aa r X i v : . [ m a t h . P R ] F e b ON POTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 N.V. KRYLOV
Abstract.
This paper is a natural continuation of [12], where strongMarkov processes are constructed in time inhomogeneous setting withBorel measurable uniformly bounded and uniformly nondegenerate dif-fusion and drift in L d +1 ( R d +1 ). Here we study some properties of theseprocesses such as the probability to pass through narrow tubes, highersummability of Green’s functions, and so on. The results seem to benew even if the diffusion is constant. Introduction
Let R d be a Euclidean space of points x = ( x , ..., x d ), d ≥
2. Fix some p , q ∈ [1 , ∞ ) such that dp + 1 q = 1 . (1.1)It is proved in [12] that Itˆo’s stochastic equations of the form x s = x + Z s σ ( t + r, x r ) dw r + Z s b ( t + r, x r ) dr (1.2)admit weak solutions, where w s is a d -dimensional Wiener process, σ is auniformly nondegenerate, bounded, Borel function with values in the set ofsymmetric d × d matrices, b is a Borel measurable R d - valued function givenon R d +1 = ( −∞ , ∞ ) × R d such that Z R (cid:16) Z R d | b ( t, x ) | p dx (cid:17) q /p dt < ∞ (1.3)if p ≥ q or Z R d (cid:16) Z R | b ( t, x ) | q dt (cid:17) p /q dx < ∞ (1.4)if p ≤ q . Observe that the case p = q = d + 1 is not excluded and in thiscase the condition becomes b ∈ L d +1 ( R d +1 ).The goal of this article is to study some properties of such solutions orMarkov processes whose trajectories are solutions of (1.2). In particular,in Section 2 for more or less general processes of diffusion type we derive Mathematics Subject Classification.
Key words and phrases.
Itˆo’s equations with singular drift, Potentials of diffusionprocesses. several estimates of Aleksandrov type by using Lebesgue spaces with mixednorms like (in case t = 0, x = 0 in (1.2)) E Z ∞ e − t f ( t, x t ) dt ≤ N k f k L p,q , (1.5)provided that d/p + 1 /q ≤ t, x t ), starting at(0 , , R ) × { x : | x | < R } is comparable to R . This playsa crucial role in Section 4 where we show a significant improvement of theAleksandrov estimates in the direction of lowering the powers of summabilityof f in (1.5) to d /p + 1 /q ≤ d < d . Time homogeneous versions ofthese estimates are also given.In the same Section 2 we give some estimates of the distribution of theexit times from cylinders, which are heavily used in the sequel. We alsoprove that, for any 0 ≤ s ≤ t < ∞ , E sup r ∈ [ s,t ] | x r − x s | n ≤ N ( | t − s | n/ + | t − s | n ) . It is to be said that instead of (1.3) or (1.4), which are not invariant underself-similar transformations, we impose a slightly stronger assumption on b ,that is invariant.As we mentioned above, in Section 4 we improve the results of Section 2in what concerns the Aleksandrov estimates, which allows us to prove Itˆo’sformula for W , p,q ( Q )-functions if d /p + 1 /q ≤ u ∈ W , p,q with d /p + 1 /q ≤ b and p = q = d + 1 in the parabolic case and p = d in the elliptic case the resultof Theorem 5.1 is “classical” (about 50 year old). It was generalized byCabr´e [2], Escauriaza [4], and Fok [6] in the elliptic case when p < d (closeto d ) again when b is bounded. In [3] a parabolic version of these results,extending some earlier results by Wang, are given for L p -viscosity solutionswith p < d + 1 (close to d + 1) when b is bounded. However, it is worthnoting that in the elliptic case it may happen that b L d and the equation isstill solvable (see, for instance, [8]). In our situation we have some freedomin choosing p, q and b ∈ L p ,q , but we only treat true solutions. Theorem5.1 covers Theorem 2.4 of [3] on the account of having mixed norms and b ∈ L p ,q .One more results in this section is aimed at applications to the theoryof fully nonlinear parabolic equations with lower order coefficients in L p,q .We prove a theorem allowing one to pass to the limit under the sign of fullynonlinear operator when the arguments (functions) converge only weaklyand give its application to linear equations. OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 It is worth mentioning that there is a vast literature about stochasticequations when (1.1) is replaced with d/p + 2 /q ≤
1. This condition ismuch stronger than ours. Still we refer the reader to the recent articles[14], [1], [18] and the references therein for the discussion of many powerfuland exciting results obtained under this stronger condition. There are alsomany papers when this condition is considerably relaxed on the account ofimposing various regularity conditions on σ and b and/or considering randominitial conditions with bounded density, see, for instance, [20], [19] and thereferences therein. Restricting the situation to the one when σ and b areindependent of time allows one to relax the above conditions significantlyfurther, see, for instance, [8] and the references therein.Introduce B R = { x ∈ R d : | x | < R } , B R ( x ) = x + B R , C T,R = [0 , T ) × B R ,C T,R ( t, x ) = ( t, x ) + C T,R , C R ( t, x ) = C R ,R ( t, x ) , C R = C R (0 , ,D i = ∂∂x i , D ij = D i D j ∂ t = ∂∂t . For p, q ∈ [1 , ∞ ] and domains Q ⊂ R d +1 we introduce the space L p,q ( Q ) asthe space of Borel functions on Q such that k f k qL p,q ( Q ) := Z R (cid:16) Z R d I Q ( t, x ) | f ( t, x ) | p dx (cid:17) q/p dt < ∞ if p ≥ q or k f k pL p,q ( Q ) := Z R d (cid:16) Z R I Q ( t, x ) | f ( t, x ) | q dt (cid:17) p/q dx < ∞ if p ≤ q with natural interpretation of these definitions if p = ∞ or q = ∞ .If Q = R d +1 , we drop Q in the above notation. Observe that p is associatedwith x and q with t and the interior integral is always elevated to the power ≤
1. In case p = q = d + 1 we abbreviate L d +1 ,d +1 = L d +1 . For the set offunctions on R d summable to the p th power we use the notation L p ( R d ).If Γ is a measurable subset of R d +1 we denote by | Γ | its Lebesgue measure.The same notation is used for measurable subsets of R d with d -dimensionalLebesgue measure. We hope that it will be clear from the context whichLebesgue measure is used. If Γ is a measurable subset of R d +1 and f is afunction on Γ we denote – Z Γ f dxdt = 1 | Γ | Z Γ f dxdt. In case f is a function on a measurable subset Γ of R d we set – Z Γ f dx = 1 | Γ | Z Γ f dx. Throughout the article ¯ R is a fixed constant, ¯ R ∈ (0 , ∞ ). N.V. KRYLOV The case of general diffusion type processes with drift in L p ,q Let (Ω , F , P ) be a complete probability space, let F t , t ≥
0, be an in-creasing family of complete σ -fields F t ⊂ F , let w t be an R d -valued Wienerprocess relative to F t . Fix δ ∈ (0 ,
1) and denote by S δ the set of d × d symmetric matrices whose eigenvalues are between δ and δ − . Assume thatwe are given an S δ -valued F t -adapted process σ t = σ t ( ω ) and an R d -valued F t -adapted process b t , such that Z T | b t | dt < ∞ for any T ∈ (0 , ∞ ) and ω . Define x t = Z t σ s dw s + Z t b s ds. Assumption 2.1.
We are given a function h ∈ L p ,q , loc such that | b t | ≤ h ( t, x t ) . Furthermore, there exists a bounded nondecreasing function ¯ b R , R ∈ (0 , ∞ ),such that for any ( t, x ) ∈ R d +1 and R ∈ (0 , ∞ ) we have k h k q L p ,q ( C R ( t,x )) ≤ ¯ b R R. (2.1)Observe that if p = q = d + 1 and h ∈ L d +2 (a typical case in thetheory of parabolic equations), then (2.1) is satisfied with ¯ b R = ¯ b = k h k d +1 L d +2 ,because by H¨older’s inequality k h k d +1 L d +1 ( C R ( t,x )) ≤ ¯ bR. On the other hand, it may happen that (2.1) is satisfied with p = q = d + 1but h L d +2 , loc . Example . Take α ∈ (0 , d ) , β ∈ (0 ,
1) such that α + 2 β = d + 1 andconsider the function g ( t, x ) = | t | − β | x | − α . Observe that Z C R ( t,x ) g ( s, y ) dyds = R Z C ( t ′ ,x ′ ) g ( s, y ) dyds, where t ′ = t/R , x ′ = x/R . Obviously, the last integral is a boundedfunction of ( t ′ , x ′ ). Hence, the function h = g / ( d +1) satisfies (2.1) with p = q = d + 1. As is easy to see for any p > d + 1 one can find α and β above such that h L p, loc .Note that if h is bounded and has compact support, (2.1) is certainlysatisfied. A condition very similar to (2.1) first appeared in [15].The following is a particular case of Theorem 4.5 of [12]. In Theorem 2.1 p = ∞ is allowed (and then q = 1). OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 Theorem 2.1.
Suppose that Assumption 2.1 is satisfied and p, q ∈ [1 , ∞ ] , dp + 1 q = 1 . (2.2) Then for any Borel f ≥ and stopping time γE Z γ f ( t, x t ) dt ≤ N ( d, p , δ ) (cid:16) A + k h k q L p ,q (cid:17) d/ (2 p ) k f k L p,q , (2.3) where A = Eγ . Our first goal is to estimate A and eliminate it from (2.3). For T, R ∈ (0 , ∞ ) introduce τ T,R ( x ) = inf { t ≥ t, x + x t ) C T,R } , τ R ( x ) = τ R ,R ( x ) , τ R = τ R (0) . Lemma 2.2.
We have A := Eτ R ( x ) ≤ R , (2.4) and consequently, assuming (2.2) , for any Borel nonnegative fE Z τ R ( x )0 f ( t, x t ) dt ≤ N ( d, p , δ )(1 + ¯ b R ) d/p R d/p k f k L p,q . (2.5)Proof. Obviously, τ R ≤ R and (2.4) follows. After that (2.5) followsfrom (2.1) and (2.3). The lemma is proved.Estimate (2.4) says that in the typical case of nondegenerate diffusion τ R is of order not more than R . A very important fact which is implied byCorollary 2.7 is that τ R is of order not less than R . To show this we needan additional assumption appearing after the following result, in which τ ′ R ( x ) = inf { t ≥ x + x t B R } , γ R ( x ) = inf { t ≥ x + x t ∈ ¯ B R } . (2.6) Theorem 2.3.
There are constants ¯ ξ = ¯ ξ ( d, δ ) ∈ (0 , and ¯ N = ¯ N ( d, p , δ )continuously depending on δ such that if, for an R ∈ (0 , ∞ ) , we have ¯ N ¯ b R ≤ , (2.7) then for | x | ≤ RP ( τ R ( x ) = R ) ≤ − ¯ ξ, P ( τ R = R ) ≥ ¯ ξ. (2.8) Moreover for n = 1 , , ... and | x | ≤ RP ( τ ′ R ( x ) ≥ nR ) = P ( τ nR ,R ( x ) = nR ) ≤ (1 − ¯ ξ ) n , (2.9) so that Eτ ′ R ( x ) ≤ N ( d, δ ) R and I := E Z τ ′ R ( x )0 h ( t, x t ) dt ≤ N ( d, p , δ )(1 + ¯ b R ) d/p ¯ b /q R R. (2.10) Furthermore, the probability starting from a point in the closed ball ofradius R/ with center in ¯ B R/ to reach the ball ¯ B R/ before exiting from B R is bigger than ¯ ξ : for any x, y with | y | ≤ R/ and | x − y | ≤ R/ P ( τ ′ R ( x ) > γ R/ ( x )) ≥ ¯ ξ. (2.11) N.V. KRYLOV
Remark . The last statement of the theorem might look awkward becauseit just says that for any x ∈ ¯ B R/ estimate (2.11) holds. Mentioning y seems superfluous. The goal of introducing y is that (2.11) shows thatstarting from any point in B R/ ( y ) the process reaches ¯ B R/ with positiveprobability without exiting from B R , thus “moves in the direction” of − y ,no matter where in B R/ ( y ) the starting point is.We first prove an auxiliary result, in which m t = Z t σ s dw s , a t = (1 / σ t σ ∗ t . Lemma 2.4. (i) There exists κ = κ ( d ) > such that for ψ ( x, t ) = R − (cid:0) R − | x | (cid:1) φ t , φ t = exp Z t κR − tr a s ds the process ψ ( m t , t ) is a local submartingale.(ii) Take a ζ ∈ C ∞ ( R ) such that it is even, nonnegative, and decreasing on (0 , ∞ ) . For T ∈ (0 , ∞ ) and x ∈ R and t ≤ T define u ( t, x ) = Eζ ( x + w T − t )) .Also take x ∈ R d and set r t = ( x + m t , a t ( x + m t )) | x + m t | (0 / , η t = 2 Z t r s ds. Then the process u ( η t , | x + m t | ) is a supermartingale before η t reaches T , inparticular, on [0 , δ T ] .(iii) There exists α = α ( d, δ ) > such that for u ( x ) = | x | − α and anynonzero x ∈ R d the process u ( | x + m t | ) is a submartingale before x + m t hitsthe origin. Proof. It is easy to see that for a κ = κ ( d ) > κµ − µ +32 d − (1 − µ ) ≥ µ , which implies that for all λκ (1 − λ ) − − λ ) + 128 d − λ ≥ . (2.12)It follows that R φ − t dψ ( m t , t ) = κ (cid:0) R − | m t | (cid:1) R − tr a t dt − (cid:0) R − | m t | (cid:1)(cid:0) m t dm t + 2tr a t dt (cid:1) + 128( m t , a t m t ) dt ≥ dM t , where M t is a local martingale. This proves (i).(ii) Observe that u is smooth, even in x , and satisfies ∂ t u + (1 / u ′′ = 0.Furthermore, as is easy to see u ′ ( t, x ) ≤ x ≥
0. It follows by Itˆo’sformula that before η t reaches T we have (dropping obvious values of somearguments) du ( η t , | x + m t | ) = r t (2 ∂ t u + u ′′ ) dt + u ′ | x + m t | (tr a t − r t ) dt + dM t , where M t is a stochastic integral. Here the second term with dt is negativesince u ′ ≤
0, and this proves that u ( η t , | x + m t | ) is a local supermartingale.Since it is nonnegative, it is a supermartingale. OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 Assertion (iii) is proved by simple application of Itˆo’s formula (see, forinstance, the proof of Lemma 2.2 in [17]). The lemma is proved.
Proof of Theorem 2.3 . Notice that by (2.5) E Z τ R | b ( t, x t ) | dt ≤ N (1 + ¯ b R ) d/p ¯ b /q R R. (2.13)Furthermore, observe that for γ defined as the minimum of R and thefirst exit time of m t from B R/ it holds that φ γ ≤ e κd/δ . Hence, by Lemma2.4 (i) 1 = ψ (0 , ≤ Eψ ( m γ , γ ) ≤ e κd/δ P ( sup t ≤ R | m t | < R/ τ R ≤ R , P ( sup t ≤ τ R | m t | < R/ ≥ ξ ( d, δ ) > . Also note that P ( τ R < R ) ≤ P (cid:0) Z τ R | b ( t t , x t ) | dt ≥ R/ (cid:1) + P ( sup t ≤ τ R | m t | ≥ R/ . Therefore we, get the right estimate in (2.8) for 2 N (1 + ¯ b R ) d/p ¯ b /q R ≤ ¯ ξ .On the other hand, take ζ such that ζ ( x ) = η ( x/R ), where η ( x ) = 1 for | x | ≤ T = δ R , in which case u (0 , x ) ≤ u (0 , < u (0 , δ (and η ). Also define µ as the first time η t reaches T , whichis certainly less than or equal to R . Now observe that u ( η µ , | x + m µ | ) = u ( T, | x + m µ | ) = ζ ( | x + m µ | ). It follows that P ( sup t ≤ R | x + m t | < R ) ≤ P ( | x + m µ | < R ) ≤ Eu ( η µ , | x + m µ | ) ≤ u (0 , x ) ≤ u (0 , . Hence, P ( τ R ( x ) < R ) ≥ P (cid:0) Z τ R | b ( t t , x t ) | dt ≤ R/ , sup t ≤ R | x + m t | ≥ R (cid:1) ≥ − P (cid:0) Z τ R | b ( t t , x t ) | dt ≥ R/ (cid:1) − P ( sup t ≤ R | x + m t | ≤ R )and it is clear how to adjust (2.7) to get both inequalities in (2.8) with per-haps ¯ ξ different from the above one. Estimate (2.9) is obtained by iterations.To prove (2.10) come back to (2.13) and denote by J its right-hand side.Then use the condition version of (2.13) to see that ( τ ,R := 0) I = ∞ X n =1 EI τ ( n − R ,R ( x ) >τ ′ R ( x ) E (cid:16) Z τ nR ,R ( x ) τ ( n − R ,R ( x ) h ( t, x t ) dt | F τ ( n − R ,R ( x ) (cid:17) ≤ J ∞ X n =1 P ( τ ( n − R ,R ( x ) = ( n − R ) ≤ J ∞ X n =1 (1 − ¯ ξ ) n − . This yields (2.10).
N.V. KRYLOV
To prove (2.11) use assertion (iii) of Lemma 2.4 to conclude that du ( | x + x t | ) ≥ b it D i u ( | x + x t | ) dt + dM t , where M t is a local martingale. For our x , on the time interval, which we de-note (0 , ν ), when x + x t ∈ B R \ ¯ B R/ we have | Du ( | x + x t | ) t ≤ N ( d, α ) R − α − Furthermore, at starting point u ( x ) ≥ (9 R/ − α . Consequently and by(2.10) (9 R/ − α ≤ N R − α − E Z τ ′ R ( x )0 h ( t, x t ) dt + P (cid:0) ν = τ ′ R ( x ) (cid:1) R − α + P (cid:0) ν = γ R/ ( x ) (cid:1) ( R/ − α , (16 / α ≤ N (1 + ¯ b R ) d/p ¯ b /q R + 1 − P (cid:0) τ ′ R ( x ) > γ R/ ( x ) (cid:1) + 16 α P (cid:0) τ ′ R ( x ) > γ R/ ( x ) (cid:1) . It follows easily that (2.11) holds with ¯ ξ perhaps different from the aboveones, once a relation like (2.7) holds. The continuity of ¯ N in (2.7) and of ξ ( d, δ ) with respect to δ is established by inspecting the above proof. Thetheorem is proved. Assumption 2.2.
There exists R ∈ (0 , ∞ ) such that¯ N ( d, p , δ )¯ b R < . (2.14)This assumption as well as Assumption 2.1 is supposed to hold throughoutthe article. Set λ = R − . Corollary 2.5.
For µ ∈ [0 , and R ≤ R we have Ee − µR − τ R ≤ e − µ ¯ ξ/ . (2.15)Indeed, the derivative with respect to µ of the left-hand side of (2.15) is − R − Eτ R e − R − µτ R ≤ − e − µ R − P ( τ R = R ) ≤ − e − µ ¯ ξ, where the last inequality follows from (2.8). By integrating we find Ee − µR − τ R − ≤ ( e − µ −
1) ¯ ξ, which after using e − µ − ≤ − µ/ , − µ ¯ ξ/ ≤ e − µ ¯ ξ/ leads to (2.15). Theorem 2.6.
For any λ, R > we have Ee − λτ R ≤ e ¯ ξ/ e − √ ˙ λR ¯ ξ/ = ( e ¯ ξ/ e −√ λR ¯ ξ/ if λ ≥ λe ¯ ξ/ e − λRR ¯ ξ/ if λ ≤ λ, (2.16) where ˙ λ = λ min(1 , λ/λ ) . OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 In particular, for any
R > and t ≤ RR ¯ ξ/ we have P ( τ R ≤ t ) ≤ e ¯ ξ/ exp (cid:16) − ¯ ξ R t (cid:17) . (2.17)Proof. Take an integer n ≥
1, introduce τ k , k = 1 , ..., n , as the first exittime of ( t t , x t ) from C R/n ( τ k − , x τ k − ) after τ k − ( τ := 0). If λ ≤ n /R , R/n ≤ R, that is n ≥ R √ λ max(1 , / ( √ λR )) , then by (2.15) with µ = ( R/n ) λ we have E (cid:16) e − λ ( τ k − τ k − ) | F τ k − (cid:17) ≤ e − ( R/n ) λ ¯ ξ/ . Hence, Ee − λτ R ≤ E n Y k =1 e − λ ( τ k − τ k − ) ≤ e − R n − λ ¯ ξ/ . (2.18)If λR ≥
1, we take n = ⌈ R √ λ ⌉ and use that R n − λ ≥ R √ λ −
1. If λR ≤
1, we take n = ⌈ R/R ⌉ and use that R n − λ ≥ RRλ −
1. This proves(2.16).To prove (2.17) observe that if λ ≥ λP ( τ R ≤ t ) = P (cid:0) exp( − λτ R ) ≥ exp( − λt ) (cid:1) ≤ exp( ¯ ξ/ λt − √ λR ¯ ξ/ . For √ λ = R ¯ ξ/ (4 t ) we get (2.17) provided R ¯ ξ/ (4 t ) ≥ R − . The theorem isproved.Recall that ¯ R is fixed throughout the article. Corollary 2.7.
Let Λ ∈ (0 , ∞ ) . Then there is a constant N = N ( R, ¯ R, Λ , ¯ ξ ) such that for any R ∈ (0 , ¯ R ] , λ ∈ [0 , Λ] N Eτ R ≥ R , N E Z τ R e − λt dt ≥ R . (2.19)Indeed, for any ν ≤ R ¯ ξ/ (4 ¯ R ) and R ∈ (0 , ¯ R ] we have νR ≤ RR ¯ ξ/ Eτ R ≥ νR P ( τ R > νR ) ≥ νR (cid:16) − e ¯ ξ/ exp (cid:16) − ¯ ξ ν (cid:17)(cid:17) ,E Z τ R e − λt dt = λ − E (1 − e − λτ R ) ≥ λ − EI τ R >νR (1 − e − λνR )= λ − P ( τ R > νR )(1 − e − λνR ) ≥ λ − (cid:16) − e ¯ ξ/ exp (cid:16) − ¯ ξ ν (cid:17)(cid:17) (1 − e − λνR ) , which yields (2.19) for an appropriate small ν = ν ( R, ¯ R, Λ , ¯ ξ ) > Corollary 2.8.
For any n > and ≤ s ≤ t we have E sup r ∈ [ s,t ] | x r − x s | n ≤ N ( | t − s | n/ + | t − s | n ) , (2.20) where N = N ( n, R, ¯ ξ ) . Indeed, clearly we may assume that s = 0. Then for ν = 4( R ¯ ξ ) − and µ ≥ tν we have t ≤ µR ¯ ξ/ P (sup r ≤ t | x r | ≥ µ ) ≤ P ( τ µ ≤ t ) ≤ e ¯ ξ/ exp (cid:16) − µ ¯ ξ t (cid:17) . Consequently, E sup r ≤ t ] | x r | n = n Z ∞ µ n − P (sup r ≤ t | x r | ≥ µ ) dµ ≤ n Z tν µ n − dµ + ne ¯ ξ/ Z ∞ µ n − exp (cid:16) − µ ¯ ξ t (cid:17) dµ, and the result follows.A few more general results are related to going through a long “sausage”. Theorem 2.9.
Let R ∈ (0 , R ] , x, y ∈ R d and | x − y | ≥ R . For r > denote by S r ( x, y ) the open convex hull of B r ( x ) ∪ B r ( y ) . Then there exist T , T , depending only on ¯ ξ , such that < T < T < ∞ and the probability π that x + x t will reach ¯ B R/ ( y ) before exiting from S R ( x, y ) and this willhappen on the time interval [ nT R , nT R ] is greater than π n , where n = j | x − y | + R R k and π = ¯ ξ/ . Proof. We may assume that y = 0. Introduce τ ( x ) as the first time x + x t reaches ¯ B R/ and γ ( x ) as the first time it exits from S R ( x, | x | ≥ R , we have n ≥ n withthe induction hypothesis that j | x | + R R k = n = ⇒ P ( γ ( x ) > τ ( x ) ∈ [ nT R , nT R ]) ≥ π n . If n = 1, 3 R/ ≤ | x | < R/
16 and by Theorem 2.3 we have P ( τ ′ R ( x ) >τ ( x )) ≥ ¯ ξ . Furthermore, in light of Theorem 2.3, there is T = T ( ¯ ξ ) suchthat P ( τ ′ R ( x ) > T R ) ≤ ¯ ξ/
3. Using (2.17) we also see that there is T = T ( ¯ ξ ) < T such that P ( τ ( x ) ≤ T R ) ≤ ¯ ξ/
3. Hence, P ( γ ( x ) > τ ( x ) ∈ [ T R , T R ]) ≥ ¯ ξ/ π . This justifies the start of the induction.Assuming that our hypothesis is true for some n ≥ n +2) R/ > | x | + R/ ≥ ( n + 1) R/
4. In that case, let z = nRx/ (4 | x | ), τ z bethe first time x + x t reaches ¯ B R/ ( z ), and let γ z be the first time it exitsfrom S R ( x, z ). As is easy to see P ( γ ( x ) > τ ( x ) ∈ [( n + 1) T R , ( n + 1) T R ]) ≥ P ( γ z > τ z ∈ [ T R , T R ] , γ ( x τ z ) > τ ( x τ z ) ∈ [ nT R , nT R ])= EI γ z >τ z ∈ [ T R ,T R ] P (cid:16) γ ( x τ z ) > τ ( x τ z ) ∈ [ nT R , nT R ] | F τ z (cid:17) . Observe that on the set τ z < ∞ we have nR/ ≤ | x τ z | + R/ < ( n + 1) R/ OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 probability above is greater than π n . Then just by shifting the origin to z and using the first part of the proof we obtain our result for n + 1 in placeof n . The theorem is proved. Remark . Observe that, for any fixed x, y , the interval [ nT R , nT R ] isas close to zero as we wish if we choose R small enough. Then, of course,the corresponding probability will be quite small but > Corollary 2.10.
Let R ≤ R , κ ∈ [0 , , and | x | ≤ κR . Then for any T > N P ( τ ′ R ( x ) > T ) ≥ e − νT/ [(1 − κ ) R ] , (2.21) where N and ν > depend only on ¯ ξ . Indeed, passing from B R to B (1 − κ ) R ( x ) shows that we may assume that x = 0 and κ = 0. In that case, consider meandering of x t between ¯ B R/ and ∂B R/ ( y ) where | y | = R/ B R . As is easy to deducefrom Theorem 2.9, given that the n th loop happened, with probability π the next loop will occur and take at least 4 R T of time. Thus the n th loopwill happen and will take at least 4 nR T of time with probability at least π n . It follows that, for any n , P ( τ ′ R ≥ nR T )) ≥ π n , and this yields (2.21) for x = 0 and κ = 0.The following complements Corollary 2.10. Corollary 2.11.
Let R ∈ (0 , ¯ R ] . Then there exists a constant N , dependingonly on ¯ ξ, ¯ R, R , such that, for any
T > , P ( τ ′ R > T ) ≤ N e − T/ ( NR ) . Indeed, if R ≤ R , the result follows from Theorem 2.3. For R ≥ R , take apoint y such that | y | = R + ¯ R , for any x define γ ( x ) as the first time x + x t hits ¯ B R/ ( y ), and set n = j R + 2 ¯ R ) + R R k . It follows from Theorem 2.9 that for any x ∈ B R P ( τ ′ R ( x ) ≤ n T R ) ≥ P ( γ ( x ) ≤ n T R ) ≥ π n . Hence P ( τ ′ R ( x ) > n T R ) ≤ − π n and the result follows from Khasminski’s lemma.3. Mixed norm estimates of potentials of stochastic processes
Here we are moving toward estimating the resolvents of Markov diffusionprocesses in L p,q . Lemma 3.1.
Assume (2.2) . Then there is a constant N , depending onlyon δ, d, p , and ¯ b ∞ , such that for any t ≥ , x ∈ R d , λ > , and Borelnonnegative f vanishing outside C ˙ λ − / ( t , x ) we have E Z ∞ e − λt f ( t, x t ) dt ≤ N ˙ λ − d/ (2 p ) Φ λ ( t , x ) k f k L p,q , (3.1) where Φ λ ( t, x ) = e − √ ˙ λ ( √ t + | x | )¯ ξ/ . Proof. Fix ρ = N ( ¯ ξ ) ˙ λ − / > / R = ρ . Then introduce τ as the first time ( t, x t ) hits¯ C ˙ λ − / ( t , x ) and set γ as the first time after τ the process ( t, x t ) exitsfrom C ˙ λ − , ˙ λ − / + ρ ( t , x ). We define recursively τ k , k = 1 , , ... , as the firsttime after γ k − the process ( t, x t ) hits ¯ C ˙ λ − / ( t , x ) and γ k as the first timeafter τ k the process ( t, x t ) exits from C ˙ λ − , ˙ λ − / + ρ ( t , x ).These stopping times are either infinite or lie between t and t + ˙ λ − .Therefore, the left-hand side of (3.1) equals E ∞ X k =0 e − λτ k I k , (3.2)where I k = I τ k >t E (cid:16) Z γ k ∧ ( t + ˙ λ − ) τ k ∧ ( t + ˙ λ − ) e − ˙ λ ( t − τ k ) f ( t, x t ) dt | F τ k (cid:17) . Here on the set where τ k > t Z γ k ∧ ( t + ˙ λ − ) τ k ∧ ( t + ˙ λ − ) dt = γ k ∧ ( t + ˙ λ − ) − τ k ≤ ˙ λ − . Using this after estimating the norm of h in C ˙ λ − , ˙ λ − / + ρ ( t , x ) we inferfrom (2.3) that I k ≤ N ˙ λ − d/ (2 p ) k f k L p,q , where N = N ( d, p , δ, ¯ b ∞ ).Next, observe that, if √ t > | x | , then τ is bigger than the first exit timeof ( t, x t ) from C √ t , and by Theorem 2.6 Ee − λτ ≤ N e − √ ˙ λ √ t ¯ ξ/ . In case √ t ≤ | x | and | x | > ˙ λ − / our τ is bigger than the first exit timeof ( t, x t ) from C | x |− ˙ λ − / , and Ee − λτ ≤ N e − √ ˙ λ ( | x |− ˙ λ − / )¯ ξ/ . The last estimate (with N = 1) also holds if | x | ≤ ˙ λ − / , so that in case √ t ≤ | x | Ee − λτ ≤ N e − √ ˙ λ | x | ¯ ξ/ and we conclude that in all cases Ee − λτ ≤ N e − √ ˙ λ ( √ t + | x | )¯ ξ/ . OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 Furthermore, by the choice of ρ and Theorem 2.6 E (cid:16) e − λ ( γ k − τ k ) | F τ k (cid:17) ≤ ,Ee − λτ k = Ee − λγ k − E (cid:16) e − λ ( τ k − γ k − ) | F γ k − (cid:17) ≤ Ee − λγ k − , so that Ee − λτ k ≤ Ee − λτ k − , Ee − λτ k ≤ − k Ee − λτ . Recalling (3.2) we see that the left-hand side of (3.1) is dominated by N Φ λ ( t , x ) k f k L p,q and the lemma is proved.The following theorem shows that the time spent by ( t, x t ) in cylinders C (0 , x ) decays very fast as | x | → ∞ . Theorem 3.2.
Suppose that p, q ∈ [1 , ∞ ] , ν := 1 − dp − q ≥ . (3.3) Then there is a constant N = N ( δ, d, p, q, p , ¯ b ∞ ) such that for any λ > and Borel nonnegative f we have I := E Z ∞ e − λt f ( t, x t ) dt ≤ N λ − ν ˙ λ − d/ (2 p ) k Ψ − νλ f k L p,q , (3.4) where Ψ λ ( t, x ) = exp( − p ˙ λ ( | x | + √ t ) ¯ ξ/ . Proof. First assume that ν = 0. Take a nonnegative ζ ∈ C ∞ ( R d +1 ) ofthe type ˙ λ ( d +2) / η ( ˙ λt, p ˙ λx ) with support in C ˙ λ − / and unit integral andfor ( t, x ) , ( s, y ) ∈ R d +1 set f s,y ( t, x ) = f ( t, x ) ζ ( t − s, x − y ) . Clearly, due to Lemma 3.1, I = Z ∞ Z R d E Z ∞ e − λt f s,y ( t, x t ) dt dyds ≤ N ˙ λ − d/ (2 p ) Z ∞ Z R d Φ λ ( s, y ) k f s,y k L p,q dyds. Case p ≥ q . Then q < ∞ and we introduce M /q − = Z ∞ Z R d Φ q/ (2 q − λ ( s, y ) dyds,M q/p − = Z ∞ Z R d Φ pq/ (4 p − q ) λ ( s, y ) dyds, p = q, M = 1 , p = q. It follows by H¨older’s inequality that˙ λ d/ (2 p ) I ≤ N M (cid:16) Z ∞ Z R d Φ q/ λ ( s, y ) Z ∞ (cid:16) Z R d f ps,y ( t, x ) dx (cid:17) q/p dt dyds (cid:17) /q = N M (cid:16) Z ∞ dt (cid:16) Z ∞ Z R d Φ q/ λ ( s, y ) (cid:16) Z R d f ps,y ( t, x ) dx (cid:17) q/p dyds (cid:17)(cid:17) /q ≤ N M M /q (cid:16) Z ∞ dt (cid:16) Z ∞ Z R d Z R d Φ p/ λ ( s, y ) f ps,y ( t, x ) dydsdx (cid:17) q/p (cid:17) /q . We replace Φ p/ λ ( s, y ) by Φ p/ λ ( t, x ) taking into account that these valuesare comparable as long as ζ ( t − s, x − y ) = 0. After that integrating over dyds and computing M , M lead immediately to (3.4). Case p < q . It follows by H¨older’s inequality that˙ λ d/ (2 p ) I ≤ N M (cid:16) Z ∞ Z R d Φ p/ λ ( s, y ) Z R d (cid:16) Z ∞ f qs,y ( t, x ) dt (cid:17) p/q dx dyds (cid:17) /p ≤ N M M (cid:16) Z R d dx (cid:16) Z ∞ Z R d Z ∞ Φ q/ λ ( s, y ) f qs,y ( t, x ) dtdyds (cid:17) p/q (cid:17) /p , where M /p − = Z ∞ Z R d Φ p/ (2 p − λ ( s, y ) dyds,M p/q − = Z ∞ Z R d Φ pq/ (4 q − p ) λ ( s, y ) dyds. This leads to (3.4) as above. The theorem is proved if ν = 0.If ν ∈ (0 , I I , where I /ν = E Z ∞ e − λt dt = 1 /λ,I / (1 − ν )2 = E Z ∞ e − λt f / (1 − ν ) ( t, x t ) dt. Here dp − pν + 1 q − qν = 1so that by the case that ν = 0 I / (1 − ν )2 ≤ N ˙ λ − d/ (2 p − pν ) k Ψ λ f / (1 − ν ) k L p − pν,q − qν = N ˙ λ − d/ (2 p − pν ) k Ψ − νλ f k / (1 − ν ) L p,q . This leads to (3.4) again. Finally, if ν = 1 so that p = q = ∞ , the left-handside of (3.4) is obviously dominated by λ − sup f , so that (3.4) holds with N = 1. The theorem is proved.By taking q = ∞ and f ( t, x ) = f ( x ) we come to the following, whichextends Corollary 2.5 of [11] to the case of time dependent drift b ∈ L p ,q , loc .It is further generalized by relaxing the restriction on p in Theorem 4.8. Corollary 3.3.
Let p ∈ [ d, ∞ ] . Then for any λ > and Borel nonnegative f ( x ) we have E Z ∞ e − λt f ( x t ) dt ≤ N λ − d/p ˙ λ − d/ (2 p ) k Ψ d/pλ f k L p ( R d ) , (3.5) where Ψ λ ( x ) = exp( − p ˙ λ | x | ¯ ξ/ and N = N ( δ, d, p, p , ¯ b ∞ ) . OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 Next results are dealing with the exit times of the process x t rather than( t, x t ). We will need them while showing an improved integrability of Green’sfunctions.Estimate (3.6) below in case b is bounded was the starting point for thetheory of time homogeneous controlled diffusion processes about fifty yearsago. Lemma 3.4.
Let p ∈ [ d, ∞ ] . Then for any Borel nonnegative f ( x ) , R ≤ ¯ R ,and x ∈ R d E Z τ ′ R ( x )0 f ( x t ) dt ≤ N ( δ, d, ¯ b ¯ R , ¯ R, R, p ) R − d/p k f k L p ( R d ) . (3.6)Proof. Define q from (2.2) and observe that for k = 1 , , ... , ∆ k = [( k − R , kR ) and f k ( t, x ) := I ∆ k ( t ) f ( x ) , according to (2.3), on the set where τ ′ R ( x ) ≥ ( k − R we have E (cid:16) Z ( kR ) ∧ τ ′ R ( x )( k − R f ( x t ) dt | F ( k − R (cid:17) = E (cid:16) Z ( kR ) ∧ τ ′ R ( x )( k − R f k ( t, x t ) dt | F ( k − R (cid:17) ≤ N (cid:16) R + k h k q L p ,q (∆ k × B R ) (cid:17) d/ (2 p ) R /q k f k L p ( R d ) ≤ N (1 + ¯ b R ) d/ (2 p ) R − d/p k f k L p ( R d ) . It follows that E Z τ ′ R ( x )0 f ( x t ) dt = ∞ X k =1 EI τ ′ R ( x ) ≥ ( k − R Z ( kR ) ∧ τ ′ R ( x )( k − R f ( x t ) dt ≤ N R − d/p k f k L p ( R d ) ∞ X k =1 P ( τ ′ R ( x ) ≥ ( k − R ) . By Corollary 2.11 each of the probabilities in the last sum is less than
N e − k/N and this proves the lemma.4. Green’s functions
Here is a straightforward consequence of (3.4).
Theorem 4.1.
Assume (3.3) and take λ > . Then there exists a constant N = N ( δ, d, p, q, p , ¯ b ∞ ) and a nonnegative Borel function G λ ( t, x ) (Green’sfunction of ( · , x · ) ) on R d +1 such that G λ ( t, x ) = 0 for t ≤ and for anyBorel nonnegative f given on R d +1 we have E Z ∞ e − λt f ( t, x t ) dt = Z R d +1 f ( t, x ) G λ ( t, x ) dxdt, k Ψ ν − λ G λ k L ′ p,q ≤ N λ − ν ˙ λ − d/ (2 p ) , (4.1) where we use the notation k u k L ′ p,q = (cid:16) Z R (cid:16) Z R d | u ( t.x ) | p ′ dx (cid:17) q ′ /p ′ dt (cid:17) /q ′ if p ≥ q, k u k L ′ p,q = (cid:16) Z R d (cid:16) Z R | u ( t.x ) | q ′ dt (cid:17) p ′ /q ′ dx (cid:17) /p ′ if p < q, and p ′ = p/ ( p − , q ′ = q/ ( q − . The highest power of pure ( p = q = d + 1) summability of G λ guaranteedby this theorem is 1 + 1 /d . It turns out that, actually, G λ is summableto a higher power. The proof of this is based on the parabolic version ofGehring’s lemma from [7].Introduce Q as the set of cylinders C R ( t, x ), R > t ≥ x ∈ R d . For Q = C R ( t, x ) ∈ Q let 2 Q = C R ( t, x ). If Q ∈ Q and Q = C R ( t, x ), we call R the radius of Q . Theorem 4.2.
Let λ ∈ (0 , ∞ ) . Then there exist d ∈ (1 , d ) and a constant N , depending only on δ, d, R, p , λ , such that for any Q ∈ Q of radius R ≤ R/ and p ≥ d + 1 , we have k G λ k L p/ ( p − ( Q ) ≤ N R − ( d +2) /p k G λ k L (2 Q ) , (4.2) which is equivalently rewritten as (cid:16) – Z Q G p/ ( p − λ dxdt (cid:17) ( p − /p ≤ N – Z Q G λ dxdt. Proof. We basically follow the idea in [5]. Take Q ∈ Q of radius R ≤ R/ γ = inf { t ≥ t, x t ) ∈ ¯ Q } , τ = inf { t ≥ γ : ( t, x t ) Q } ,γ n +1 = inf { t ≥ τ n : ( t, x t ) ∈ ¯ Q } , τ n +1 = inf { t ≥ γ n +1 : ( t, x t ) Q } . Then for any nonnegative Borel f vanishing outside Q with k f k L d +1 ( Q ) = 1we have Z Q f G λ ( t, x ) dxdt = E Z ∞ e − λt f ( t, x t ) dt = ∞ X n =1 Ee − λγ n E (cid:16) Z τ n γ n e − λ ( t − γ n ) f ( t, x t ) dt | F γ n (cid:17) . Next we use the conditional version of (2.5) to see that the conditional ex-pectations above are less than
N R d/ ( d +1) . After that we use the conditionalversion of Corollary 2.7 to get that R ≤ N E (cid:16) Z τ n γ n e − λ ( t − γ n ) dt | F γ n (cid:17) . Then we obtain Z Q f G λ ( t, x ) dxdt ≤ N R − ( d +2) / ( d +1) ∞ X n =1 Ee − λγ n E (cid:16) Z τ n γ n e − λ ( t − γ n ) dt | F γ n (cid:17) OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 = N R − ( d +2) / ( d +1) ∞ X n =1 E Z τ n γ n e − λt dt ≤ N R − ( d +2) / ( d +1) E Z ∞ e − λt I Q ( t, x t ) dt = N R − ( d +2) / ( d +1) Z Q G λ ( t, x ) dxdt. The arbitrariness of f implies that (cid:16) – Z Q G ( d +1) /dλ ( t, x ) dxdt (cid:17) d/ ( d +1) ≤ N – Z Q G λ ( t, x ) dxdt. Now the assertion of the theorem for p = d follows directly from theparabolic version of the famous Gehring’s lemma stated as Proposition 1.3in [7]. For larger p it suffices to use H¨older’s inequality. The theorem isproved.The parameters d and N in Theorem 4.2 may depend on δ , d , R , p , λ .What is important for the future is that d below is independent of λ . Theorem 4.3.
There exists d ∈ (1 , d ) , depending only on δ , d , R , p , suchthat for any p ≥ d + 1 and λ > Z ∞ Z R d G p/ ( p − λ ( t, x ) dxdt ≤ N ( δ, d, R, p , λ, p ) . Furthermore, the above constant N ( δ, d, R, p , λ, p ) can be taken in the form N ( δ, d, R, p , p )¨ λ ( d +2) / (2 p ) − p , where ¨ λ p = λ (1 ∧ λ ) d/ (2 p − d − . Proof. Represent R d +1+ = [0 , ∞ ) × R as the union of countably many Q , Q , ... ⊂ Q of radius R/ R d +1+ belongs to no morethan m ( d ) of the 2 Q i ’s and let d be taken from Theorem 4.2 with λ = 1.Then for λ ≥ k G λ k L p/ ( p − ( R d +1+ ) ≤ k G k L p/ ( p − ( R d +1+ ) ≤ k X i I Q i G k L p/ ( p − ( R d +1+ ) ≤ X i k G k L p/ ( p − ( Q i ) ≤ N X i k G k L (2 Q i ) ≤ N k G k L ( R d +1+ ) = N . If λ ∈ (0 ,
1) we take nonnegative f ∈ L p ( R d +1+ ) and observe that J := E Z ∞ e − λt f ( t, x t ) dt = ∞ X n =0 e − λn E Z n +1 n e − λ ( t − n ) f ( t, x t ) dt ≤ ∞ X n =0 e − λ e − λn E Z n +1 n e − ( t − n ) f ( t, x t ) dt. By the first case each expectation in the sum is dominated by N k f I [ n,n +1) k L p .Therefore J ≤ N ∞ X n =0 e − λn k f I [ n,n +1) k L p ≤ N (1 − e − λ ) − ( p − /p k f k L p ( R d + ) , where the second inequality follows from H¨olders inequality. This takes careof the case that λ ∈ (0 ,
1) in both statements of the theorem.To prove the second statement in case λ ≥ t, y t ),where y t = √ λx t/λ . We have y t = Z t σ s/λ d ( √ λw s/λ ) + Z t λ − / b s/λ ds, where √ λw s/λ is a Wiener process and | λ − / b s/λ | ≤ λ − / h ( s/λ, x t/λ ) =: ˜ h ( s, y s ) . Observe that k ˜ h k q L p ,q ( C R ( t,x )) = √ λ k h k q L p ,q ( C R/ √ λ ( tλ,x/ √ λ )) ≤ ¯ b R/ √ λ R ≤ ¯ b R R. It follows that the above theory is applicable to ( t, y t ) and provides estimateswith the same constants as for ( t, x t ). In particular, for any p ≥ d + 1 (with d found above) and Borel nonnegative f ( t, x ) I := E Z ∞ e − t f ( t, y t ) dt ≤ N N ( δ, d, R, p , p ) k f k L p ( R d +1+ ) . After that it only remains to note that I = λE Z ∞ e − λt g ( t, x t ) dt, where g ( t, x ) = f ( λt, √ λx ) and k f k L p ( R d +1+ ) = λ ( d +2) / (2 p ) k g k L p ( R d +1+ ) . The theorem is proved.Similar improvement of integrability occurs for the Green’s function of x t rather than ( t, x t ). Here is a straightforward consequence of (3.5). Theorem 4.4.
Let p ∈ [ d, ∞ ) . Then for any λ > there exists a nonneg-ative Borel function g λ ( x ) (Green’s function of x · ) on R d such that for anyBorel nonnegative f given on R d we have E Z ∞ e − λt f ( x t ) dt = Z R d f ( x ) g λ ( x ) dx, k Ψ − d/pλ g λ k L p ′ ( R d ) ≤ N λ − d/p ˙ λ − d/ (2 p ) , (4.3) where Ψ λ ( x ) = exp( − p ˙ λ | x | θ/ , p ′ = p/ ( p − , and N depends only on δ , d , R, p, p , ¯ b ∞ . OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 According to this theorem this Green’s function is summable to the power d/ ( d − B is anopen ball in R d by 2 B we denote the concentric open ball of twice the radiusof B . Theorem 4.5.
Let λ ∈ (0 , ∞ ) . Then there exist d ∈ (1 , d ) and a constant N , depending only on d, δ , R , λ , such that for any ball B of radius R ≤ R/ and p ≥ d , we have k g λ k L p/ ( p − ( B ) ≤ N R − d/p k g λ k L (2 B ) , (4.4) which is equivalently rewritten as (cid:16) – Z B g p/ ( p − λ dx (cid:17) ( p − /p ≤ N – Z B g λ dx. Proof. We again follow the idea in [5]. Take a ball B of radius R ≤ R/ γ = inf { t ≥ x t ∈ ¯ B } , τ = inf { t ≥ γ : x t B } ,γ n +1 = inf { t ≥ τ n : x t ∈ ¯ B } , τ n +1 = inf { t ≥ γ n +1 : x t B } . Then for any nonnegative Borel f vanishing outside B with k f k L d ( B ) = 1we have Z B f g λ ( x ) dx = E Z ∞ e − λt f ( x t ) dt = ∞ X n =1 Ee − λγ n E (cid:16) Z τ n γ n e − λ ( t − γ n ) f ( x t ) dt | F γ n (cid:17) . Next we use the conditional version of (3.6) to see that the conditionalexpectations above are less than
N R . After that we use the conditionalversion of Corollary 2.7 to get that R ≤ N E (cid:16) Z τ n γ n e − λ ( t − γ n ) dt | F γ n (cid:17) . Then we obtain Z B f g λ ( x ) dx ≤ N R − ∞ X n =1 Ee − λγ n E (cid:16) Z τ n γ n e − λ ( t − γ n ) dt | F γ n (cid:17) = N R − ∞ X n =1 E Z τ n γ n e − λt dt ≤ N R − E Z ∞ e − λt I B ( x t ) dt = N R − Z B g λ ( x ) dx. The arbitrariness of f implies that (cid:16) – Z B g d/ ( d − λ ( x ) dx (cid:17) ( d − /d ≤ N – Z B g λ ( x ) dx, and again it only remains to use Gehring’s lemma in case p = d . For larger p it suffices to use H¨older’s inequality. The theorem is proved. By mimicking the proof of Theorem 4.3 one gets its “elliptic” counterpart.
Theorem 4.6.
There exists d ∈ (1 , d ) , depending only on δ , d , R , p , suchthat for any p ≥ d and λ > Z R d g p/ ( p − λ ( x ) dx ≤ N ( δ, d, R, p , p )¨ λ d/ (2 p ) − e,p , where ¨ λ e,p = λ (1 ∧ λ ) d/ (2 p − d ) . Remark . Below by d we denote the largest of the d ’s from Theorems4.3 and 4.6 and observe that, as the simple example of a ij = δ ij and b ≡ d > d/ Lemma 4.7.
Suppose that p, q ∈ [1 , ∞ ] , d p + 1 q = 1 . (4.5) Then for any Borel f ( t, x ) ≥ I := E Z ∞ e − λt f ( t, x t ) dt ≤ N ¨ λ − (2 d − d ) / (2 p ) d +1 k f k L p,q , (4.6) where N = N ( δ, d, R, p, p ) . Proof. If p = d + 1, then q = d + 1 and (4.6) follows from Theorem 4.3.In other terms, for any Borel f ( t, x ) ≥ E Z ∞ e − λt f ( t, x t ) dt = Z Q G λ ( t, x ) f ( t, x ) dxdt ≤ N ¨ λ − (2 d − d ) / (2 d +2) d +1 k f k L d . If p = d and q = ∞ estimate (4.6) follows from Theorem 4.6 since I ≤ E Z ∞ e − λt sup s ≥ f ( s, x t ) dt = Z R d g λ ( x ) sup s ≥ f ( s, x ) dx ≤ N (cid:16) Z R d sup s ≥ f d ( s, x ) dx (cid:17) /d = N ¨ λ − (2 d − d ) / (2 d ) e,d k f k L d , ∞ and as is easy to check ¨ λ e,d = ¨ λ d +1 .If p = ∞ and q = 1 I ≤ Z ∞ sup x f ( t, x ) dt = k f k L ∞ , . We will use these facts in an interpolation argument. In case ∞ > p >d + 1 we have p > q and set β = p/ ( d + 1) and α = β/ ( β − OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 a nonnegative g ( t ) such that (cid:0) f ( t, x ) g ( t ) (cid:1) /g ( t ) = f ( t, x ) (0 / I ≤ I I , where I = (cid:16) Z ∞ g − α ( t ) dt (cid:17) /α ,I = (cid:16) E Z ∞ e − λt g β ( t ) f β ( t, x t ) dt (cid:17) /β ≤ N ¨ λ − (2 d − d ) / (2 p ) d +1 (cid:16) Z ∞ g ( d +1) β ( t ) (cid:16) Z R d f ( d +1) β ( t, x ) dx (cid:17) dt (cid:17) / ( d β + β ) . For g found from g − α ( t ) = g ( d +1) β ( t ) Z R d f ( d +1) β ( t, x ) dx we get (4.6) and this takes care of the case that ∞ > p > d + 1.If ∞ > q > d + 1 we have p < q and set β = q/ ( q − d −
1) and α = β/ ( β − g ( x ) such that (cid:0) f ( t, x ) g ( x ) (cid:1) /g ( x ) = f ( t, x )(0 / I ≤ I I , where I = (cid:16) E Z ∞ e − λt g − β ( x t ) dt (cid:17) /β ≤ N ¨ λ ( d − d ) / (2 d β ) d +1 (cid:16) Z R d g − d β ( x ) dx (cid:17) / ( d β ) ,I = (cid:16) E Z ∞ e − λt g α ( x t ) f α ( t, x t ) dt (cid:17) /α ≤ N ¨ λ ( d − d ) / (2 d α +2 α ) d +1 (cid:16) Z R d g ( d +1) α ( x ) (cid:16) Z ∞ f ( d +1) α ( t, x ) dt (cid:17) dx (cid:17) / ( αd + α ) . For g found from g − d β ( x ) = g ( d +1) α ( x ) Z ∞ f ( d +1) α ( t, x ) dt we get (4.6) after simple manipulations and this proves the lemma.Using this lemma instead of (2.5) and just repeating the proof of Lemma3.1 we come to a natural counterpart of the latter and then by literallyrepeating the proof of Theorem 3.2 we come to the following result, thatis a version of Theorem 4.1 of Nazarov [16] in which d/p + 1 /q ≤ h is weaker. A properprobabilistic version of Theorem 4.1 of Nazarov [16] is found in [12]. Recallthat R d +1+ = { t ≥ } ∩ R d +1 . Theorem 4.8.
Suppose p, q ∈ [1 , ∞ ] , ν := 1 − d p − q ≥ . (4.7) Then there is N = N ( δ, d, R, p, q, p , ¯ b ∞ ) such that for any λ > and Borelnonnegative f we have E Z ∞ e − λt f ( t, x t ) dt ≤ N ¨ λ − ν +( d − d ) / (2 p ) d +1 k Ψ − νλ f k L p,q ( R d +1+ ) , (4.8) where Ψ λ ( t, x ) = exp( − p ˙ λ ( | x | + √ t ) ¯ ξ/ . In particular, if f is independentof t , p ≥ d , and q = ∞ E Z ∞ e − λt f ( x t ) dt ≤ N ¨ λ − d/ (2 p ) d +1 k ¯Ψ d /pλ f k L p ( R d ) , where ¯Ψ λ ( x ) = exp( − p ˙ λ | x | ¯ ξ/ . Theorem 4.9.
Assume that (4.7) holds. Then(ii) for any n = 1 , , ... , nonnegative Borel f on R d +1+ , and T ≤ we have E h Z T f ( t, x t ) dt i n ≤ n ! N n T nχ k Ψ (1 − ν ) /n /T f k nL p,q ( R d +1+ ) , (4.9) where N = N ( δ, d, R, p, q, p , ¯ b ∞ ) and χ = ν + (2 d − d ) / (2 p ) ;(ii) for any nonnegative Borel f on R d +1+ , and T ≥ we have I := E Z T f ( t, x t ) dt ≤ N T − /q k Ψ − ν f k L p,q ( R d +1+ ) , (4.10) where N = N ( δ, d, R, p, q, p , ¯ b ∞ ) . Proof. The proof of (i) proceeds by induction on n and is achieved byalmost literally repeating the proof of Theorem 2.7 of [11]. The inductionhypothesis is that for all ( t, x ) ∈ R d +1+ and κ ∈ [0 , /n ] E h Z T f ( t + s, x + x s ) ds i n ≤ n ! N n T nχ Ψ ( ν − κn /T ( t, x ) k Ψ (1 − ν ) κ /T f k nL p,q ( R d +1+ ) . (4.11)We will discuss in detail only the case of n = 1. In that case observethat λ := 1 /T ≥ κ ∈ [0 , e − λT E Z T f ( t + s, x + x s ) ds ≤ E Z ∞ e − λs f ( t + s, x + x s ) ds ≤ N λ − χ k f ( t + · , x + · )Ψ (1 − ν ) κλ k L p,q ( R d +1+ ) ≤ N λ − χ Ψ ( ν − κλ ( x ) k f Ψ (1 − ν ) κλ k L p,q ( R d +1+ ) , where the last inequality is due to the fact that, for x, y ∈ R d , Ψ λ ( s, y ) ≤ Ψ λ ( t + s, x + y )Ψ − λ ( t, x ). Since λT = 1, we get (4.11) with n = 1.While proving (4.10) we may assume that T = k , where k ≥ n ≥ E (cid:16) Z n +1 n f ( t, x t ) dt | F n (cid:17) ≤ N k Ψ − ν f I [ n,n +1] k L p,q . Hence, I ≤ N k − X n =0 k Ψ − ν f I [ n,n +1] k L p,q =: N J.
OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 If p ≥ q , we have by H¨older’s inequality J ≤ k ( q − /q k Ψ − ν f k L p,q . If p < q J ≤ k ( p − /p (cid:16) Z R d k − X n =0 (cid:16) Z n +1 n Ψ q (1 − ν )1 f q ( t, x ) dt (cid:17) p/q dx (cid:17) /p ≤ k ( p − /p +( q − p ) / ( qp ) k Ψ − ν f k L p,q , which yields (4.10) since ( p − /p + ( q − p ) / ( qp ) = 1 − /q . The theorem isproved.Next theorem improves estimate (2.5) in what concerns the restrictionson p, q . Theorem 4.10.
Assume that (4.7) holds with ν = 0 . Then for any R ∈ (0 , ¯ R ] , x , and Borel nonnegative f given on C R , we have E Z τ R ( x )0 f ( t, x + x t ) dt ≤ N R (2 d − d ) /p k f k L p,q ( C R ) , (4.12) where N = N ( δ, d, R, p, p , ¯ b ¯ R , ¯ R ) . Proof. Since τ R ( x ) ≤ R , the left-hand side of (4.12) is smaller than e λR E Z ∞ e − λt I C R f ( t, x + x t ) dt for any λ >
0. We estimate the last expectation by using Theorem 4.8,observe that, for λ = R − we have N ¨ λ d +1 ≥ λ owing to R ≤ ¯ R , and thenimmediately come to (4.12), however, with N depending on ¯ b ∞ in place of¯ b ¯ R . To see that this replacement is not needed, it suffices to note that theleft-hand side of (4.12) will not change if we replace b t with b t I t<τ ¯ R ( x ) whichadmits the estimate | b t I t<τ ¯ R ( x ) | ≤ hI C ¯ R (0 , − x ) ( t, x t ) . The theorem is proved.Theorem 4.9 allows us to prove Itˆo’s formula for functions u ∈ W , p,q ( Q ),where Q is a domain in R d +1 and W , p,q ( Q ) = { v : v, ∂ t v, Dv, D v ∈ L p,q ( Q ) } with norm introduced in a natural way. Before, the formula was known onlyfor (smooth, Itˆo, and) W , d +1 -functions and processes with bounded drifts orfor W d -functions in case the drift of the process is dominated by h ( x t ) with h ∈ L d (see [11]).The following extends Theorem 2.10.1 of [9]. Theorem 4.11.
Assume that (4.7) holds with ν = 0 and p < ∞ , q < ∞ . Let Q be a bounded domain in R d +1 , ∈ Q , b be bounded , and u ∈ W , p,q ( Q ) ∩ C ( ¯ Q ) . Then, for τ defined as the first exit time of ( t, x t ) from Q and for all t ≥ , u ( t ∧ τ, x t ∧ τ ) = u (0 ,
0) + Z t ∧ τ D i u ( s, x s ) dm is + Z t ∧ τ [ ∂ t u ( s, x s ) + a ijs D ij u ( s, x s ) + b is D i u ( s, x s )] ds (4.13) and the stochastic integral above is a square-integrable martingale. Proof. First assume that u is smooth and its derivatives are bounded.Then (4.13) holds by Itˆo’s formula and, moreover, by denoting τ n = n ∧ τ for any n ≥ E Z τ n +1 τ n | Du ( s, x s ) | ds ≤ N E (cid:16) Z τ n +1 τ n D i u ( s, x s ) dm is (cid:17) = N E (cid:16) u ( τ n +1 , x τ n +1 ) − u ( τ n , x τ n ) − Z τ n +1 τ n [ ∂ t u ( s, x s ) + a ijs D ij u ( s, x s ) + b is D i u ( s, x s )] ds (cid:17) ≤ N sup ¯ Q | u | + N E (cid:16) Z τ n +1 τ n I Q (cid:0) | ∂ t u | + | Du | + | D u | (cid:1) ( s, x s ) ds (cid:17) . Since Q is bounded, τ is bounded as well and in light of Theorem 4.9 weconclude that E Z τ | Du ( s, x s ) | ds ≤ N sup ¯ Q | u | + N k ∂ t u, Du, D u k L p,q ( Q ) , (4.14)where N are independent of u and Q as long as the size of Q in the t -directionis under control. Owing to Fatou’s theorem, this estimate is also true forthose u ∈ W , p,q ( Q ) ∩ C ( ¯ Q ) that can be approximated uniformly and in the W , p,q ( Q )-norm by smooth functions with bounded derivatives (recall that p < ∞ , q < ∞ ). For our u there is no guarantee that such approximationis possible. However, mollifiers do such approximations in any subdomain Q ′ ⊂ ¯ Q ′ ⊂ Q . Hence, (4.14) holds for our u if we replace Q by Q ′ (containing(0 , Q ′ ↑ Q proves (4.14) in the generals case and proves the lastassertion of the theorem.After that (4.13) with Q ′ in place of Q is proved by routine approximationof u by smooth functions. Setting Q ′ ↑ Q finally proves (4.13). The theoremis proved. 5. Application to parabolic equations
Fix a constant δ ∈ (0 ,
1) and recall that by S δ we denote the set of d × d -symmetric matrices whose eigenvalues are between δ and δ − . In this sectionwe impose the following. OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 Assumption 5.1. (i) On R d +1 we are given Borel measurable σ ( t, x ) and b ( t, x ) with values in S δ and in R d respectively.(ii) We are given p , q ∈ [1 , ∞ ) satisfying (1.1) and a function h ( t, x )satisfying (2.1) and such that | b | ≤ h .(iii) Assumption 2.2 is satisfied.Introduce a = σ and set Lu ( t, x ) = (1 / a ij ( t, x ) D ij u ( t, x ) + b i ( t, x ) D i u ( t, x ) . For a domain Q ⊂ R d +1 one denotes by ∂ ′ Q its parabolic boundary definedas the set of all points on ∂Q which are endpoints of continuous curves oftype ( t, x t ), t ∈ [ a, b ], which start in Q and belong to Q for all t < b .The following has the same flavor as Nazarov’s Theorem 4.1 of [16] orTheorem 4.3 of [12]. We get wider range of p, q on the account of restricting b . Here is a qualitative form of the maximum principle. Theorem 5.1.
Let < R ≤ ¯ R , domain Q ⊂ C R , and assume that (4.7) holds with ν = 0 , p < ∞ , q < ∞ , and that we are given a function u ∈ W , p,q, loc ( Q ) ∩ C ( ¯ Q ) . Take a function c ≥ on Q . Then on Qu ≤ N R (2 d − d ) /p k I Q,u> ( ∂ t u + Lu − cu ) − k L p,q + sup ∂ ′ Q u + , (5.1) where N = N ( δ, d, R, ¯ R, p, p , ¯ b ¯ R ) . In particular (the maximum principle),if ∂ t u + Lu − cu ≥ in Q and u ≤ on ∂ ′ Q , then u ≤ in Q . Proof. Obviously the right-hand side of (5.1) decreases if we replace c with zero. Hence we may assume that c = 0. Also, we need to prove (5.1)only in Q ∩ { u > } on the parabolic boundary of which either u = 0 or u ≤ sup ∂ ′ Q u + . Therefore, we may assume that u > Q .Then for ε > Q ε as the collection of ( t, x ) ∈ Q such that theclosed ball in R d +1 centered at ( t, x ) with radius ε lies in Q . Obviously Q ε isopen. It is not hard to prove (see, for instance, the proof of Lemma 3.1.13 in[10]) that dist ( ∂ ′ Q, ∂ ′ Q ε ) = ε . It follows, owing to the continuity of u andthe monotone convergence theorem, that it suffices to prove (5.1) with Q ε in place of Q . As a consequence of that we may assume that u ∈ W , p,q ( Q ).This gives us the opportunity to replace L in (5.1) with L n := I | b |≥ n ∆ + I | b | Assume that (4.7) holds with p < ∞ , q < ∞ and take R ∈ (0 , ∞ ] . Then there exists constants N, κ > , depending only on d, δ, R, p, q, p , ¯ b ∞ , such that for any λ ≥ and u ∈ W , p,q, loc ( C R ) ∩ C ( ¯ C R ) ( C ∞ = R d +1+ , C ( R d +1+ ) is the set of bounded continuous functions on R d +1+ )we have λ k u + k L p,q ( C R/ ) ≤ N k ( λu − Lu − ∂ t u ) + k L p,q ( C R ) + N λR d/p +2 /q e − κR √ λ sup ∂ ′ C R u + , (5.3) where the last term should be dropped if R = ∞ . Proof. By having in mind the possibility to approximate C R from insideby similar domains, we see that we may assume that u ∈ W , p,q ( C R ) and R < ∞ . Then as in the proof of Theorem 5.1 we reduce the general caseto the one in which b is bounded and after that we may assume that u issmooth. In that case, for ( t, x ) ∈ C R/ , similarly to (5.2) u ( t, x ) = Ee − λτ R u ( t + τ R , x τ R ) − E Z τ R e − λt f ( t + s, x s ) ds =: I ( t, x ) + J ( t, x ) , where f = λu − Lu − ∂ t u , τ R is the first exit time of ( t + s, x s ) from C R and x s is a solution of (1.2).Here, thanks to (2.16) I ( t, x ) ≤ N e − κR √ λ sup ∂ ′ C R u + , k I C R/ I + k L p,q ≤ N R d/p +2 /q e − κR √ λ sup ∂ ′ C R u + , (5.4)where N, κ > d, δ, R, p, q, p .To estimate J we define f as zero outside C R and observe that J ( t, x ) ≤ E Z ∞ e − λt f + ( t + s, x s ) ds =: ¯ J ( t, x )By Theorem 4.8 we have¯ J ( t, x ) ≤ N λ − η k Ψ − νλ f + ( t + · , x + · ) k L p,q ( R d +1+ ) , (5.5)where η = ν + (2 d − d ) / (2 p ). OTENTIALS OF IT ˆO’S PROCESSES WITH DRIFT IN L d +1 If p ≥ q , (5.5) implies that Z R d | λ η ¯ J ( t, x ) | p dx ≤ N Z R d (cid:16) Z ∞ F q/p ( t, s, x ) ds (cid:17) p/q dx, (5.6)where F ( t, s, x ) = Z R d Ψ (1 − ν ) pλ ( s, y ) f p ( t + s, x + y ) dy. By Minkowski’s inequality the integral on the right in (5.6) is dominated by (cid:16) Z ∞ (cid:16) Z R d F ( t, s, x ) dx (cid:17) q/p ds (cid:17) p/q , where Z R d F ( t, s, x ) dx = Z R d f p ( t + s, y ) dy Z R d Ψ (1 − ν ) pλ ( s, y ) dy ≤ N λ − d/ e − µ √ λs Z R d f p ( t + s, y ) dy, with µ = µ ( δ, p, q, R ) > 0. Below by µ we denote all such constants. Itfollows that Z R d | λ ηd +1 ¯ J ( t, x ) | p dx ≤ N λ − d/ (cid:16) Z ∞ e − µ √ λs (cid:16) Z R d f p ( t + s, y ) dy (cid:17) q/p ds (cid:17) p/q , k λ η ¯ J k qL p,q ( R d +1+ ) ≤ N λ − − qd/ (2 p ) k f k qL p,q ( R d +1+ ) , which along with (5.4) yield (5.3).If q ≥ p , Z ∞ | λ η ¯ J ( t, x ) | q dt ≤ N Z ∞ (cid:16) Z R d F p/q ( t, x, y ) dy (cid:17) q/p dt where F ( t, x, y ) = Z ∞ Ψ (1 − ν ) qλ ( s, y ) f q ( t + s, x + y ) ds. By Minkowski’s inequality (cid:16) Z ∞ | λ η ¯ J ( t, x ) | q dt (cid:17) p/q ≤ N Z R d (cid:16) Z ∞ F ( t, x, y ) dt (cid:17) p/q dy, where Z ∞ F ( t, x, y ) dt ≤ Z ∞ f q ( s, x + y ) ds Z ∞ Ψ (1 − ν ) qλ ( s, y ) ds ≤ N λ − e − µ √ λ | y | Z ∞ f q ( s, x + y ) ds. Hence, (cid:16) Z ∞ | λ η ¯ J ( t, x ) | q dt (cid:17) p/q ≤ N λ − p/q Z R d e − µ √ λ | y | (cid:16) Z ∞ f q ( s, x + y ) ds (cid:17) p/q dy, k λ η ¯ J k pL p,q ( R d +1+ ) ≤ N λ − p/q − d/ k f k pL p,q ( R d +1+ ) and we again come to (5.3). The theorem is proved. The full strength of Theorem 5.2 is seen in the theory of fully nonlinearequations. But even for linear ones one gets a nontrivial information as,for instance, in the following theorem which, in particular, implies that theoperator L + ∂ t with the domain { u ∈ W , p,q, loc ( C R ) ∩ C ( ¯ C R ) : Lu + ∂ t u ∈ L p,q ( C R ) , u (cid:12)(cid:12) ∂ ′ C R = 0 } is a closed operator in L p,q ( C R ). Theorem 5.3. Assume that (4.7) holds with p < ∞ , q < ∞ and take R ∈ (0 , ∞ ) . Suppose we are given u , u , ... ∈ W , p,q, loc ( C R ) ∩ C ( ¯ C R ) and f ∈ L p,q ( C R ) such that f n := Lu n + ∂ t u n ∈ L p,q ( C R ) for n ≥ , sup n ≥ sup ∂ ′ C R | u n | < ∞ , k f n − f k L p,q ( C R ) + k u n − u k L p,q ( C R ) → as n → ∞ . Then Lu + ∂u = f in C R . Proof. Take a smooth ψ on C R and apply (5.3) to u n − u + ψ/λ in placeof u . 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