convergence theorem to get (3.1) as is. Therefore, in the restof the proof we assume that (5.12) holds with τ in place of τ n , that is,ˆ u ∈ W ( τ ).The next argument is standard (see, for instance, Lemma 3.3 and Corol-lary 3.2 of [14]). Itˆo’s formula implies that E k u k L + E Z τ Z R d I t dxdt ≥ , (5.13)where I t := 2ˆ u t ( ˆ f t − λ ˆ u t − c t ˆ u t ) − a ijt D j ˆ u t + ˆ f it ) D i ˆ u t + | ˆ σ i · t D i ˆ u t + ˆ g t | ℓ . We use the inequality | ˆ σ i · t D i ˆ u t + ˆ g t | ℓ ≤ (1 + ε ) | ˆ σ i · t D i ˆ u t | ℓ + 2 ε − | ˆ g t | ℓ , ε ∈ (0 , , and Assumption 3.1. Then for ε = ε ( δ ) > I t ≤ − δ | D ˆ u t | − c t + λ )ˆ u t + 2ˆ u t ˆ f t − f it D i ˆ u t + N | ˆ g t | ℓ . Once again using 2ˆ u t ˆ f t ≤ λ ˆ u t + λ − | ˆ f t | and similarly estimating 2 ˆ f it D i ˆ u t we conclude that I t ≤ − ( δ/ | D ˆ u t | − ( c t + λ )ˆ u t + N (cid:0) d X i =1 | ˆ f it | + | ˆ g t | ℓ (cid:1) + N λ − | ˆ f t | . By coming back to (5.13) we obtain k ˆ u p c t + λ k L ( τ ) + k D ˆ u k L ( τ ) ≤ N (cid:0) d X i =1 k ˆ f i k L ( τ ) + k ˆ g k L ( τ ) (cid:1) + N λ − k ˆ f k L ( τ ) + N E k u k L . This is equivalent to (3.1) and the lemma is proved.To proceed further we need a construction. Take ¯ b , ¯ b , and ¯ c from (5.3).From Lemma 4.2 of [13] and Assumption 3.2 it follows that, for h t = ¯ b t , ¯ b t , ¯ c t ,it holds that | D n h t | ≤ κ n , where κ n = κ n ( n, γ, d, ρ ) ≥ D n h t is anyderivative of h t of order n ≥ x . By Corollary 4.3 of [13]we have | h t ( x ) | ≤ K ( t )(1 + | x | ), where for each ω the function K ( t ) = K ( ω, t ) is locally (square) integrable with respect to t on R + . Owing tothese properties the equation x t = x − Z tt (¯ b s + ¯ b s )( x s ) ds, t ≥ t , (5.14) for any ( ω and) ( t , x ) ∈ R d +1+ has a unique solution x t = x t ,x ,t . Obviously,the process x t ,x ,t , t ≥ t , is F t -adapted.Next, for i = 1 , χ ( i ) ( x ) to be the indicator function of B ρ /i andintroduce χ ( i ) t ,x ,t ( x ) = χ ( i ) ( x − x t ,x ,t ) I t ≥ t . Here is a crucial estimate.
Lemma 5.5.
Assume that u is a solution of (2.2) with some f j , g ∈ L ( τ ) .Then for ( t , x ) ∈ R d +1+ and λ > we have k χ (2) t ,x u √ c + λ k L ( τ ) + k χ (2) t ,x Du k L ( τ ) ≤ N (cid:0) d X i =1 k χ (1) t ,x f i k L ( τ ) + k χ (1) t ,x g k L ( τ ) (cid:1) + N λ − k χ (1) t ,x f k L ( τ ) + N E k u t I B ρ ( x ) I t ≤ τ k L + N γ /q k χ (1) t ,x Du k L ( τ ) + N ∗ λ − k χ (1) t ,x Du k L ( τ ) + N ∗ (1 + λ − ) k χ (1) t ,x u k L ( τ ) + N ∗ λ − d X i =1 k χ (1) t ,x f i k L ( τ ) , (5.15) where and below in the proof by N we denote generic constants dependingonly on d, δ , and K and by N ∗ constants depending only on the same objectsand ρ . Proof. Since we are only concerned with the values of u t if t ≤ t ≤ τ , wemay start considering (2.2) on [ t , τ ∨ t ) and then shifting time allows usto assume that t = 0. Obviously, we may also assume that x = 0. Withthis stipulations we will drop the subscripts t , x . Then, we can include theterm ν k u into g k and obtain (5.15) by the triangle inequality if we assumethat this estimate is true in case ν k ≡
0. Thus, without losing generality weassume t = 0 , x = 0 , ν k ≡ . Fix a ζ ∈ C ∞ with support in B ρ and such that ζ = 1 on B ρ / and0 ≤ ζ ≤
1. Set x t = x , ,t ,ˆ b t = ¯ b t ( x t ) , ˆ b t = ¯ b t ( x t ) , ˆ c t = ¯ c t ( x t ) η t ( x ) = ζ ( x − x t ) , v t ( x ) = u t ( x ) η t ( x ) . The most important property of η t is that dη t = (ˆ b it + ˆ b it ) D i η t dt. Also observe for the later that we may assume that χ (2) t ≤ η t ≤ χ (1) t , | Dη t | ≤ N ρ − χ (1) t , (5.16)where χ ( i ) t = χ ( i )0 , ,t and N = N ( d ). PDES WITH GROWING COEFFICIENTS 15
By Corollary 4.2 (also see the argument before (5.10)) we obtain that for t ≤ τ dv t = (cid:2) D i ( η t a ijt D j u t + b it v t ) − ( a ijt D j u t + b it u t ) D i η t + b it η t D i u t − ( c t + λ ) v t + D i ( f it η t ) − f it D i η t + f t η t +(ˆ b it + ˆ b it ) u t D i η t (cid:3) dt + (cid:2) σ ik D i v t − σ ik u t D i η t + g kt η t (cid:3) dw kt . We transform this further by noticing that η t a ijt D j u t = a ijt D j v t − a ijt u t D j η t . To deal with the term b it η t D i u t we use Corollary 5.2 and find the corre-sponding functions V jt . Then simple arithmetics show that dv t = ( σ ik D i v t + ˆ g kt ) dw kt + (cid:2) D i (cid:0) a ijt D j v t + ˆ b it v t (cid:1) − (ˆ c t + λ ) v t + ˆ b it D i v t + D i ˆ f it + ˆ f t (cid:3) dt, whereˆ f t = f t η t − f it D i η t − a ijt ( D j u t ) D i η t + (ˆ b it − b it ) u t D i η t + (ˆ c t − c t ) u t η t + V t , ˆ f it = f it η t − a ijt u t D j η t + ( b it − ˆ b it ) u t η t + V it , i = 1 , .., d, ˆ g kt = − σ ik u t D i η t + g kt η t . It follows by Lemma 5.4 that for λ > k v √ ˆ c + λ k L ( τ ) + k Dv k L ( τ ) ≤ N λ − k ˆ f k L ( τ ) + N (cid:0) d X i =1 k ˆ f i k L ( τ ) + k ˆ g k L ( τ ) + E k v k L (cid:1) . (5.17)Recall that here and below by N we denote generic constants dependingonly on d, δ , and K .Now we start estimating the right-hand side of (5.17). First we deal withˆ f it and ˆ g kt . Recall (5.16) and observe that obviously, if η t ( x ) = 0, then | x − x t | ≤ ρ . Therefore, k ˆ g k L ( τ ) ≤ N ∗ k uχ (1) · k L ( τ ) + N k gχ (1) · k L ( τ ) (5.18)(we remind the reader that by N ∗ we denote generic constants dependingonly on d, δ, K , and ρ ). By Corollary 5.2 k ( b it − ˆ b it ) u t η t k L ≤ N γ /q ( ρ − d/q )0 k χ (1) t Du t k L + ρ − d/q k χ (1) t u t k L ) . (5.19)Here ρ − d/q )0 ≤ q ≥ d . By adding that k a ij uD j η k L ( τ ) ≤ N ∗ k χ (1) · u k L ( τ ) , we derive from (5.6), (5.18), and (5.19) that d X i =1 k ˆ f i k L ( τ ) + k ˆ g k L ( τ ) ≤ N (cid:0) d X i =1 k χ (1) · f i k L ( τ ) + k χ (1) · g k L ( τ ) (cid:1) + N γ /q k χ (1) · Du k L ( τ ) + N ∗ k χ (1) · u k L ( τ ) . (5.20) While estimating ˆ f we use (5.6) again and observe that we can deal with(ˆ b it − b it ) u t D i η t as in (5.19) this time without paying much attention to thedependence of our constants on ρ and obtain that k (ˆ b i − b i ) uD i η k L ( τ ) ≤ N ∗ ( k χ (1) · Du k L ( τ ) + k χ (1) · u k L ( τ ) ) . By estimating also roughly the remaining terms in ˆ f and combining thiswith (5.20) and (5.17), we see that the left-hand side of (5.17) is less thanthe right-hand side of (5.15). However, | χ (2) t Du t | ≤ | η t Du t | ≤ | Dv t | + | u t Dη t | ≤ | Dv t | + N ρ − | u t χ (1) t | and also | χ (2) t u t | ( c t + λ ) ≤ | η t u t | ( c t + λ ) ≤ | v t | (ˆ c t + λ ) + | η t u t | (1 + | c t − ˆ c t | ) . By combining this with the fact that by Corollary 5.2 k (ˆ c i − c ) uη k L ( τ ) ≤ N γ /q k χ (1) · Du k L ( τ ) + N ∗ k χ (1) · u k L ( τ ) )we obtain (5.15). The lemma is proved.Next, from the result giving “local” in space estimates we derive global inspace estimates but for functions having, roughly speaking, small “future”support in the time variable. Lemma 5.6.
Assume that u is a solution of (2.2) with some f j , g ∈ L ( τ ) and assume that u t = 0 if t + κ − ≤ t ≤ τ with κ = κ ( γ, d, ρ ) ≥ introduced before (5.14) and some (nonrandom) t ≥ (nothing is requiredfor those ω for which τ < t + κ − ). Then for λ > and I t := I [ t , ∞ ) k I t u √ c + λ k L ( τ ) + k I t Du k L ( τ ) ≤ N (cid:0) d X i =1 k I t f i k L ( τ ) + k I t g k L ( τ ) (cid:1) + N λ − k I t f k L ( τ ) + N E k u t I t ≤ τ k L + N γ /q k I t Du k L ( τ ) + N ∗ λ − k I t Du k L ( τ ) + N ∗ (1 + λ − ) k I t u k L ( τ ) + N ∗ λ − d X i =1 k I t f i k L ( τ ) , (5.21) where and below in the proof by N we denote generic constants dependingonly on d, δ , and K and by N ∗ constants depending only on the same objectsand ρ . Proof. Take x ∈ R d and use the notation introduced before Lemma 5.5.One knows that for each t ≥ t , the mapping x → x t ,x ,t is a diffeomor-phism with Jacobian determinant given by (cid:12)(cid:12)(cid:12)(cid:12) ∂x t ,x ,t ∂x (cid:12)(cid:12)(cid:12)(cid:12) = exp (cid:0) − Z tt d X i =1 D i [¯ b is + ¯ b is ]( x t ,x ,s ) ds (cid:1) . PDES WITH GROWING COEFFICIENTS 17
By the way the constant κ is introduced, we have e − Nκ ( t − t ) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂x t ,x ,t ∂x (cid:12)(cid:12)(cid:12)(cid:12) ≤ e Nκ ( t − t ) , where N depends only on d . Therefore, for any nonnegative Lebesgue mea-surable function w ( x ) it holds that e − Nκ ( t − t ) Z R d w ( y ) dy ≤ Z R d w ( x t ,x ,t ) dx ≤ e Nκ ( t − t ) Z R d w ( y ) dy. In particular, since Z R d | χ ( i ) t ,x ,t ( x ) | dx = Z R d | χ ( i ) ( x − x t ,x ,t ) | dx , we have e − Nκ ( t − t ) = N ∗ i e − Nκ ( t − t ) Z R d | χ ( i ) ( x − y ) | dy ≤ N ∗ i Z R d | χ ( i ) t ,x ,t ( x ) | dx ≤ N ∗ i e Nκ ( t − t ) Z R d | χ ( i ) ( x − y ) | dy = e Nκ ( t − t ) , where N ∗ i = | B | − ρ − d i d and | B | is the volume of B . It follows that Z R d | χ (1) t ,x ,t ( x ) | dx ≤ ( N ∗ ) − e Nκ ( t − t ) , ( N ∗ ) − e − Nκ ( t − t ) ≤ Z R d | χ (2) t ,x ,t ( x ) | dx . Furthermore, since u t = 0 if τ ≥ t ≥ t + κ − and χ ( i ) t ,x ,t = 0 if t < t , inevaluating the norms in (5.15) we need not integrate with respect to t suchthat κ ( t − t ) ≥
1, so that for all t really involved we have Z R d | χ (1) t ,x ,t ( x ) | dx ≤ ( N ∗ ) − e N , ( N ∗ ) − e − N ≤ Z R d | χ (2) t ,x ,t ( x ) | dx . After this observation it only remains to integrate (5.15) through with re-spect to x and use the fact that N ∗ = 2 − d N ∗ . The lemma is proved. Proof of Theorem 3.1 . First we show how to choose γ = γ ( d, δ, K ) > N the constant factor of γ /q k I t Du k L ( τ ) in (5.21). We know that N = N ( d, δ, K ) and we choose γ ∈ (0 ,
1] so that N γ /q ≤ /
2. Thenunder the conditions of Lemma 5.6 for λ ≥ k I t u √ c + λ k L ( τ ) + k I t Du k L ( τ ) ≤ N (cid:0) d X i =1 k I t f i k L ( τ ) + k I t g k L ( τ ) (cid:1) + N λ − k I t f k L ( τ ) + N E k u t I t ≤ τ k L + N ∗ λ − k I t Du k L ( τ ) + N ∗ k I t u k L ( τ ) + N ∗ λ − d X i =1 k I t f i k L ( τ ) . (5.22) After γ has been fixed we have κ = κ ( d, δ, K, ρ ) and we take a ζ ∈ C ∞ ( R )with support in (0 , κ − ) such that Z ∞−∞ ζ ( t ) dt = 1 . (5.23)For s ∈ R define ζ st = ζ ( t − s ), u st ( x ) = u t ( x ) ζ st . Obviously u st = 0 if s + + κ − ≤ t ≤ τ . Therefore, we can apply (5.22) to u st with t = s + observing that du st = ( σ ikt D i u st + ν t u st + ζ st g k ) dw kt + (cid:0) D i ( a ijt D j u st + b it u st ) + b it u st − ( c t + λ ) u st + D i ( ζ st f it ) + ( ζ st f t + ( ζ st ) ′ u t (cid:1) dt. Then from (5.22) for λ ≥ k I s + ζ s u √ c + λ k L ( τ ) + k I s + ζ s Du k L ( τ ) ≤ N (cid:0) d X i =1 k I s + ζ s f i k L ( τ ) + k I s + ζ s g k L ( τ ) (cid:1) + N λ − (cid:0) k I s + ζ s f k L ( τ ) + k I s + ( ζ s ) ′ u k L ( τ ) (cid:1) + N E k u s + ζ ss + I s + ≤ τ k L + N ∗ λ − k I s + ζ s Du k L ( τ ) + N ∗ k I s + ζ s u k L ( τ ) + N ∗ λ − d X i =1 k I s + ζ s f i k L ( τ ) . (5.24)Here I s + can be dropped since I s + I [0 ,τ ) = I s I [0 ,τ ) and I s ζ s = ζ s . Afterdropping I s + we integrate through (5.24) with respect to s ∈ R , use (5.23),and observe that, since κ depends only on d, δ, K, ρ , we have Z ∞−∞ | ζ ′ ( s ) | ds = N ∗ . We also use the fact that ζ ss + = 0 only if s + = 0 and − κ − ≤ s ≤ Z − κ − ( ζ s ) ds = 1 . Then we conclude λ k u k L ( τ ) + k u √ c k L ( τ ) + k Du k L ( τ ) ≤ N (cid:0) d X i =1 k f i k L ( τ ) + k g k L ( τ ) + E k u k L (cid:1) + N λ − (cid:0) k f k L ( τ ) + k u k L ( τ ) (cid:1) + N ∗ λ − k Du k L ( τ ) + N ∗ k u k L ( τ ) + N ∗ λ − d X i =1 k f i k L ( τ ) . Without losing generality we assume that N ≥ λ = λ ( d, δ, K, ρ ). We take it so that λ ≥ N ∗ , λ ≥ N . Thenwe obviously come to (3.1) with N = 4 N . The theorem is proved. PDES WITH GROWING COEFFICIENTS 19 Proof of Theorem 3.2
We may assume in this section that F t = F t + for all t ∈ R + . This doesnot restrict generality because replacing F t with F t + makes our assumptionsweaker and does not affect our assertions because the solutions are continu-ous in time. Furthermore, having in mind setting all data equal to zero for t > τ , we see that without loss of generality we may assume that τ = ∞ .Set L = L ( ∞ ) , W = W ( ∞ ) , W = W ( ∞ ) . We need a few auxiliary results.
Lemma 6.1.
For any
T, R ∈ R + , and ω ∈ Ω we have sup t ≤ T Z B R ( | b t ( x ) | q + | b t ( x ) | q + c qt ( x )) dx < ∞ . (6.1)Proof. Obviously it suffices to prove (6.1) with B ρ ( x ) in place of B R forany x . In that case, for instance, Z B ρ ( x ) | b t ( x ) | q dx ≤ q Z B ρ ( x ) | b t ( x ) − ¯ b t ( x ) | q dx + N | ¯ b t ( x ) | q and we conclude estimating the left-hand side as in (5.4) also relying onAssumption 3.3. Similarly, b t and c t are treated. The lemma is proved. Lemma 6.2.
For any R ∈ R + there exists a sequence of stopping times τ n ↑ ∞ such that for any n = 1 , , ... and ω for almost any t ≤ τ n we have Z B R ( | b t | q + | b t | q + | c t | q ) dx ≤ n. (6.2)Proof. For each t, R >
0, and ω define β t,R = Z B R ( | b t | q + | b t | q + | c t | q ) dx,ψ t,R = lim ≤ s n } are stopping times with respect to F t + (= F t ) and ψ t,R ≤ n for t < τ n .Furthermore, by Lemma 6.1 we have τ n ↑ ∞ as n → ∞ . By Lebesguedifferentiation theorem we conclude that (for any ω ) for almost all t ≤ τ n we have (6.2). This proves the lemma.By combining this lemma with Lemma 5.1 we obtain the following. Corollary 6.3. If ψ ∈ C ∞ has support in B R , then for τ n from Lemma 6.2for each n = 1 , , ... , for almost all t ≤ τ n , for any u ∈ W and v ∈ W wehave | ( b it D i ( vψ ) , u ) | ≤ N k v k W k u k W , | ( b it D i u, vψ ) | ≤ N k v k W k u k W , | ( c t vψ, u ) | ≤ N k v k L k u k W , (6.3) where the constant N = N ( n, d ) . Since bounded linear operators are continuous we obtain the following.
Corollary 6.4. If φ ∈ C ∞ has support in B R , then for τ n from Lemma 6.2and each n the operators u · → ( b i · D i u · , φ ) , u · → ( b i · u · , D i φ ) , u · → ( c · u · , φ ) ,u · → Z · ( b it D i u t , φ ) dt, u · → Z · ( b it u t , D i φ ) dt, u · → Z · ( c · u · , φ ) dt are continuous as operators from W to L ( | (0 , n ∧ τ n ]]) = L ( | (0 , n ∧ τ n ]] , R ) . In the proof of Theorem 3.2 we are going to use sequences which convergeweakly in W . Therefore, the following result is relevant. Lemma 6.5.
Assume that for some f j ∈ L , j = 0 , ..., d , g = ( g k ) ∈ L , u ∈ W , and any φ ∈ C ∞ equation (2.4) with u ∈ L (Ω , F , L ) holds foralmost all ( ω, t ) . Then there exists a function ˜ u ∈ W solving equation (2.2) (for all t ) with initial data u in the sense of Definition 3.1. Proof. We split the proof into two steps.
Step 1. Modifying u t ψ . We recall some facts from the theory of Itˆostochastic integrals in a separable Hilbert space, say H and some otherresults, which can be found, for instance, in [19] and [12]. Integrating H -valued processes with respect to a one-dimensional Wiener process presentsno difficulties and leads to strongly continuous H -valued locally square-integrable martingales with natural isometry. If g = ( g k ) ∈ L , then byDoob’s inequality E sup t (cid:13)(cid:13) m X k = n Z t g ks dw ks (cid:13)(cid:13) L ≤ E Z ∞ m X k = n k g ks k L ds → m ≥ n → ∞ . Therefore, m t = ∞ X k =1 Z t g ks dw ks is well defined as a continuous L -valued square-integrable martingale. Fur-thermore, for any φ ∈ C ∞ with probability one we have( m t , φ ) = ∞ X k =1 Z t ( g ks , φ ) dw ks PDES WITH GROWING COEFFICIENTS 21 for all t and the series on the right converges uniformly in probability on R + . If g ∈ L ( τ n ), n = 1 , , ... , and stopping times τ n ↑ ∞ , then m t = ∞ X k =1 Z t g ks dw ks is well defined as a locally square-integrable L -valued continuous martin-gale. Again for any φ ∈ C ∞ with probability one we have( m t , φ ) = ∞ X k =1 Z t ( g ks , φ ) dw ks (6.4)for all t and the series on the right converges uniformly in probability onevery finite interval of time.We fix a ψ ∈ C ∞ and apply the above to h ψt := ∞ X k =1 Z t ψ ( σ iks D i u s + ν ks u s + g ks ) dw ks . Observe that, by assumption, for any v ∈ C ∞ for almost all ( ω, t )( u t ψ, v ) = ( u ψ, v ) + Z t h F s , v i ds + ( h ψt , v ) , (6.5)where h F t , v i = ( b it D i u t − ( c t + λ ) u t + f t , vψ ) − ( a ijt D j u t + b it u t + f it , D i ( vψ )) . We also define V = W , and notice that if k v k V ≤
1, then by Corollary 6.3for any T ∈ R + for almost any ( ω, t ) ∈ Ω × [0 , T ] we have |h F t , v i| ≤ N (cid:0) d X j =0 k f jt k L + k u t k W (cid:1) , where N is independent of v, t (but may depend on ω and T ). It followsthat, for V ∗ defined as the dual of V , the V ∗ -norm of F t is in L ([0 , T ])(a.s.) for every T ∈ R + . It also follows that (6.5) holds for almost all ( ω, t )for each v ∈ V rather than only for v ∈ C ∞ .By Theorem 3.1 of [19] there exists a set Ω ψ of full probability and an L -valued function ˜ u ψt on Ω × R + such that ˜ u ψt is F t -measurable, ˜ u ψt is L -continuous in t for every ω and ˜ u ψt = u t ψ for almost all ( ω, t ). Furthermore,for ω ∈ Ω ψ , t ≥
0, and φ ∈ C ∞ we have(˜ u ψt , φ ) = ( h ψt , φ ) + Z t ( b is D i u s − ( c s + λ ) u s + f s , φψ ) ds − Z t (cid:0) a ijs D j u s + b is u s + f is , D i ( φψ ) (cid:1) ds. (6.6) Step 2. Constructing ˜ u t . Let ψ ∈ C ∞ be such that ψ = 1 on B and set ψ n ( x ) = ψ ( x/n ), n = 1 , , ... . Define ˜ u nt = ˜ u ψ n t and notice that by the abovefor m ≥ n and almost all ( ω, t )˜ u mt I B n = u t ψ m I B n = u t I B n = ˜ u nt I B n as L -elements. Since the extreme terms are L -continuous functions of t ,there exist sets Ω nm , m ≥ n , of full probability such that for ω ∈ Ω nm wehave ˜ u mt I B n = ˜ u nt I B n as L -elements for all t .Then for t ≥ ω ∈ Ω ′ := T m ≥ n Ω nm the formula˜ u t = I Ω ′ ∞ X n =0 ˜ u n +1 t I B n +1 \ B n defines a distribution such that ˜ u t I B n = ˜ u nt I B n as L -elements for any ω ∈ Ω ′ , t ≥
0, and n . It follows that ˜ u t = u t as distributions for almost any( ω, t ), hence, ˜ u ∈ W and there exists an event Ω ′′ ⊂ Ω ′ of full probabilitysuch that for any ω ∈ Ω ′′ and almost any t ≥ u t = u t . Now(6.6) implies that if φ ∈ C ∞ is such that φ ( x ) = 0 for | x | ≥ n , then for ω ∈ Ω ′′ ∩ Ω ψ n and all t ≥ u t , φ ) = (˜ u nt , φ ) = ( h ψ n t , φ ) + Z t ( b is D i ˜ u s − ( c s + λ )˜ u s + f s , φ ) ds − Z t (cid:0) a ijs D j ˜ u s + b is ˜ u s + f is , D i φ (cid:1) ds. (6.7)By recalling what was said about (6.4) and using Corollary 6.3, we seethat indeed the requirements of Definition 3.1 are satisfied with ˜ u and ∞ inplace of u and τ , respectively. The lemma is proved. Lemma 6.6.
Let φ ∈ C ∞ be supported in B R and take τ n from Lemma6.2. Let u n , u ∈ W , n = 1 , , ... , be such that u n → u weakly in W . For n = 1 , , ... define χ n ( t ) = ( − n ) ∨ t ∧ n , b int = χ n ( b it ) , b int = χ n ( b it ) and set c ns = n ∧ c s . Then for any m = 1 , , ... Z t [( b ins D i u ns , φ ) − ( b ins u ns , D i φ ) − ( c ns u ns , φ )] ds → Z t [( b is D i u s , φ ) − ( b is u s , D i φ ) − ( c s u s , φ )] ds (6.8) weakly in the space L ( | (0 , m ∧ τ m ]]) as n → ∞ . Proof. By Corollary 6.4 and by the fact that (strongly) continuous oper-ators are weakly continuous we obtain that Z t [( b is D i u ns , φ ) − ( b is u ns , D i φ ) − ( c s u ns , φ )] ds → Z t [( b is D i u s , φ ) − ( b is u s , D i φ ) − ( c s u s , φ )] ds PDES WITH GROWING COEFFICIENTS 23 as n → ∞ weakly in the space L ( | (0 , m ∧ τ m ]]) for any m . Therefore, itsuffices to show that Z t [( D i u ns , ( b is − b ins ) φ ) − ( u ns , ( b is − b ins ) D i φ + ( c s − c ns ) φ )] ds → L ( | (0 , m ∧ τ m ]]) for any m . In other words, it suffices to show thatfor any ξ ∈ L ( | (0 , m ∧ τ m ]]) E Z m ∧ τ m ξ t (cid:0) Z t [( D i u ns , ( b is − b ins ) φ ) − ( u ns , ( b is − b ins ) D i φ + ( c s − c ns ) φ )] ds (cid:1) dt → . This relation is rewritten as E Z m ∧ τ m [( D i u ns , η s ( b is − b ins ) φ ) − ( η s u ns , ( b is − b ins ) D i φ + ( c s − c ns ) φ )] ds → , (6.9)where the process η s := Z m ∧ τ m s ξ t dt is of class L ( | (0 , m ∧ τ m ]]) since m ∧ τ m is bounded ( ≤ m ).However, by the choice of τ m and the dominated convergence theorem, η s ( b is − b ins ) D i φ → , η s ( b is − b ins ) φ → , η s ( c s − c ns ) φ → n → ∞ strongly in L ( | (0 , m ∧ τ m ]]) (use the fact that q ≥
2) and byassumption u n → u and Du n → Du weakly in L ( | (0 , τ m ]]). This implies(6.9) for any m and the lemma is proved. Proof of Theorem 3.2 . Define b nt , b nt , and c nt as in Lemma 6.6 andconsider equation (2.2) with b nt , b nt , and c nt in place of b t , b t , and c t ,respectively, and with τ = n . By a classical result there exists a unique u n ∈W ( n ) satisfying the modified equation with initial condition u . Obviously, b nt , b nt , and c nt satisfy Assumption 3.2 with the same γ as b t , b t , and c t do.By Theorem 3.1 for λ ≥ λ ( d, δ, K, ρ ) we have k u n k L ( n ) + k Du n k L ( n ) ≤ N, where N is independent of n . Hence the sequence of functions u nt I t ≤ n isbounded in the Hilbert space W and consequently has a weak limit point u ∈ W . For simplicity of presentation we assume that the whole sequence u nt I t ≤ n converges weakly to u . Take a φ ∈ C ∞ . Then by Lemma 6.6 forappropriate τ m we have that (6.8) holds weakly in L ( | (0 , m ∧ τ m ]]) for any m . Since u = u t → ∞ X k =1 Z t (Λ ks u s , φ ) dw ks is a continuous operator from W to L ( | (0 , m ]]), it is weakly continuous, sothat ∞ X k =1 Z t (Λ ks u ns , φ ) dw ks → ∞ X k =1 Z t (Λ ks u s , φ ) dw ks weakly in L ( | (0 , m ]]) for any m . Obviously, the same is true for ( u nt , φ ) → ( u t , φ ) and the remaining terms entering the equation for u ns . Hence bypassing to the weak limit in the equation for u nt we see that u satisfies theassumptions of Lemma 6.5 applying which finishes the proof of the theorem. References [1] S. Assing and R. Manthey,
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