Probab. Theory Relat. Fields (2010) 147:583–605 DOI 10.1007/s00440-009-0217-7
Itô’s formula for the L p -norm of stochastic W p1 -valued processes N. V. Krylov
Received: 12 July 2008 / Revised: 26 March 2009 / Published online: 23 April 2009 © Springer-Verlag 2009
Abstract We prove Itô’s formula for the L p -norm of a stochastic W p1 -valued processes appearing in the theory of SPDEs in divergence form. Keywords Stochastic partial differential equations · Divergence equations · Itô’s formula Mathematics Subject Classification (2000)
60H15 · 35R60
1 Introduction Let (, F, P) be a complete probability space with an increasing filtration {Ft , t ≥ 0} of complete with respect to (F, P) σ -fields Ft ⊂ F. Denote by P the predictable σ -field in × (0, ∞) associated with {Ft }. Let wtk , k = 1, 2, . . ., be independent one-dimensional Wiener processes with respect to {Ft }. Let D be the space of generalized functions on the Euclidean d-dimensional space Rd of points x = (x 1 , . . . , x d ). We consider processes with values in D whose stochastic differential is given by du t = Di f ti + f t0 dt + gtk dwtk ,
(1.1)
j
where f t , gtk are L p -valued processes, u t is a W p1 -valued process, and the summation convention over repeated indices is enforced. Our main goals are to give conditions on
The work was partially supported by NSF Grant DMS-0653121. N. V. Krylov (B) University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455, USA e-mail:
[email protected]
123
584
N.V. Krylov
u, f j , and g k , which are sufficient to assert that u t is a continuous L p -valued process, p and to derive Itô’s formula for u t L p . This was never done before, no matter how strange it may look. The hardest step is showing that u t is continuous as an L p -valued function. A more or less standard fact is that under natural conditions one can estimate p
E sup u t L p
(1.2)
t
and from here and Eq. (1.1), implying that (u t , ϕ) is continuous in t for any test function ϕ ∈ C0∞ , one used to derive that u t is only a weakly continuous L p -valued process. Even though the above mentioned Itô’s formula was not proved, the fact that, actually, u t is indeed continuous as an L p -valued process was known and proved by different methods for p = 2, on the basis of abstract results for SPDEs in Hilbert spaces, and for p > 2, on the basis of embedding theorems for stochastic Banach spaces (see, for p instance [3,7]). In this way of arguing proving the continuity of u t L p required a full blown theory of SPDEs with constant coefficients (cf. [3,8]). We present a “direct” and self-contained proof of the formula and the continuity. Our results are used in paper [5] which in turn is the basis for [6]. Finally, we mention that there are many situations in which Itô’s formula is known for Banach space valued processes. See, for instance, [1] and the references therein. These formulas could be more general in some respects but they do not cover our situation and are closer to our Lemma 5.1 where the term Di f i is not present in (1.1). 2 Main result We take a stopping time τ and fix a number p ≥ 2. Denote L p = L p (Rd ). We use the same notation L p for vector- and matrix-valued or else 2 -valued functions such as gt = (gtk ) in (1.1). For instance, if u(x) = (u 1 (x), u 2 (x), . . .) is an 2 -valued measurable function on Rd , then p u L p
=
p |u(x)|2
Rd
dx =
∞ Rd
p/2 |u (x)| k
2
d x.
k=1
Introduce Di =
∂ , i = 1, . . . , d. ∂ xi
By Du we mean the gradient with respect to x of a function u on Rd .
123
Itô’s formula for the L p -norm
585
As usual, W p1 = {u ∈ L p : Du ∈ L p }, uW p1 = u L p + Du L p . If τ is a stopping time, then ¯ L p , W1p (τ ) := L p (| 0, τ ]], P, ¯ W p1 , L p (τ ) := L p (| 0, τ ]], P, where P¯ is the completion of P with respect to P(dω) × dt. We also need the space W 1p (τ ), which is the space of functions u t = u t (ω, ·) on {(ω, t) : 0 ≤ t ≤ τ, t < ∞} with values in the space of generalized functions on Rd and having the following properties: (i) We have u 0 ∈ L p (, F0 , L p ); (ii) We have u ∈ W1p (τ ); (iii) There exist f i ∈ L p (τ ), i = 0, . . . , d, and g = (g 1 , g 2 , . . .) ∈ L p (τ ) such that for any ϕ ∈ C0∞ with probability 1 for all t ∈ [0, ∞) we have
(u t∧τ , ϕ) = (u 0 , ϕ) +
∞
t
Is≤τ (gsk , ϕ) dwsk
k=1 0
t +
Is≤τ ( f s0 , ϕ) − ( f si , Di ϕ) ds.
(2.1)
0
In particular, for any φ ∈ C0∞ , the process (u t∧τ , φ) is Ft -adapted and continuous (a.s.). In case that property (iii) holds, we say that (1.1) holds for t ≤ τ . The reader can find in [3] a discussion of (ii) and (iii), in particular, the fact that the series in (2.1) converges uniformly in probability on every finite subinterval of [0, τ ]. This will also be seen from the proof of Lemma 4.2. By the way, the necessity to consider infinitely many Wiener processes is also explained in [3] in applications to superdiffusions. Here is our main result. Theorem 2.1 Let u ∈ W 1p (τ ), f j ∈ L p (τ ), g = (g k ) ∈ L p (τ ) and assume that (1.1) holds for t ≤ τ in the sense of generalized functions. Then there is a set ⊂ of full probability such that
123
586
N.V. Krylov
(i) u t∧τ I is a continuous L p -valued Ft -adapted function on [0, ∞); (ii) for all t ∈ [0, ∞) and ω ∈ Itô’s formula holds:
t∧τ
|u t∧τ | d x =
|u 0 | d x + p
p
Rd
|u s | p−2 u s gsk d x dwsk
p
Rd
0 Rd
⎛
t∧τ ⎜ p|u s | p−2 u s f s0 − p( p − 1)|u s | p−2 f si Di u s + ⎝ Rd
0
⎞
⎟ + (1/2) p( p − 1)|u s | p−2 |gs |22 d x ⎠ ds.
(2.2)
Furthermore, for any T ∈ [0, ∞) p
p
p
E sup u t L p ≤ 2Eu 0 L p + N T p−1 f 0 L p (τ ) t≤τ ∧T
+N T
( p−2)/2
d
i p
p
p
f L p (τ ) + gL p (τ ) + DuL p (τ ) ,
(2.3)
i=1
where N = N (d, p). Formula (2.2) admits a slight generalization discussed in Remark 5.2. We also note that the powers of T entering (2.3) are optimal which one sees first taking T = 1 and then using parabolic dilations of the type (t, x) → (c2 t, cx) with arbitrary constants c = 0. We prove Theorem 2.1 in Sect. 6 after we prepare the necessary tools in Sects. 3–5. Here is an “energy” estimate which we use in [5]. Corollary 2.2 Under the conditions of Theorem 2.1
τ |u 0 | p d x + E
E Rd
0
⎛ ⎜ ⎝
p|u t | p−2 u t f t0 − p( p − 1)|u t | p−2 f ti Di u t
Rd
⎞
⎟ + (1/2) p( p − 1)|u t | p−2 |gt |22 d x ⎠ dt ≥ E Iτ <∞
|u τ | p d x.
(2.4)
Rd
Furthermore, if τ is bounded then there is an equality instead of inequality in (2.4). The proof of the corollary is given in Sect. 6.
123
Itô’s formula for the L p -norm
587
3 Auxiliary results We need two well-known results (see, for instance, Lemma 6.1 and Corollary 6.2 in [4]), which we prove for completeness of presentation. Lemma 3.1 Let (E, , µ) be a measure space, r ∈ (0, ∞), take some measurable functions u n , u, and assume that u n → u in measure. Finally, let
|u n | µ(d x) →
|u|r µ(d x) < ∞.
r
E
E
Then |u n − u|r µ(d x) → 0.
(3.1)
E
Proof We have r |u| − |u n |r = 2 |u|r − |u n |r − |u|r − |u n |r . + Upon integrating through this equation and observing that (|u|r − |u n |r )+ ≤ |u|r we conclude by the dominated convergence theorem that
r |u| − |u n |r µ(d x) → 0.
(3.2)
E
Next, if |u n − u| ≥ 3|u|, then |u n | + |u| ≥ 3|u|, |u| ≤ (1/2)|u n |, |u|r ≤ 2−r |u n |r , |u n |r − |u|r ≥ 1 − 2−r |u n |r , |u n − u| ≤ |u n | + |u| ≤ 2|u n |, −1 |u n |r − |u|r , |u n − u|r ≤ 2r |u n |r ≤ 2r 1 − 2−r which along with (3.2) imply that |u n − u|r I|u n −u|≥3|u| µ(d x) → 0. E
Furthermore, |u n − u|r I|u n −u|<3|u| µ(d x) → 0 E
by the dominated convergence theorem. By combining the last two relations we come to (3.1). The lemma is proved.
123
588
N.V. Krylov
Corollary 3.2 Let (E, , µ) be a measure space, r, s ∈ (1, ∞), r −1 + s −1 = 1, u n , u ∈ L r (µ), vn , v ∈ L s (µ), u n → u and vn → v in measure. Finally, let u n L r (µ) → u L r (µ) , vn L s (µ) → u L s (µ) . Then
|u n vn − uv| µ(d x) → 0, E
u n vn µ(d x) →
E
uv µ(d x). E
Indeed, it suffices to use Hölder’s inequality and the formula u n vn − uv = (u n − u)v + (vn − v)u + (u n − u)(vn − v). 4 Integrating L p functions Most likely a big part of what follows in this section can be obtained from some abstract constructions in [1]. However, it does not look easy to obtain estimate (4.2). In any case, it is worth giving all rather simple arguments for completeness. Set L p = L p (∞) and for Borel subsets of a Euclidean space denote by B( ) the σ -field of Borel subsets of . Definition 4.1 By U p we denote the set of functions u = u t (x) = u t (ω, x) on × [0, ∞) × Rd such that (i) (ii) (iii) (iv)
u is measurable with respect to F ⊗ B([0, ∞)) ⊗ B(Rd ); for each x, the function u t (x) is Ft -adapted; u t (x) is continuous in t ∈ [0, ∞) for each (ω, x); the function u t (ω, ·) as a function of (ω, t) is L p -valued, Ft -adapted, and continuous in t for any ω.
Lemma 4.2 Let g = (g k ) ∈ L p . Then there exists a function u ∈ U p such that for any φ ∈ C0∞ the equation (u t , φ) =
∞ t
gsk , φ dwsk
(4.1)
k=1 0
holds for all t ∈ [0, ∞) with probability one. Furthermore, for any T ∈ [0, ∞) we have sup |u t (x)| d x ≤ N T p
E Rd
where N = N ( p).
123
t≤T
( p−2)/2
T
p
gs L p ds,
E 0
(4.2)
Itô’s formula for the L p -norm
589
Proof First assume that there is an integer j ≥ 1, (nonrandom) functions g ik ∈ C0∞ , and bounded stopping times τ0 ≤ τ1 ≤ · · · ≤ τ j such that g k ≡ 0 for k > j and gtk (x) =
j
g ik (x)I(τi−1 ,τi ] (t)
i=1
for k ≤ j. Then define
u t (x) =
j
k k g ik (x) wt∧τ − w t∧τi−1 . i
i,k=1
Obviously, u ∈ U p . Furthermore, (4.1) holds for any φ ∈ C0∞ for all t with probability one since its right-hand side equals j
g ik , φ
k k wt∧τ − wt∧τ i i−1
i,k=1
for all t with probability one. Next, by the Burkholder–Davis–Gundy inequalities for each x p ⎛ T ⎞ p/2 t E sup |u t (x)| p = E sup gsk (x) dwsk ≤ N E ⎝ |gs (x)|22 ds ⎠ , t≤T t≤T k 0
0
which after applying Hölder’s inequality ( p ≥ 2) yields
E sup |u t (x)| ≤ N T p
t≤T
( p−2)/2
T
p
|gs (x)|2 ds.
E 0
We integrate this inequality over Rd and use the fact that the measurability properties of g, u and the continuity of u t in t allow us to use Fubini’s theorem. Then we come to (4.2). By Theorem 3.10 of [3] the set of g’s like the one above is dense in L p . Therefore, to prove the lemma it suffices to show that the set of g’s for which the statements of the lemma are true is closed in L p . Take a sequence g n = (g nk ) ∈ L p , n = 1, 2, . . ., such that for each n there is a function u n corresponding to g n and possessing the asserted properties. Assume that
123
590
N.V. Krylov
for a g ∈ L p we have g n → g in L p as n → ∞. Using a subsequence of g n we may assume that for any T ∈ [0, ∞)
sup |u n+1 (x) − u nt (x)| p d x ≤ T ( p−2)/2 2−n . t
E
(4.3)
t≤T
Rd
Introduce n −2 . (x) − u (x)| ≥ n An = (ω, x) : sup |u n+1 t t t≤n
Then ∞ n=1
≤
sup |u n+1 (x) − u nt (x)|I An (x) d x t
E Rd ∞
t≤n
n
2( p−1)
E
n=1
≤
∞
Rd
p sup u n+1 (x) − u nt (x) d x t t≤n
n 2( p−1) n ( p−2)/2 2−n < ∞,
n=1
implying that ∞ n=1
sup u n+1 (x) − u nt (x) I An (x) < ∞ t t≤n
for almost all (ω, x). The series with the complements of An in place of An obviously converges everywhere. We conclude that the F ⊗ B(Rd )-measurable set G = (ω, x) :
∞ n=1
n+1 sup u t (x) − u nt (x) < ∞ t≤n
has full measure. By Fubini’s theorem the function P((ω, x) ∈ G) is a Borel function of x equal to 1 for almost all x. Accordingly we introduce a Borel set of full measure
= {x : P((ω, x) ∈ G) = 1} and the F ⊗ B(Rd )-measurable set G of full measure by
G = (ω, x) : x ∈ ,
∞ n=1
123
n+1 n sup u t (x) − u t (x) < ∞ . t≤n
Itô’s formula for the L p -norm
591
Now define u t (x) = lim u nt (x) n→∞
(4.4)
for (ω, x) ∈ G , t ≥ 0 and set u t (x) ≡ 0 for (ω, x) ∈ G . Also set vt (x) =
limn→∞ u nt (x) if the limit exists, 0 otherwise.
Obviously, u t (ω, x) = vt (ω, x)IG (ω, x).
(4.5)
Furthermore, v is known to be F ⊗ B([0, ∞)) ⊗ B(Rd )-measurable since the u n possess this property. It follows that u is F ⊗B([0, ∞))⊗B(Rd )-measurable. For each x, the functions u nt (x) are Ft -adapted and so is vt (x). Also IG (ω, x) is F0 -measurable (and hence Ft -adapted) for each x since the Ft are complete and P(IG (ω, x) = 1) = P((ω, x) ∈ G, x ∈ ) equals zero if x ∈ and one if x ∈ by the choice of . Now Eq. (4.5) allows us to conclude that u t (x) is Ft -adapted for each x. Since the limit in (4.4) is uniform in t on any finite interval, we see that u t is continuous in t for any (ω, x). In particular, sup |u t (x)|
t≤T
is F ⊗ B(Rd )-measurable, estimate (4.2) with u in place of u makes sense and holds owing to Fatou’s lemma and the assumption on u n . Estimate (4.2) shows that there is a set ∈ F0 of full probability such that u t (ω, ·) ∈ L p for all t if ω ∈ and moreover Rd
sup |u t (ω, x)| p d x < ∞
t≤T
for any T ∈ [0, ∞) if ω ∈ . This fact, the continuity of u t in t, and the dominated convergence theorem imply that u t is continuous as an L p -valued function of t for any ω ∈ . We now set u t (ω, x) = u t (ω, x)I (ω). Then we see that to show that u ∈ U p it suffices to prove that u t (ω, ·) is Ft -adapted as an L p -valued function.
123
592
N.V. Krylov
Obviously, to do this step it suffices to prove the assertion of the lemma related to (4.1). Since :=
m nt
t k
gsk − gsnk , φ dwsk
0
are local martingales starting from zero, by the Burkholder–Davis–Gundy inequalities for any T ∈ [0, ∞) p/2
E sup |m nt | p ≤ N EmT , t≤T
where N = N ( p), that is p ⎛ T ⎞ p/2 t ∞ 2 k In := E sup gs − gsnk , φ dwsk ≤ N E ⎝ gsk − gsnk , φ ds ⎠ . t≤T k k=1 0
0
Here by Cauchy’s inequality
gsk − gsnk , φ
2
≤ |gsk − gsnk |2 , |φ| φ L 1 ,
so that ⎛ ⎜ In ≤ N E ⎝
T
⎞ p/2 gs − g n 2 |φ| d xds ⎟ ⎠ s 2
0 Rd
T ≤ NE
gs − g n p d x ds ≤ N g − g n L , s p 2
(4.6)
0 Rd
where the constants N are independent of n (but depend on φ among other things). In addition, estimate (4.3) easily implies that E sup (u t − u nt , φ) → 0 t≤T
as n → ∞ for any T ∈ [0, ∞). By combining these facts and passing to the limit in (4.1) with u n in place of u we get the desired result and the lemma is proved. Remark 4.3 It is tempting to assert that u t (x) is P ⊗ B(Rd )-measurable since it is F ⊗ B([0, ∞)) ⊗ B(Rd )-measurable and, for each x, it is predictable. However, we do not know if this assertion is true. M. Veraar kindly pointed out to the author that at least in a more general setting the assertion is false (see [9]).
123
Itô’s formula for the L p -norm
593
In a similar way, without using the Burkholder–Davis–Gundy inequalities, the following result is established. Lemma 4.4 Let f ∈ L p (∞). Then there exists a function u ∈ U p such that for any φ ∈ C0∞ the equation t (u t , φ) =
( f s , φ) ds
(4.7)
0
holds for all t ∈ [0, ∞) with probability one. Furthermore, for any T ∈ [0, ∞) we have
T sup |u t (x)| d x ≤ N T p
E Rd
p−1
p
f s L p ds,
E
t≤T
(4.8)
0
where N = N ( p). Remark 4.5 Observe that the integral on the right in (4.7) need not exist for each ω since ( f t , φ) is generally only measurable with respect to the completion of P and the function ( f t , φ) need not be Lebesgue measurable in t for each ω. In case of Lemma 4.2 this problem does not arise because of freedom in defining the stochastic integrals.
5 Itô’s formula in a simple case The goal of this section is to prove the following result. As we have pointed out in the Introduction this result can be obtained from much more general results of [1] on Itô’s formula in UMD Banach spaces. We give it an independent and more elementary proof for completeness. Lemma 5.1 Let f ∈ L p (τ ), g = (g k ) ∈ L p (τ ) and assume that we are given a function u t on × [0, ∞) with values in the space of distributions on Rd such that u 0 ∈ L p (, F0 , L p ) and for any φ ∈ C0∞ with probability one for all t ∈ [0, ∞) we have t (u t∧τ , φ) = (u 0 , φ) +
Is≤τ ( f s , φ) ds + 0
∞ t
gsk , φ Is≤τ dwsk .
(5.1)
k=1 0
Then, there is a set ∈ F0 of full probability such that
123
594
N.V. Krylov
(i) u t∧τ I is an L p -valued Ft -adapted continuous process on [0, ∞), (ii) for all t ∈ [0, ∞) and ω ∈ ⎡ t∧τ ⎢ p p p u t∧τ L p = u 0 L p + |u s | p−2 u s f s d x ⎣ 0
Rd
+ (1/2) p( p − 1)
⎤ ⎥ |u s | p−2 |gs |22 d x ⎦ ds
Rd
t∧τ +p
|u s | p−2 u s gsk d x dwsk . 0
(5.2)
Rd
Proof First observe that the right-hand sides of (5.1) and (5.2) will be affected only on a set of probability zero independent of t if we replace f and g with L p -valued predictable functions fˆ and gˆ such that f t − fˆt + gt − gˆ t L = 0 p
Lp
for almost all (ω, t). It follows that without losing generality we may assume that f and g are predictable as L p -valued functions. Lemmas 4.2 and 4.4 allow us to find a v ∈ U p such that for any φ ∈ C0∞ equation (5.1) with vt in place of u t∧τ holds for all t with probability one. It follows that for any countable set A ⊂ C0∞ there exists a set of full probability such that for any ω ∈ , φ ∈ A, and t ≥ 0 we have (u t∧τ , φ) = (vt , φ). If the set A is chosen appropriately, then we conclude that u t∧τ = vt in the sense the distributions, whenever ω ∈ and t ≥ 0. In particular, assertion (i) holds with this . This argument allows us to assume that τ = ∞ and u ∈ U p and concentrate on proving (5.2). This argument also shows that for any T ∈ [0, ∞) E sup |u t (x)| p d x < ∞ Rd
t≤T
implying that there exists a set of full probability such that for any ω ∈ we have sup |u t (x)| p d x < ∞, ∀T ∈ [0, ∞). (5.3) Rd
t≤T
Now, take a nonnegative function ζ ∈ C0∞ (Rd ) with unit integral, for ε > 0, define ζε = ε−d ζ (x/ε), and for any locally summable h given on Rd introduce the notation h (ε) = h ∗ ζε . Then (5.1) implies that for each x almost surely for all t ∈ [0, ∞) (ε) u t (x)
=
(ε) u 0 (x) +
t 0
123
f s(ε) (x) ds
t +
gsk(ε) (x) dwsk . 0
(5.4)
Itô’s formula for the L p -norm
595
By Itô’s formula, for each x (ε) |u t | p
t p−2 (ε) p k(ε) = u 0 + p u (ε) u (ε) dwsk s s gs 0
t p−2 (ε) p−2 (ε) 2 (ε) (ε) + u f ds + (1/2) p( p − 1) p u (ε) ds u g s s s s s 2
0
(5.5) (a.s.), where we dropped the argument x for simplicity. We want to integrate this equality over Rd and use the stochastic and deterministic Fubini’s theorems. We will see that there is no difficulties with the integral with respect to ds. However, in order to be able to apply the stochastic version of Fubini’s theorem we need at least that the resulting stochastic integral make sense, that is we need at least the inequality ⎞2 ⎛ t ∞ p−1 k(ε) ⎟ ⎜ (ε) gs d x ⎠ ds < ∞ ⎝ u s 0 k=1
Rd
to hold (a.s.). The computations below show that, actually, for a sequence of stopping times τn ↑ ∞,
E
t∧τ n ∞ 0
k=1
⎞2 p−1 k(ε) ⎟ ⎜ (ε) gs d x ⎠ ds < ∞ ⎝ u s ⎛
(5.6)
Rd
and this is known to be sufficient to apply the stochastic version of Fubini’s theorem (see, for instance, Lemma 3.3 of [2] or [9] and the references therein for more sophis(ε) ticated results). By the way, notice that u t (x) is continuous (infinitely differentiable) in x for any (ω, t). Therefore, it is Ft ⊗ B(Rd )-measurable. Since it is also continuous (ε) in t for each (ω, x), the function u t (x) if P ⊗ B(Rd )-measurable and there is no measurability obstructions in applying Fubini’s theorems. To deal with the integral with respect to s observe that by Young’s inequality for any t ∈ [0, ∞) t t t γ p/( p−1) (ε) p t p−1 (ε) p (ε) p−1 (ε) u s f s ds ≤ u s ds + p f s ds, t γ 0
0
0
t t t γ p/( p−2) (ε) p t ( p−2)/2 (ε) p (ε) p−2 (ε) 2 u s gs ds ≤ u s ds + gs ds, 2 2 t γ p/2 0
0
0
(5.7)
123
596
N.V. Krylov
where γ > 0 is any number (however, if p = 2 we set γ = 1 in the second inequality). Actually, below in this proof we only need (5.7) with γ = 1. More general γ ’s will appear in the proof of Theorem 2.1. By Minkowski’s inequality ∞ k=1
⎞2 ⎛ p−1 p−1 k(ε) ⎟ (ε) ⎜ (ε) ⎜ gs d x ⎠ ≤ ⎝ u (ε) g ⎝ u s s s ⎛
Rd
Rd
⎞2 2
⎟ dx⎠ .
(5.8)
By Hölder’s inequality the right-hand side of (5.8) is less than ⎛
⎞2( p−1)/ p ⎛ ⎞2/ p p p ⎜ (ε) ⎜ (ε) ⎟ ⎟ ⎝ u s d x ⎠ ⎝ gs d x ⎠ Rd
2
Rd
⎞2/ p ⎛ ⎞2( p−1)/ p p p ⎜ ⎟ ⎜ ⎟ ≤ ⎝ gs(ε) d x ⎠ ⎝ sup u (ε) . s dx⎠ ⎛
2
Rd
Rd
s≤t
Here p (ε) (ε) p p , sup u (ε) u s ≤ |u s | p s ≤ sup |u s | sup |u s | p Rd
s≤t
s≤t
(ε)
s≤t
dx =
(ε)
,
sup |u s | p d x, Rd
s≤t
It follows from here and (5.3) that the process ⎛ t ∞ ⎜ (ε) p−1 k(ε) u g ξtε := ⎝ s s 0 k=1
Rd
⎞2 2
⎟ d x ⎠ ds
is well defined, Ft adapted, and is continuous in t (a.s.). Hence, the stopping times ! " τn = τ ∧ inf t ≥ 0 : ξtε ≥ n , n = 1, 2, . . ., are well defined, τn ↑ ∞, and, obviously, (5.6) holds. Estimates (5.7) show that there is no trouble in applying the deterministic Fubini’s theorem to the integrals with respect to ds in (5.5). Estimate (5.6) implies that for each fixed t we can apply the stochastic Fubini’s theorem to the stochastic term in (5.5)
123
Itô’s formula for the L p -norm
597
with t ∧ τn in place of t. Hence we obtain that with probability one ⎡ t∧τ n p p (ε) (ε) (ε) p−2 (ε) (ε) ⎢ u s f s d x ds u t∧τn d x = u 0 d x + ⎣ p u s Rd
Rd
0
Rd
⎤ p−2 2 (ε) ⎥ + (1/2) p( p − 1) u (ε) gs d x ⎦ds s 2
Rd t∧τ n
+p
(ε) p−2 (ε) k(ε) u s gs d x dwsk . u s
0 Rd
Since τn ↑ ∞, we have also ⎡ t p p p−2 (ε) (ε) ⎢ (ε) u (ε) u t d x = u 0 d x + ⎣ p u (ε) s s f s d x ds Rd
Rd
0
Rd
⎤ p−2 2 (ε) ⎥ + (1/2) p( p−1) u (ε) gs d x ⎦ds s 2
Rd
+p
t (ε) p−2 (ε) k(ε) u s gs d x dwsk u s
(5.9)
0 Rd
(a.s.) for each t. We now pass to the limit as ε → 0 in (5.9). Observe that for any h ∈ L p we have h (ε) L p ≤ h L p and h (ε) → h in L p as ε → 0. Therefore, the left-hand side of (5.9) tends to the left-hand side of (5.2) with (τ = ∞) for all (ω, t). The same is true (a.s.) for the first term on the right in (5.9). To prove the convergence in probability of stochastic integrals it suffices to prove that the quadratic variation of their difference goes to zero, that is ⎛ t ∞ (ε) p−2 (ε) k(ε) ⎜ u s gs − |u s | p−2 u s gsk u s ⎝ 0 k=1
⎞2 ⎟ d x ⎠ ds → 0
(5.10)
Rd
as ε → 0 in probability. Notice that for s ≤ t by Minkowski’s and Hölder’s inequalities
123
598
∞ k=1
N.V. Krylov
⎛ ⎜ ⎝
⎞2
⎛
⎟ ⎜ |u s | p−2 u s gsk(ε) − gsk d x ⎠ ds ≤ ⎝
Rd
|u s | p−1 gs(ε) − gs
Rd r ≤t
2( p−1) (ε) gs
≤ sup u r L p
⎞2 2
⎟ dx⎠
2 − gs . Lp
The integral of the last term in s tends to zero as ε → 0 (a.s.) owing to the above mentioned properties of mollifiers and the fact that supr ≤t u r L p < ∞ (for all ω). Hence to prove (5.10) it suffices to show that ⎛ ⎞2 t ∞ p−2 (ε) ⎜ ⎟ p−2 u (ε) u s gsk(ε) d x ⎠ ds J ε := u s ⎝ s − |u s | 0 k=1
Rd
tends to zero (a.s.). By Minkowski’s inequality
ε
t
J ≤
ε 2 Is ds,
(5.11)
0
where Isε
p−2 (ε) := |u (ε) u s − |u s | p−2 u s |gs(ε) |2 d x. s | Rd
Observe that on a set of full probability for almost any s we have (ε) gs − gs
2
→ 0, gs(ε)
2
→ |gs |2 , u (ε) s → us
in L p . In particular, (ε) p−2 (ε) u us s
p−1 p−2 |u = u (ε) → | u s s s L p/( p−1)
Lp
By Lemma 3.1 (ε) p−2 (ε) u s − |u s | p−2 u s → 0 u s
123
L p/( p−1)
.
Itô’s formula for the L p -norm
599
in L p/( p−1) . It follows that Isε → 0 as ε → 0 for almost all s if ω ∈ with P( ) = 1. Furthermore, p−1 (ε) Isε ≤ sup u r(ε) gs r ≤t
≤
Lp
Lp
p−1 + sup u r L p gs(ε) r ≤t
Lp
p−1 2 sup u r L p gs L p , r ≤t
which is square integrable over [0, t] (a.s.). By the dominated convergence theorem and (5.11) we have J ε → 0 (a.s.) yielding the desired convergence in probability of the stochastic integrals in (5.9) as ε → 0. The integral with respect to ds on the right in (5.9) presents no difficulty owing to Corollary 3.2 and is treated similarly to what is done above after (5.11). Thus, for each t equation (5.2) holds with probability one. Since both parts are continuous in t, it also holds for all t at once on the set of full probability and this finally brings the proof of the lemma to an end. Remark 5.2 As M. Veraar pointed out to the author one can write Itô’s formulas for p functions κ(u t L p ) other than κ(x) = x, for instance, κ(x) = x α with α ≥ 2/ p. Of course, this is a trivial matter if κ is twice continuously differentiable. However, one can deal with less regular functions as well. For instance, let κ(x), x ∈ [0, ∞), be a continuous function which is twice continuously differentiable for x = 0 and assume that the functions Aε (x) = κ (x + ε)x 1−2/ p ,
Bε (x) := κ (x + ε)x 2−2/ p
are bounded as functions of x on each finite subinterval [0, N ] of [0, ∞) uniformly with respect to ε ∈ (0, 1) and the limits A(x) := lim Aε (x), ε↓0
B(x) := lim Bε (x) ε↓0
exist for all x ∈ [0, ∞). Then it turns out that with probability one for all t t∧τ κ(vt∧τ ) = κ(v0 ) +
κ (vs )σsk dwsk
0
t∧τ +
κ (vs )bs + (1/2)κ (vs )as ds,
(5.12)
0
where we set κ (0) = κ (0) := 0 and
123
600
N.V. Krylov
vs :=
p u s L p ,
:= p
|u s | p−2 u s gsk d x, as :=
Rd
bs := p
σsk
k=1
|u s | p−2 u s f s d x + (1/2) p( p − 1)
Rd
∞ 2 σsk ,
|u s | p−2 |gs |22 d x.
Rd
As an example of κ satisfying the above conditions one can take κ(x) = x α with α ≥ 2/ p. The proof of (5.12) is based on the fact that dvt = bt dt + σtk dwtk . Having in mind stopping vt at time τ or the first time when it reaches a constant level we may assume that vt∧τ is bounded. Then one uses Itô’s formula for κ(vt + ε), where ε ∈ (0, 1) and passes to the limit as ε ↓ 0 observing that κ (vs + ε)bs = Aε (vt )b˜t , κ (vs + ε)σsk = Aε (vt )σ˜ tk κ (vs + ε)as = Bε (vs )a˜ s , where (0 · 0−1 := 0) 2/ p−1 2/ p−1 2/ p−2 b˜s = bs vs , σ˜ tk = σtk vs , a˜ s = as vs .
It turns out that t∧τ ∞ 2 k ˜bs + a˜ s + σ˜ s ds < ∞ 0
(5.13)
k=1
for any finite t (a.s.), which is proved by using estimates similar to the ones from the proof of Lemma 5.1 and this justifies the possibility to pass to the limit. In case of Theorem 2.1 we also have (5.12) under the same assumptions as above but in the definition of bs we have to replace f t with f t0 and add the term − p( p − 1)It := − p( p − 1)
|u t | p−2 f ti Di u t d x.
Rd
In this situation one proves (5.13) by the same argument as above combined with the following corollary of Hölder’s inequality: p−2
|It | ≤ u t L p f ti L p Di u t L p . 6 Proof of Theorem 2.1 and Corollary 2.2 First we prove the theorem. We use the notation h (ε) as in the proof of Lemma 5.1 taking there a nonnegative ζ ∈ C0∞ with unit integral. By substituting φ ∗ ζ¯ε , where (ε) ζ¯ε (x) = ζε (−x), in place of φ in (2.1) we see that u t satisfies
123
Itô’s formula for the L p -norm
601
∞ t (ε) (ε) u t∧τ , φ = (u 0 , φ) + Is≤τ gsk(ε) , φ dwsk k=1 0
t +
Is≤τ − f si(ε) , Di φ + f s0(ε) , φ ds,
(6.1)
0
(ε)
k(ε)
which is (5.1) with u 0 and gt
in place of u 0 and gtk , respectively, and with
Di
fi
(ε)
(ε) + f0
in place of f t . From Lemma 5.1 we obtain that, for an ε with P(ε ) = 1, u (ε) t∧τ Iε is a continuous L p -valued Ft -adapted process on [0, ∞) and the corresponding counterpart of (5.2) holds, which after integrating by parts leads to the fact that with probability one for all t ≥ 0 t∧τ (ε) p (ε) p (ε) p−2 (ε) k(ε) u s gs d x dwsk u t∧τ d x = u 0 d x + p u s Rd
Rd
⎛
0 Rd
t∧τ p−2 (ε) p−2 i(ε) ⎜ (ε) 0(ε) u p u (ε) + u f − p( p − 1) f s Di u (ε) ⎝ s s s s s 0
Rd
⎞ p−2 2 (ε) ⎟ d x ⎠ ds. + (1/2) p( p − 1) u (ε) gs s 2
(6.2)
We take the supremums with respect to t of both parts and repeat a standard argument which was introduced by E. Pardoux. We will be using (5.7) and the fact that, by the inequality a p−2 bc ≤ a p + b p + c p , a, b, c ≥ 0, we have γ p/( p−2) (ε) p (ε) p−2 i(ε) f s Di u (ε) d x ≤ u s u s d x s T
Rd
Rd
+
T ( p−2)/2 γ p/2
(ε) p (ε) p f s + Du s d x, (6.3) Rd
123
602
N.V. Krylov
where f (ε) = ( f 1(ε) , . . . , f d(ε) ) and γ > 0 is any number. We also use (5.8), the Burkholder–Davis–Gundy inequalities, and the simple observation that
1 T
τ∧T
p (ε) p . u s ds ≤ sup u (ε) s Lp
Lp
t≤τ ∧T
0
Then for an appropriate choice of the parameter γ we find from (6.2) that
(ε) p (ε) p (ε) p E sup u t ≤ E u 0 + (1/4)E sup u t t≤τ ∧T
Lp
Lp
+N T
t≤τ ∧T
T ∧τ
( p−2)/2
E
Lp
p p (ε) p gs + f s(ε) + Du (ε) s Lp
Lp
Lp
0 T ∧τ
0(ε) p f s ds
+N T p−1 E ⎛ ⎜ +N1 E ⎝
Lp
0 T ∧τ
0
⎛ ⎜ ⎝
⎞2 ⎞1/2 ⎟ ⎟ p−1 (ε) |u (ε) gs d x ⎠ ds ⎠ , s | 2
Rd
where the last expectation is estimated by
⎛ ⎞1/2 p−1 T∧τ 2 (ε) (ε) ⎝ E sup u t gs ds ⎠ Lp
t≤τ ∧T
Lp
0
⎛ ⎞1/ p p−1 T∧τ p (ε) ⎝ ≤ T ( p−2)/(2 p) E sup u (ε) gs ds ⎠ t Lp
t≤τ ∧T
≤ (4N1 )
123
−1
Lp
0
T ∧τ (ε) p (ε) p ( p−2)/2 E sup u t + N T E gs ds. t≤τ ∧T
Lp
Lp
0
ds
Itô’s formula for the L p -norm
603
Hence, p (ε) p (ε) p u u ≤ E + (1/2)E sup E sup u (ε) t t 0 t≤τ ∧T
Lp
Lp
Lp
t≤τ ∧T
T ∧τ
+N T ( p−2)/2 E
p p (ε) p gs + f s(ε) + Du (ε) s Lp
Lp
Lp
ds
0 T ∧τ
0(ε) p f s ds.
+N T p−1 E
Lp
0
Upon collecting like terms we come to T ∧τ (ε) p 0(ε) p (ε) p p−1 E sup u t ≤ 2E u 0 + N T E f s ds t≤τ ∧T
Lp
Lp
Lp
0
+N T
( p−2)/2
T ∧τ
p p (ε) p gs + f s(ε) + Du (ε) s
E
Lp
Lp
Lp
ds.
0
(6.4) One can lawfully object that the last step leads to estimate (6.4) only if its left-hand (ε) p side is finite. However, by Lemma 5.1 the process u t L p is a continuous Ft -adapted
process which starts at u (ε) 0 L p and we can stop it at time τn when it first reaches the p
(ε) p
level u 0 L p + n with n > 0 or at time τ whichever comes first. Then (ε) p (ε) p E sup u t ≤ E u 0 + n < ∞. Lp
t≤τn ∧T
Lp
Hence, the left-hand side of (6.4) will be finite if we replace there τ with τn . Therefore, thus modified (6.4) holds and sending n to infinity yields (6.4) as is. (ε ) (ε ) By applying this result to u t 1 − u t 2 we conclude that (ε ) (ε ) p E sup u t 1 − u t 2 → 0 t≤τ ∧T
Lp
as ε1 , ε2 → 0. It follows that there exists a function vt = vt (ω, x), 0 ≤ t ≤ τ (ω), t < ∞, x ∈ Rd , such that vt∧τ is continuous in t and Ft -adapted as an L p -valued function and p (ε) E sup u t − vt → 0 t≤τ ∧T
Lp
(6.5)
123
604
N.V. Krylov
as ε → 0. In particular, in probability, for any φ ∈ C0∞ , we have
(ε) u t∧τ , φ → (vt∧τ , φ)
(6.6)
uniformly on [0, T ] for any T ∈ [0, ∞). Also in probability, p T T d p i(ε) j (ε) 0(ε) i 0 i f f f + ds − f , D ϕ − f , ϕ ≤ N − f ds i s s s s s s Lp j=0 0
0
→ 0, and (cf. (4.6)) T p/2 ∞ T p 2 k(ε) k gs − gs , ϕ ds ≤ N gs(ε) − gs ds → 0. Lp k=1 0
0
Therefore, we can pass to the limit in (6.1) and conclude that (1.1) holds with v in place of u. The same argument as in the proof of Lemma 5.1 now shows that with probability one the generalized functions vt∧τ and u t∧τ coincide for all t ∈ [0, ∞). This proves the assertion (i) of the theorem. After that (2.3) is obtained by sending ε → 0 in (6.4). Finally, the argument in the proof of Lemma 5.1 can also be repeated almost literally to obtain formula (2.2) from (6.2). The theorem is proved. Proof of Corollary 2.2 Denote by J (τ ) the left-hand side of (2.4). Estimates similar to (5.7) and (6.3) show that J (τ ) < ∞ and if we have a sequence of stopping times τn ↑ τ , then J (τn ) → J (τ ) as n → ∞. By taking a sequence which localizes the stochastic integral in (2.2), then taking expectations of both sides of (2.2), and, finally, using Fatou’s lemma, we obtain the first assertion of the corollary. If τ is bounded, then the described procedure will yield the second assertion as well, due to (2.3) and the dominated convergence theorem. Acknowledgments The author is sincerely grateful to M. Veraar for the discussion of the first draft of the article and useful suggestions. I also owe my sincere gratitude to the referees of the paper for careful reading and many useful comments.
References 1. Brze´zniak, Z., van Neerven, J.M.A.M., Veraar, M.C., Weis, L.: Ito’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differ. Equ. 245, 30–58 (2008) 2. Kallianpur, G., Striebel, C.: Stochastic differential equations occurring in the estimation of continuous parameter stochastic processes. Teor. Verojatnost. i Primenen. 14(4), 597–622 (1969) 3. Krylov, N.V.: An analytic approach to SPDEs. In: Stochastic Partial Differential Equations: Six Perspectives, Mathematical Surveys and Monographs, vol. 64, pp. 185–242. AMS, Providence, RI (1999) 4. Krylov, N.V.: On parabolic PDEs and SPDEs in Sobolev spaces W p2 without and with weights. In: Chow, P.-L., Mordukhovich, B., Yin, G. (eds.) Topics in Stochastic Analysis and Nonparametric Estimation. IMA Volumes in Mathematics and its Applications, vol. 145, pp. 151–198. Springer, New York (2008)
123
Itô’s formula for the L p -norm
605
5. Krylov, N.V.: On divergence form SPDEs with VMO coefficients. SIAM J. Math. Anal. 40(6), 2262– 2285 (2009) 6. Krylov, N.V.: Filtering equations for partially observable diffusion processes with Lipschitz continuous coefficients. In: The Oxford Handbook of Nonlinear Filtering. Oxford University Press, Oxford (To appear) 7. Krylov, N.V., Rozovsky, B.L.: Stochastic evolution equations, “Itogy nauki i tekhniki” 14, VINITI, Moscow, 71–146 (1979). In Russian; English translation in J. Soviet Math. 16(4), 1233–1277 (1981) 8. Rozovskii, B.L.: Stochastic Evolution Systems. Kluwer, Dordrecht (1990) 9. van Neerven, J.M.A.M., Veraar, M.: On the stochastic Fubini theorem in infinite dimensions. In: Stochastic Partial Differential Equations and Applications—VII. Volume 245 of Lect. Notes Pure Appl. Math., pp. 323–336. Chapman & Hall/CRC, Boca Raton, FL (2006)
123
