aa r X i v : . [ m a t h . P R ] F e b ON DIFFUSION PROCESSES WITH DRIFT IN L d +1 N.V. KRYLOV
Abstract.
This paper is a natural continuation of [11] and [12] wherestrong Markov processes are constructed in time inhomogeneous settingwith Borel measurable uniformly bounded and uniformly nondegeneratediffusion and drift in L d +1 ( R d +1 ) and some properties of their Green’sfunctions and probability of passing through narrow tubes are investi-gated. On the basis of this here we study some further properties ofthese processes such as Harnack inequality, H¨older continuity of poten-tials, Fanghua Lin estimates and so on. 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 [11] that Itˆo’s stochastic equations of the form x t = x + Z t σ ( t + s, x s ) dw s + Z t b ( t + s, x s ) ds (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 the properties of such solutions orMarkov processes whose trajectories are solutions of (1.2). In particular, inSection 2 for more or less general processes of diffusion type we derive thepower lower estimate for the probability to reach at time T a given ball of Mathematics Subject Classification.
Key words and phrases.
Itˆo’s equations with singular drift, Markov diffusion processes,Harnack inequality. radius ρ (in case t = 0, x = 0 in (1.2)). This estimate plays a crucial role inproving the Harnack inequality in Section 4. In Section 2 we also prove thatthe probability to reach sets of almost full measure are strictly bigger thanzero. This seemingly weak statement is also crucial for proving the Harnackinequality.Section 3 is devoted to proving estimates from below for the average timespent in space-time sets of small measure in terms of a power of their mea-sure. We also extract some consequences of these estimates, which helpproving the H¨older continuity of potentials and harmonic functions in Sec-tion 4 and also allow us to establish the Fanghua Lin estimates in Section5. Section 4 is devoted to the case when our process is, actually, not just ofdiffusion type, but a Markov (time-inhomogeneous) process, whose existenceis shown in [11]. We prove that their resolvent operators are bounded in L p,q . We prove Harnack’s inequality for the caloric functions associatedwith these processes, establish that their resolvent operators map L p,q tothe set of H¨older continuous functions, and give some other estimates forthe resolvent in the whole space and in domains. These results extend someof those in [18].In Section 5 we give some applications of our results to the theory oflinear parabolic equations. In particular, we prove the Harnack inequalityand H¨older continuity of their solutions, which in case p = q = d + 1 areknown as Krylov-Safonov estimates and played an enormous role in thetheory of linear and fully nonlinear elliptic and parabolic equations withbounded coefficients. The solutions we are dealing with are of class W , p,q with d /p + 1 /q ≤ d < d . We now have the opportunity to considerlower order coefficients in L p,q -spaces and develop W , p,q -solvability theory.As is mentioned already, in this section we also derive the Fanghua Linestimate, which is one of the starting point of the regularity theory of fullynonlinear equations as presented in [8].The final Section 6 is an appendix in which we collected some results from[12] frequently used in the main text.To the best of the author’s knowledge our results are new even if p = q = d +1, with such low integrability of b and general σ H¨older and Harnackproperties were unknown.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[15], [1], [17] 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 are IFFUSION PROCESSES WITH DRIFT IN L d +1 independent of time allows one to relax the above conditions significantlyfurther, see, for instance, [7] 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. 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 < ∞ N.V. KRYLOV for any T ∈ (0 , ∞ ) and ω . Define x t = Z t σ s dw s + Z t b s ds and for R ∈ (0 , ∞ ) define τ R ( x ) as the first time ( t, x + x t ) exits from C R , τ ′ R ( x ) as the first time x t exits from B R , τ R = τ R (0), τ ′ R = τ ′ R (0). 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) Assumption 2.2.
We take ¯ N = ¯ N ( d, p , δ ) introduced in Theorem 6.1 andsuppose that there exists R ∈ (0 , ∞ ) such that¯ N ¯ b R < . (2.2)This assumption as well as Assumption 2.1 is supposed to hold throughoutthe article. Remark . Throughout the article we fix a number¯ R ∈ [ R, ∞ ) . In some places we write that certain constants depend “only on... ¯ R and¯ b ¯ R ” and the reader might notice that, actually, the constants depend “onlyon... ¯ R and ¯ b R ”. In such situations it is useful to note that, as is easyto see, ¯ b R can always be chosen less than or equal to N ( d )¯ b ¯ R . Also notethat if we take ¯ R = R , then mentioning ¯ b ¯ R becomes unnecessary, because¯ b R ≤ ¯ N − .Our first big project is to prove a version of Theorem 4.17 of [9], which pro-vides an important step toward establishing Harnack’s inequality for caloricfunctions. It is worth saying that in the case of bounded b Theorem 2.1 isproved by constructing a rather simple barrier, see the PDE argument inthe proof of Lemma 9.2.1 (“lemma on an oblique cylinder”) of [8] or theprobabilistic argument in the proof of Lemma 2.3 of [18]. In our case for thesame purpose, we needed a rather tedious argument like in Theorem 4.17 of[9] just to get a good control of the spatial process x t .Below ¯ ξ = ¯ ξ ( d, δ ) ∈ (0 ,
1) is taken from Theorem 6.1.
Theorem 2.1.
Let R ∈ (0 , ¯ R ] , κ, η ∈ (0 , , x, y ∈ B κR , and η − R ≥ t ≥ ηR . Then there exist N, ν > , depending only on κ, η, ¯ ξ, R , and ¯ R , suchthat, for any ρ ∈ (0 , , N P ( x + x t ∈ B ρR ( y ) , τ ′ R ( x ) > t ) ≥ ρ ν . (2.3) IFFUSION PROCESSES WITH DRIFT IN L d +1 The proof of this theorem, given below after appropriate preparations,follows that of Theorem 4.17 of [9] and, roughly speaking, consists of splittingthe interval [0 , t ] into several parts, estimating the probability that on thefirst part the process will reach a neighborhood of y without exiting from B R , and then on the consecutive time intervals shrink the neighborhood withconstant coefficient in such a way as to arrive at time t in B ρR ( y ) withoutexiting from B R . Lemma 2.2.
Let R ∈ (0 , ¯ R ] and let ρ ∈ (0 , , θ ∈ (0 , ∞ ) , and κ ∈ [1 / , .Then there exists µ = µ ( ¯ ξ, ¯ R, R, κ, ρ , θ ) > such that P ( x + x θR ∈ B ρ κR ( y ) , τ R ( x ) > θR ) ≥ µ, (2.4) whenever x, y ∈ B κR . Furthermore, one can take µ > the same if ¯ ξ, ¯ R, κ are fixed and ρ and θ vary in compact subsets of their ranges. Proof. Observe that (2.4) becomes stronger if ρ becomes smaller. There-fore we may assume that ρ ≤ min (cid:0) R/ (16 ¯ R ) , p θ/ (144 T ) , κ − − (cid:1) ( ≤ / , (2.5)where T = T ( ¯ ξ ) is taken from Theorem 6.3. Then we split the proof intotwo cases. Case 1: | x − y | ≤ ρ κR . In that case, owing to R ≤ ¯ R and ρ ≤ R/ (6 ¯ R ),we have | x − y | ≤ (1 / ρ κ ( R ∧ R ) . By Corollary 6.11, which is applicable since ρ κ ( R ∧ R ) ≤ R , N P ( sup t ≤ θR | x + x t − y | < ρ κ ( R ∧ R )) ≥ exp( − νθR / [ ρ κ ( R ∧ R )] ) . The probability here is less than the probability in (2.4) since R ∧ R ≤ R and ρ κR ≤ (1 − κ ) R . Furthermore, R / ( R ∧ R ) ≤ ¯ R /R and this proves (2.4) in the first case. Case 2: | x − y | ≥ ρ κR . Set R = 16 ρ κR and note that | x | + R <κR + (1 − κ )16 ρ R < R . Similarly, | y | + R < R . Therefore, the sausage S R ( x, y ), defined as the open convex hull of B R ( x ) ∪ B R ( y ), belongs to B R . By Theorem 6.3 with probability not less than π n before time nT R the process x + x t will hit ¯ B R / ( y ) without exiting from S R ( x, y ), where n ≤ | x − y | R + 14 . Since R < R , | x − y | < R , and also thanks to 144 T ρ ≤ θ , we have nT R ≤ T R (4 | x − y | + R / ≤ T R R = 144 T ρ κR ≤ θR . By introducing γ as the first time x + x t hits ¯ B R / ( y ) we conclude that P ( τ ′ R ( x ) > γ, γ ≤ θR ) ≥ π n . (2.6) N.V. KRYLOV
Observe also that R /
16 = ρ R , where R = ρ κR ≤ R and at time γ thepoint x + x γ is in ¯ B ρ R ( y ). It follows from Corollary 6.11 that, given that γ < ∞ , with probability π > ξ , ρ , and θ = θR − R ( ≤ θρ − κ − ) the process x + x t will stay in B R ( y ) on the time interval[ γ, γ + θ R ]. Notice that γ + θ R ≥ θ R = θR . Along with (2.6) thisimplies that P ( x + x θR ∈ B R ( y ) , τ R ( x ) > θR ) ≥ π n π > . To prove (2.4), it only remains to recall that R = ρ κR .This proves the first assertion of the lemma. The second one is obtainedby just inspecting the above proof. The lemma is proved.The following is a particular case of Theorem 2.1 for t = ηR . Lemma 2.3.
Let κ, η ∈ (0 , . Then there are constants N, ν > , dependingonly on κ, η, ¯ ξ, R , and ¯ R , such that, for any R ∈ (0 , ¯ R ] , ρ ∈ (0 , , and x ∈ B κR , N P (cid:0) τ R ( x ) > ηR , x + x ηR ∈ B ρR (cid:1) ≥ ρ ν . (2.7)Proof. We may assume that κ ∈ (1 / , ρ be a positive solution of ρ = min (cid:16) R
16 ¯
R , p η (1 − ρ )12 √ T , κ − (cid:17) , observe that it suffices to prove (2.7) for ρ ≤ κ , and set n ( ρ ) = j ln( ρ/κ )ln ρ k + 1 ( ≥ , ¯ η = η − ρ − ρ n ( ρ )0 . Note that ¯ η ∈ [ η (1 − ρ ) , η (1 + ρ ) − ] and ρ ≤ min (cid:16) R
16 ¯
R , √ ¯ η √ T , κ − (cid:17) so that by Lemma 2.2 estimate (2.4), with θ = ¯ η , y = 0, is valid and meansthat P (cid:0) x + x ¯ ηR ∈ B ρ κR , sup s ≤ ¯ ηR | x + x s | < R (cid:1) ≥ µ, (2.8)whenever R ∈ (0 , ¯ R ] and x ∈ B κR . For n = 1 , , ... introduce ( t := 0) R n = ρ n − R, s n = ¯ ηR n = ¯ ηR ρ n − , t n = n X k =1 s k ,A n = { sup s ≤ s n | x + x s + t n − | < R n } , Π n = n \ k =1 A k and observe that by the conditional version of (2.8) on the set { y := x + x t n − ∈ B κR n } we have (a.s.) P (cid:16) y + ( x t n − x t n − ) ∈ B κR n +1 , sup s ≤ s n | y + ( x t n − + s − x t n − ) | < R n | F t n − (cid:17) ≥ µ. (2.9) IFFUSION PROCESSES WITH DRIFT IN L d +1 Furthermore, obviously, for n ≥ P n := P ( x + x t n ∈ B κR n +1 , Π n ) ≥ P ( x + x t n − ∈ B κR n , Π n − ,x + x t n − + ( x t n − x t n − ) ∈ B κR n +1 , sup s ≤ s n | x + x t n − + ( x t n − + s − x t n − ) | < R n ) , which in light of (2.9) yields P n ≥ µP n − and, since for | x | < κR we have P ≥ µ by (2.8), it holds that for | x | < κR and all n ≥ P (cid:0) x + x t n ∈ B κR n +1 , sup s ≤ t n | x + x s | < R (cid:1) ≥ µ n . (2.10)Observe that t n ( ρ ) = ¯ ηR − ρ n ( ρ )0 − ρ = ηR , κR n ( ρ )+1 = κρ R n ( ρ ) ≤ ρR. Therefore, (2.10) implies that P ( τ ( x ) > ηR , | x + x ηR | ≤ ρR ) ≥ µ n ( ρ ) . Now to finish the proof, it only remains to note that µ n ( ρ ) ≥ µ exp (cid:16) ln( ρ/κ )ln ρ ln µ (cid:17) = N ρ ν . The lemma is proved.
Proof of Theorem 2.1 . Let R = (1 − κ ) R and note that ξ := t/R − η satisfies η − (1 − κ ) − > ξ ≥ η (cid:2) (1 − κ ) − − (cid:3) . By the conditional version of Lemma 2.3 on the set { z := x + x ξR ∈ B κR ( y ) } we have (a.s.) N P (cid:16) sup s ∈ [ ξR ,ξR + ηR ] | z + x s − x ξR − y | < R ,x + x ξR + ηR ∈ B ρR ( y ) | F ξR (cid:17) ≥ ρ ν . By Lemma 2.2, where we take ρ = R /R and replace θ there with ξ (1 − κ ) , P ( sup s ≤ ξR | x + x s | < R, x + x ξR ∈ B κR ( y )) ≥ µ. By combining these two facts and using that ξR + ηR = t , we obviouslycome to (2.3). The theorem is proved.Theorem 2.4 originated in [13] in case b is bounded and is one of two mostimportant ingredients in the proof of the Harnack inequality. Observe thatin this theorem we do not claim that q ( ξ ) = 0 for ξ not close to one. Thisfact will be proved later. N.V. KRYLOV
Theorem 2.4.
For any κ ∈ (0 , there is a function q ( ξ ) , ξ ∈ (0 , ,depending only on κ, δ, d, R, p , ¯ R , and, naturally, on ξ , such that for any R ≤ ¯ R , x ∈ B κR , and closed Γ ⊂ ¯ C R satisfying | Γ | ≥ ξ | C R | we have P ( τ Γ ( x ) < τ R ( x )) ≥ q ( ξ ) , (2.11) where τ Γ ( x ) is the first time the process ( t, x + x t ) hits Γ (and τ R ( x ) is itsfirst exit time from C R ). Furthermore, q ( ξ ) → as ξ ↑ . Finally, for anyclosed Γ ′ ⊂ B R satisfying | Γ ′ | ≥ ξ | B R | we have P ( τ ′ Γ ( x ) < τ ′ R ( x )) ≥ q ( ξ ) , (2.12) where τ ′ Γ ( x ) is the first time the process x + x t hits Γ (and τ ′ R ( x ) is its firstexit time from B R ). Proof. By considering what is going on in B (1 − κ ) R ( x ) we convince our-selves that we may assume that x = 0. Also, obviously we may assume that R ≤ R .In that case for any ε ∈ (0 , ¯ ξ/
4) we have ( τ Γ = τ Γ (0)) P ( τ Γ > τ R ) ≤ P (cid:16) τ R = Z τ R I C R \ Γ ( t, x t ) dt (cid:17) ≤ P ( τ R ≤ εR ) + ε − R − E Z τ R I C R \ Γ ( t, x t ) dt. In light of Theorems 6.2 and Lemma 6.10 we can estimate the right-handside and then obtain P ( τ Γ > τ R ) ≤ N e − / ( Nε ) + N ε − R d/ ( d +1) − | C R \ Γ | / ( d +1) ≤ N e − / ( Nε ) + N ε − (1 − ξ ) / ( d +1) where the constants N depend only on d, δ, p . By denoting q ( ξ ) = 1 − inf ε ∈ (0 , ¯ ξ/ (cid:0) N e − / ( Nε ) + N ε − (1 − ξ ) / ( d +1) (cid:1) , we get what we claimed about (2.11).Estimate (2.12) follows from (2.11) if one takes in the latter Γ = [0 , R ] × Γ ′ and observes that { τ Γ ( x ) < τ R ( x ) } ⊂ { τ ′ Γ ( x ) < τ ′ R ( x ) } . The lemma is proved.3.
Estimating time spent in space-time sets of small measure
The central result of this section is Theorem 3.5 which needs some auxil-iary constructions and assertions.We present extensions to the case that b ∈ L d +1 of probabilistic versionsof some PDE results found in [14], [16], [8]. Recall the notation introducedin the Introduction and also introduce C oT,R = (0 , T ) × B R , C oT,R ( t, x ) = ( t, x ) + C oT,R , C oR ( t, x ) = C oR ,R ( t, x ) , IFFUSION PROCESSES WITH DRIFT IN L d +1 C oR = C oR (0 , q, η, κ ∈ (0 , . For cylinders Q = C oρ ( t, x ) define Q ′ = ( t, x ) − C oη − ρ ,ρ , Q ′′ = (cid:0) t − η − ρ , x (cid:1) + C oη − ρ κ ,ρκ ,Q ′ + = Q ∪ Q ′ ∪ (cid:0) { t } × B ρ ( x ) (cid:1) . Imagine that the t -axis is pointed up vertically. Then Q ′ is adjacent to Q from below, the two cylinders have a common base, and along the t -axis Q ′ is η − times longer than Q . The cylinder Q ′′ is obtained by contracting Q ′ to the center of its lower base with the contraction factor κ − for the t -axisand κ − for the spatial axes. Remark . If Q = C oρ ( t, x ), then the distance between Q and Q ′′ along the t axis is η − ρ − η − ρ κ = η − ρ (1 − κ ) , (3.1)which is 1 if η = 1 − κ .Let Γ be a measurable subset of C and introduce B = B (Γ , q ) as thefamily of open cylinders Q of type C oρ ( t , x ) such that Q ⊂ C and | Q ∩ Γ | ≥ q | Q | . Finally, define Γ ′′ = [ Q ∈B Q ′′ , Γ ′′ ε = [ Q ∈B : | Q |≥ ε Q ′′ . Observe that for Q ∈ B the set Q ′′ is open. Hence, Γ ′′ is open andmeasurable. Lemma 3.1. If | Γ | ≤ q | C | , then | Γ ′′ | ≥ (cid:16) − − q d +1 (cid:17) − (1 + η ) − κ d +2 | Γ | and, if the factor of | Γ | above is strictly bigger than one, there exists θ = θ ( d, q, η, κ ) > such that for any sufficiently small ε > there exists aclosed Γ ε ⊂ Γ ′′ ε such that | Γ ε | ≥ θ | Γ | . (3.2)The first assertion of the lemma originated in [14], [16], is presented, forinstance as Lemma 9.3.6 in [8]. The second one is proved in the same wayas the second assertion of Lemma 4.8 of [9]. Lemma 3.2.
There is a constant q = q ( d, δ, p , R, ¯ R ) ∈ (0 , such thatfor any R ≤ ¯ R , Borel set Γ ⊂ C R satisfying | Γ | ≥ q | C R | , and x ∈ ¯ B R/ wehave E Z τ R ( x )0 I Γ ( t, x + x t ) dt ≥ µ R , (3.3) where µ = µ ( d, δ, p , ¯ R, R, ¯ b ¯ R ) ∈ (0 , . Proof. Note that in light of Corollary 6.12 we have Eτ R ( x ) ≥ νR , where ν = ν ( d, δ, ¯ R, R ) >
0. By using Theorem 6.7 we get that Eτ R ( x ) − E Z τ R ( x )0 I Γ ( t, x + x t ) dt = E Z τ R ( x )0 I C R \ Γ ( t, x + x t ) dt ≤ N R (2 d − d ) / ( d +1) ( | C R | − | Γ | ) / ( d +1) ≤ N R (1 − q ) / ( d +1) ≤ N (1 − q ) / ( d +1) Eτ R ( x ) , where the constants N depend only on d, δ , ¯ R , R , p , and ¯ b ¯ R and d = d ( d, δ, R, p ) ∈ ( d/ , d ) is taken from [12]. We see how to choose q to getthe desired result. The lemma is proved.In Lemma 3.3 by q we mean the one from Lemma 3.2. Lemma 3.3.
Take Q = C oρ ( s, y ) with ρ ≤ ¯ R , use the notation Q ′ , Q ′′ , Q ′ + introduced above, assume than η = 1 − κ , and suppose that Borel Γ ⊂ Q issuch that | Γ | ≥ q | Q | . Then there is a constant ν > , depending only on κ , d, δ , ¯ R , R , p , and ¯ b ¯ R , such that for any ( t , x ) ∈ Q ′′ E Z τ I Γ ( t + t, x + x t ) dt ≥ ν Eτ, (3.4) where τ is the first exit time of ( t + t, x + x t ) from Q ′ + . Proof. Thanks to Remark 3.1 we have s − t ∈ ( ρ , η − ρ ). Also | y − x | <κρ . It follows by Theorem 2.1 that P (cid:0) sup r ∈ [0 ,s − t ] | x + x r − y | < ρ, | x + x s − t − y | < (1 / ρ (cid:1) ≥ ν, where ν = ν ( κ, d, δ, R, ¯ R ) > γ defined as the first exit time of ( t + t, x + x t ) from Q ′ inlight of Lemma 3.2 we have E Z τ I Γ ( t + t, x + x t ) dt = EI γ>s − t Z τγ I Γ ( t + t, x + x t ) dt ≥ EI γ>s − t , | x + x s − t − y | <ρ/ E (cid:16) Z τ I Γ ( t + t, x + x t ) dt | F s − t (cid:17) ≥ µ ρ P (cid:0) sup r ∈ [0 ,s − t ] | x + x r − y | < ρ, | x + x s − t − y | < ρ/ (cid:1) ≥ µ νρ . On the other hand, the height of Q ′ + is (1 + η − ) ρ , so that ( t + t, x + x t )cannot spend in Q ′ + more time than (1 + η − ) ρ . This proves the lemma. Lemma 3.4.
Denote G R (Γ , x ) := E Z τ ′ R ( x ) ∧ (2 R )0 I Γ ( t, x + x t ) dt, fix q, κ ∈ (0 , , and introduce µ R ( q ) as R − times the infimum of G R (Γ , x ) over all Borel Γ ⊂ C R ( R , satisfying | Γ | ≥ q | C R ( R , | over all x ∈ B κR ,and over all processes x t satisfying our assumptions with the same δ and ¯ b R .Then µ R ( q ) is a decreasing function of R . IFFUSION PROCESSES WITH DRIFT IN L d +1 The proof of this lemma, left to the reader, is easily achieved by using theself-similar dilations: x t → cx t/c , which preserves (actually, makes smaller)¯ b R (see, for instance, the proof of our Theorem 6.4 in [12]). Theorem 3.5.
For any κ ∈ (0 , there exist γ ∈ (0 , and N , dependingonly on κ, d, δ, p , R , with N also depending on ¯ R and and ¯ b ¯ R , such that forany R ∈ (0 , ¯ R ] , q ∈ (0 , , Borel Γ ⊂ C R ( R , satisfying | Γ | ≥ q | C R ( R , | ,and x ∈ B κR we have E Z τ ′ R ( x ) ∧ (2 R )0 I Γ ( t, x + x t ) dt ≥ N − q /γ R . (3.5)Proof. By using the notation from Lemma 3.4, our assertion is rewrittenas µ R ( q ) ≥ N − q /γ R . (3.6)Fix κ ∈ (0 , η = 1 − κ and q = q the factor of | Γ | in Lemma 3.1 isstrictly bigger than one and take θ = θ ( d, q , − κ , κ ) > µ ∈ (0 , κ, d, δ, p , R, ¯ R , and ¯ b ¯ R , such that µ R ( q ) ≥ µ for q ∈ [ q ,
1] and R ≤ ¯ R .We will be comparing µ R ( q ′ ) and µ R ( q ′′ ) for 0 < q ′ < q ′′ < θ ) q ′ ≥ q ′′ . (3.7)We take a Borel Γ ⊂ C R ( R ,
0) satisfying | Γ | ≥ q ′ | C R ( R , | and in theconstruction before Lemma 3.1 we replace C by C R ( R , , κ, η , and q (not q ′ ) we build up the set Γ ε and take ε so small that (3.2) holds. There are two cases:(i) (cid:12)(cid:12) Γ ε \ C R ( R , (cid:12)(cid:12) ≤ ( q ′′ − q ′ ) | C R ( R , | ,(ii) (cid:12)(cid:12) Γ ε \ C R ( R , (cid:12)(cid:12) > ( q ′′ − q ′ ) | C R ( R , | . Case (i ) . Our goal is to show that G R (Γ , x ) ≥ min (cid:0) µ , ν µ R ( q ′′ ) (cid:1) R , | x | ≤ κR, (3.8)where ν depends only on κ, d, p , δ, R, ¯ R , and ¯ b ¯ R .Observe that, if | Γ | ≥ q | C R | , by definition G R (Γ , x ) ≥ µ ( q ) R ≥ µ R for | x | ≤ κR . Hence, we may assume that | Γ | < q | C | . In that case define ˆΓ ε = Γ ε ∩ C R ( R , . Notice that by definition and Lemma 3.1 q ′ | C R | ≤ | Γ | ≤ θ − | Γ ε | . Moreover, by assumption | Γ ε | = (cid:12)(cid:12) Γ ε \ C R ( R , (cid:12)(cid:12) + | ˆΓ ε | ≤ ( q ′′ − q ′ ) | C R | + | ˆΓ ε | . Due to (3.7), it follows that | ˆΓ ε | ≥ q ′′ | C R | , so that G R (ˆΓ ε , x ) ≥ µ R ( q ′′ ) R , | x | ≤ κR. We now estimate G R (Γ , x ) from below by means of G R (ˆΓ ε , x ) using Lemma3.3. Since Γ ε ⊂ Γ ′′ ε , the closed set Γ ε is covered by the family { Q ′′ : Q ∈B , | Q | ≥ ε } . Then there is finitely many Q (1) , ..., Q ( n ) ∈ B such that | Q ( i ) | ≥ ε , i = 1 , ..., n , and Γ ε ⊂ n [ i =1 Q ′′ ( i ) =: Π ε . Then for ( t, x ) ∈ Π ε define i ( t, x ) as the first i ∈ { , ..., n } for which( t, x ) ∈ Q ′′ ( i ). Also set Q ′ + (0) = C R ,R and i ( t, x ) = 0 if ( t, x ) ∈ ∂C R ,R .Now define recursively γ = 0, τ as the first time after γ when ( t, x + x t )exits from C R ,R \ Γ ε , γ as the first time after τ when ( t, x + x t ) exits from Q ′ + ( i ( τ , x + x τ )), and generally, for k = 2 , , ... define τ k as the first timeafter γ k − when ( t, x + x t ) exits from C R ,R \ Γ ε , γ k as the first time after τ k when ( t, x t ) exits from Q ′ + ( i ( τ k , x + x τ k )). It is easy to check that so defined τ k and γ k are stopping times and, since | Q ( i ) | ≥ ε and the trajectories of( t, x + x t ) are continuous, τ k ↑ τ ′ R ( x ) ∧ R as k → ∞ . Furthermore, (a.s.)all the τ k ’s equal τ ′ R ( x ) ∧ R for all large k .For a domain Q ⊂ R d +1 we denote by γ ( s, y, Q ) the first exit time of( s + t, y + x s + t − x s ) from Q and obtain G R (Γ , x ) ≥ ∞ X k =1 E Z γ k τ k I Γ ( t, x + x t ) dt = ∞ X k =1 EE (cid:16) Z γ ( s,y,Q ′ + ( i ))0 I Γ ( s + t, x s + t − x s + y ) dt | F s (cid:17)(cid:12)(cid:12)(cid:12) i = i ( s,y ) ,y = x + x s ,s = τ k . We estimate the interior expectation from below by Lemma 3.3 and get that G R (Γ , x ) /ν is greater than or equal to ∞ X k =1 EE (cid:16) Z γ ( s,y,Q ′ + ( i ))0 I Π ε ( s + t, x s + t − x s + y ) dt | F s (cid:17)(cid:12)(cid:12)(cid:12) i = i ( s,y ) ,y = x + x s ,s = τ k ≥ ∞ X k =1 EE (cid:16) Z γ ( s,y,Q ′ + ( i ))0 I Γ ε ( s + t, x s + t − x s + y ) dt | F s (cid:17)(cid:12)(cid:12)(cid:12) i = i ( s,y ) ,y = x + x s ,s = τ k = ∞ X k =1 E Z γ k τ k I Γ ε ( t, x + x t ) dt = G R (Γ ε , x ) ≥ G R (ˆΓ ε , x ) ≥ µ R ( q ′′ ) R . IFFUSION PROCESSES WITH DRIFT IN L d +1 This proves (3.8).
Case (ii) . Here the goal is to prove that G R (Γ , x ) ≥ µ νη n ( q ′′ − q ′ ) n R , | x | ≤ κ, (3.9)where ν > n ≥ d, δ, ¯ R , R , and κ .First we claim that for some ( t, x ) ∈ Γ ε it holds that t < ( q ′ − q ′′ + 1) R .Indeed, otherwiseΓ ε \ C R ( R , ⊂ C ( q ′′ − q ′ ) R ,R (( q ′ − q ′′ + 1) R , | Γ ε \ C R ( R , | ≤ ( q ′′ − q ′ ) | C R | . It follows that there is a cylinder Q = C oρ ( s, y ) ∈ B such that Q ′ contains points in the half-space t < ( q ′ − q ′′ + 1) R . Since q ′ < q ′′ , we have q ′ − q ′′ + 1 <
1, and since Q ′ is adjacent to Q ⊂ C R ( R , Q ′ is at least ( q ′′ − q ′ ) R , that is, ρ η − ≥ ( q ′′ − q ′ ) R , ρ ≥ η ( q ′′ − q ′ ) R . (3.10)On the other hand, Q ⊂ C R ( R , s > R , and ρ < R .Moreover, by construction, | Γ ∩ Q | ≥ q | Q | and by Lemma 3.2 on the setwhere | z − y | ≤ ρ/ I ( z ) := E (cid:16) Z τ I Γ ( s + t, z + x s + t − x s ) dt | F s (cid:17) ≥ µ ρ ≥ µ η ( q ′′ − q ′ ) R , where τ is the first exit time of ( s + t, z + x s + t − x s ) from C ρ ( s, y ). Now byTheorem 2.1 for x ∈ B κR E Z τ ′ R ( x ) ∧ (2 R )0 I Γ ( t, x + x t ) dt ≥ EI τ ′ R ( x ) >s, | x + x s − y |≤ ρ/ I ( x + x s ) ≥ µ η ( q ′′ − q ′ ) R P x (cid:0) τ ′ R ( x ) > s, | x + x s − y | ≤ κρ (cid:1) ≥ N − ( ρ/R ) ν µ η ( q ′′ − q ′ ) R . This proves (3.9).By combining the two cases (i) and (ii) we conclude that G R (Γ , x ) ≥ min (cid:0) µ , ν µ R ( q ′′ ) , µ νη n ( q ′′ − q ′ ) n (cid:1) R , | x | ≤ κR, and the arbitrariness of Γ allows us to conclude that µ R ( q ′ ) ≥ min (cid:0) µ , ν µ R ( q ′′ ) , µ νη n ( q ′′ − q ′ ) n (cid:1) , (3.11)whenever (3.7) holds. Observe that (3.11) is identical to (9.3.10) of [8] andby literally repeating what is in [8], just replacing ξ there with our θ , wecome to (3.6) for any R . By Lemma 3.4 the right-hand side in (3.6) can betaken the same for R ≤ ¯ R . The theorem is proved.The following four results are derived from Theorem 3.5 in the same wayas similar results are derived from Theorem 4.1 of [9]. Corollary 3.6.
For any κ ∈ (0 , there exists N , depending only on κ, d, δ, p , R, ¯ R , and ¯ b ¯ R , such that, for any R ∈ (0 , ¯ R ] , x ∈ B κR , and closedset Γ ⊂ C R ( R , , the probability that the process ( t, x + x t ) reaches Γ beforeexiting from C R ,R is greater than or equal to N − ( | Γ | / | C R | ) µ − / ( d +1) : P ( τ Γ ( x ) < τ R ,R ( x )) ≥ N − ( | Γ | / | C R | ) µ − / ( d +1) , (3.12) where τ Γ ( x ) is the first time ( t, x + x t ) hits Γ , τ R ,R ( x ) is the first exit timeof ( t, x + x t ) from C R ,R , µ = 1 /γ , and γ is taken from Theorem 3.5. Corollary 3.7.
For any R ∈ (0 , ¯ R ] , Borel nonnegative f vanishing outside C R ( R , , and x ∈ B κR Z C R ( R , f / (2 µ ) ( t, y ) dydt ≤ N R d +2 − /µ (cid:16) E Z τ R ,R ( x )0 f ( t, x + x t ) dt (cid:17) / (2 µ ) , where N depends only on κ, d, δ, p , R, ¯ R , and ¯ b ¯ R , Corollary 3.8.
Let R ∈ (0 , ¯ R ] , γ ∈ (0 , , and assume that a closed set Γ ⊂ B R is such that, for any r ∈ (0 , R ) , | Br ∩ Γ | ≥ γ | B r | . Then there existconstants α ∈ (0 , and N , depending only on κ, d, δ, p , R, ¯ R , ¯ b ¯ R , and γ ,such that, for any x ∈ B R/ , P ( τ R ( x ) < τ Γ ( x )) ≤ N ( | x | /R ) α , (3.13) where τ Γ ( x ) is the first time x + x t hits Γ . The fourth result has the same spirit as Theorem 4.11 of [9] and canbe used in investigating the boundary behavior of solutions of parabolicequations with drift in L p,q .We are going to use the following condition p, q ∈ [1 , ∞ ] , ν := 1 − d p − q ≥ , (3.14)where d ∈ ( d/ , d ), depending only on δ , d , R , p , is taken from [12]. Theorem 3.9.
Let (3.14) be satisfied with ν = 0 , T ∈ (0 , ∞ ) , and let D be a bounded domain in R d with ∈ ∂D . Assume that for some constants ρ, γ > and any r ∈ (0 , ρ ) we have | B r ∩ D c | ≥ γ | B r | . Then there exist β > and N , depending only on d, δ, p , R , ¯ b ∞ , γ with N also dependingon ρ and the diameter of Q := (0 , T ) × D , such that, for any nonnegative f ∈ L p,q ( Q ) , u ( x ) := E Z τ ( x )0 f ( t, x + x t ) dt ≤ N | x | β k f k L p,q ( Q ) , (3.15) where τ ( x ) is the first exit time of ( t, x + x t ) from Q . In the next section we will need the following fact of crucial importance,the origin of which lies in [14] and [16]. A few other related results belowalso have their origin in [14] and [16] where the drift is bounded.
IFFUSION PROCESSES WITH DRIFT IN L d +1 Theorem 3.10.
Let κ, η, ζ, q ∈ (0 , , R ∈ (0 , ¯ R ] , T ∈ [ ηR , η − R ] , andclosed Γ ⊂ C T,R be such that | Γ ∩ C ζT,R ((1 − ζ ) T, | ≥ q | C ζT,R | . Thenthere exists π = π ( κ, η, ζ, q, d, δ, p , R, ¯ R, ¯ b ¯ R ) > , such that, for ( t, x ) ∈ C (1 − ζ ) T,κR , P ( τ Γ ( t, x ) < τ T,R ( t, x )) ≥ π , (3.16) where τ Γ ( t, x ) is the first time ( t + s, x + x s ) hits Γ and τ T,R ( t, x ) is its firstexit time from C T,R . Proof. Observe that one can choose ρ ∈ (0 , d, η, ζ ,and q , and one can find ( t , x ) ∈ C T,R with t ≥ ρ R + (1 − ζ ) T suchthat C ρR ( t + ρ R , x ) ⊂ C T,R and | Γ ∩ C ρR ( t + ρ R , x ) | ≥ ¯ q | C ρR | ,where ¯ q > d, η, ζ , and q . Then by Corollary 3.6, for x ∈ B κR ( x ) the probability that the process ( t + s, x + x s ) will hit Γ beforeexiting from C ρ R ,ρR ( t , x ) is estimated from below by a strictly positiveconstant depending only on κ, ¯ q, d, δ, p , R, ¯ R , and ¯ b ¯ R . After that it onlyremains to invoke Theorem 2.1 recalling that t ≥ ρ R + (1 − ζ ) T and t < (1 − ζ ) T . The theorem is proved.4. The case of diffusion processes
In this section, among other things, we generalize some recent results in[18] and extend them to processes with singular drift.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. Assumption 4.1. (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.Let Ω be the set of R d +1 -valued continuous function ( t + t, x t ), t ∈ R ,defined for t ∈ [0 , ∞ ). For ω = { ( t + t, x t ) , t ≥ } , define t t ( ω ) = t + t , x t ( ω ) = x t , and set N t = σ (( t s , x s ) , s ≤ t ), N = N ∞ . In the followingtheorem which is Theorem 6.1 of [11] we use the terminology from [4]. Theorem 4.1. On R d +1 there exists a strong Markov process X = { ( t t , x t ) , ∞ , N t , P t,x ) such that the process X = { ( t t , x t ) , ∞ , N t + , P t,x ) is Markov and for any ( t, x ) ∈ R d +1 there exists a d -dimensional Wienerprocess w t , t ≥ , which is a Wiener process relative to ¯ N t , where ¯ N t is the completion of N t with respect to P t,x , and such that with P t,x -probabilityone, for all s ≥ , t s = t + s and x s = x + Z s σ ( t + r, x r ) dw r + Z s b ( t + r, x r ) dr. (4.1) Remark . To be completely rigorous, to refer to [11] we should have b ∈ L p ,q (globally), and (2.1) is not needed. But with our b , owing toCorollary 2.8 and Theorems 6.6, the arguments in [11] only simplify anddo not require b ∈ L p ,q . Still it is worth saying that the author believesthat under only conditions in [11] Harnack’s inequality is true. Regardingthe H¨older continuity of caloric functions in the same setting we have noguesses. The H¨older continuity seems to require some sort of self-similarityand the L p ,q -norm is not preserved under such transformations if p , q aresubject to (1.1). Theorem 4.2.
For any λ ≥ , p, q satisfying (3.14) , and Borel nonnegative f ( t, x ) and for R λ f ( t, x ) := E t,x Z ∞ e − λs f ( t + s, x s ) ds. we have k R λ f k L p,q ( R d +1+ ) ≤ N λ − k f k L p,q ( R d +1+ ) , (4.2) where N = N ( δ, d, R, p, q, p , ¯ b ∞ ) . Proof. By Theorem 6.5 we have R λ f ( t, x ) ≤ N λ − η k Ψ − νλ f ( t + · , x + · ) k L p,q ( R d +1+ ) , (4.3)where η = ν + (2 d − d ) / (2 p ) and Ψ λ ( t, x ) = exp( −√ λ ( | x | + √ t ) ¯ ξ/ Definition 4.1. If Q is a set in R d +1 and u is a bounded Borel functionon Q , we call it caloric (relative to the process X ) if for any ( s, y ) and T, R ∈ (0 , ∞ ) such that ¯ C T,R ( s, y ) ⊂ Q and any ( t, x ) ∈ C := C T,R ( s, y ) wehave u ( t, x ) = E t,x u ( t + τ C , x τ C ) , where τ C is the first exit time of ( t + s, x s ) from C .First, we deal with H¨older norm estimates for harmonic functions andpotentials. If z = ( t , x ) and z = ( t , x ), we define ρ ( z , z ) = | x − x | + | t − t | / (4.4)and call ρ ( z , z ) the parabolic distance between z and z . IFFUSION PROCESSES WITH DRIFT IN L d +1 Lemma 4.3.
Let R ∈ (0 , ¯ R ] and let u be a caloric function in ¯ C R . Thenthere exist constants N and α ∈ (0 , , depending only on δ, d, p , R, ¯ R, ¯ b ¯ R , such that, for any α ∈ (0 , α ] and z , z ∈ C R , we have (cid:12)(cid:12) u ( z ) − u ( z ) (cid:12)(cid:12) ≤ N R − α ρ α ( z , z ) sup (cid:0) | u | , ¯ C R (cid:1) . (4.5) Furthermore, sup( | u | , ¯ C R ) in (4.5) can be replaced by osc( u, ¯ C R ) , wherewe use the notation osc( g, Γ) = osc Γ g = sup Γ g − inf Γ g. Proof. For r such that C r ⊂ C R , set w ( r ) = osc( u, ¯ C r ) , m ( r ) = inf ¯ C r u, M ( r ) = sup ¯ C r u,µ ( r ) = (1 / (cid:0) m ( r ) + M ( r ) (cid:1) . Take r ≤ R/ (cid:12)(cid:12) C r ∩ (cid:8) u ≤ µ ( r ) (cid:9)(cid:12)(cid:12) ≥ (1 / | C r | . Then there is a closed Γ ⊂ C r ∩ (cid:8) u ≤ µ ( r ) (cid:9) such that (cid:12)(cid:12) C r ,r ( r , ∩ Γ (cid:12)(cid:12) ≥ (1 / | C r ,r | (4.6)By Theorem 3.10 for any ( t, x ) ∈ ¯ C r we have P t,x ( τ Γ < τ r ) ≥ π , where π > δ, d, p , R, ¯ R, ¯ b ¯ R , τ Γ is the first time ( t + s, x s )hits Γ, τ r is its first exit time from C r . Then by definition and the strongMarkov property for τ = τ Γ ∧ τ r we have u ( t, x ) = E t,x u ( t + τ r , x τ r )= E t,x u ( t + τ r , x τ r ) I τ Γ <τ r + E t,x u ( t + τ r , x τ r ) I τ Γ ≥ τ r = E t,x u ( t + τ Γ , x τ Γ ) I τ Γ <τ r + E t,x u ( t + τ r , x τ r ) I τ Γ ≥ τ r ≤ µ ( r ) π + M (2 r )(1 − π )(we used that µ ( r ) ≤ M (2 r )). It follows that M ( r ) ≤ π (cid:0) m ( r ) + M ( r ) (cid:1) + (1 − π ) M (2 r ) , (cid:0) − π M ( r ) ≤ π m ( r ) + (1 − π ) M (2 r ) . Adding to this the obvious inequality (cid:0) π − m ( r ) ≤ − π m ( r ) + ( π − m (2 r ) , we get (cid:0) − π (cid:1) w ( r ) ≤ (1 − π ) w (2 r ) , w ( r ) ≤ εw (2 r ) , (4.7)where ε < ε = ε ( π ). We may, certainly, assume that ε > / We have proved (4.7) assuming that (4.6) is true. However if (4.6) is false,then − u satisfies an inequality similar to (4.6) and this leads to (4.7) again.Therefore, w ( r ) ≤ εw (2 r ) for all r ≤ R/
2. Iterations then yield w ( r ) ≤ ε w (4 r ) for r ≤ R/ , ..., w ( r ) ≤ ε n w (2 n r ) for r ≤ − n R. If r ≤ R/ n := ⌊ log ( R/r ) ⌋ , then r ≤ − n R and w ( r ) ≤ ε n w (2 n r ) ≤ ε − ( r/R ) α w ( R ) ≤ ε − ( r/R ) α sup (cid:0) | u | , ¯ C R (cid:1) , where α = − log ε ∈ (0 , u in any C r with r ≤ R/
2. The same estimate obviously holds for theoscillation of u in any C r ( t, x ) ⊂ C R as long as r ≤ R/ t, x ) ∈ C R .Now take z = ( t , x ) , z = ( t , x ) ∈ C R such that r := ρ ( z , z ) ≤ R/ t = t ∧ t , x = ( x + x ) / . Then we have z i ∈ ¯ C R ( t, x ), i = 1 ,
2, and (cid:12)(cid:12) u ( z ) − u ( z ) (cid:12)(cid:12) ≤ ε − ( r/R ) α sup (cid:0) | u | , ¯ C R ( t, x ) (cid:1) ≤ ε − ρ α ( z , z ) R − α sup (cid:0) | u | , ¯ C R (cid:1) . In the case that ρ ( z , z ) ≥ R/ (cid:12)(cid:12) u ( z ) − u ( z ) (cid:12)(cid:12) ≤ (cid:0) | u | , ¯ C (cid:1) ≤ α ρ α ( z , z ) R − α sup (cid:0) | u | , ¯ C (cid:1) . Thus, N = 2 α + 2 ε − in (4.5) is always a good choice with α foundabove. One can take any smaller α as well since ρ ( z , z ) ≤ N ( d ) R . Thelemma is proved. Remark . The constant N in (4.5), generally, depends on ¯ R . However,if ¯ b ∞ ≤ ε , where ε > d and δ , then this constant isindependent of ¯ R . This is proved by using self-similar transformations whichchange the process but allow us to take any R we wish. In such situationthe Liouville theorem is valid: If u is bounded and caloric in R d +1+ , then u is constant (just send R → ∞ in (4.5)).Here is the statement of the Harnack inequality. Theorem 4.4.
Let R ∈ (0 , ¯ R ] , and let u be a nonnegative caloric func-tion in ¯ C R ,R . Then there exists a constant N , which depends only on δ, d, R, ¯ R, p , ¯ b ¯ R , such that u ( R , ≤ N u (0 , x ) whenever | x | ≤ R/ . Proof. We basically repeat the proof of Theorem 6.1 in [10] and, toexclude a trivial situation, additionally assume that u ( R , > . IFFUSION PROCESSES WITH DRIFT IN L d +1 For κ = 1 / , η = 1 /
2, we take N and ν from Theorem 2.1, call this NN , and, having in mind Theorem 2.4, find γ ∈ (0 ,
1) close to 1 and ε > − ε ≥ q ( γ )2 − + (cid:2) − q ( γ ) (cid:3) ν . (4.8)Next, for r ∈ [0 , R ), introduce µ ( r ) = u ( R , − r/R ) − ν , n ( r ) = sup { u, ¯ C r ( R , } ( n (0) = u ( R , , and define r as the greatest number in r ∈ [0 , R ) satisfying n ( r ) = µ ( r ) . Such a number does exist because n (0) = µ (0), µ ( r ) → ∞ as r ↑ R , and n ( r ) is bounded, increasing, and (H¨older) continuous. Choose ( t , x ) ∈ ¯ C r ( R ,
0) such that n ( r ) = u ( t , x ) and consider the cylinder Q := n ( t, x ) : 0 ≤ t − t < ( R − r ) , | x − x | < R − r o . As is easy to see ¯ Q ⊂ ¯ C r ( R , r = ( R + r ) /
2. By the definitionof r , this implies thatsup ¯ Q u < µ ( r ) = u ( R , (cid:16) R − r R (cid:17) − ν ≤ ν n ( r ) . We claim that owing to this and (4.8), (cid:12)(cid:12) Q ∩ (cid:8) u > n ( r ) / (cid:9)(cid:12)(cid:12) ≥ (1 − γ ) | Q | . (4.9)To argue by contradiction, assume (4.9) is false. Then (cid:12)(cid:12) Q ∩ (cid:8) u ≤ n ( r ) / (cid:9)(cid:12)(cid:12) > γ | Q | and there is a closed set Γ ⊂ Q ∩ (cid:8) u ≤ n ( r ) / (cid:9) such that | Γ | > γ | Q | .Introduce τ Γ as the first time the process ( t + s, x t ) hits Γ and τ Q as the firsttime it exits from Q . It follows by definition, the strong Markov property asin the proof of Lemma 4.3, and from Theorem 2.4 that (note that n ( r ) / ≤ sup ¯ Q u ) u ( t , x ) = E t ,x I τ Γ <τ Q u ( t + τ Γ , x τ Γ ) + E t ,x I τ Γ ≥ τ Q u ( t + τ Q , x τ Q ) ≤ P t ,x ( τ Γ < τ Q ) n ( r ) / − P t ,x ( τ Γ < τ Q )) sup ¯ Q u ≤ q ( γ ) n ( r ) / − q ( γ )) sup ¯ Q u ≤ q ( γ ) n ( r ) / − q ( γ ))2 ν n ( r ) . Owing to (4.8) we now have n ( r ) ≤ (1 + ε ) n ( r ) (cid:2) q ( γ )2 − + (1 − q ( γ ))2 ν (cid:3) ≤ (1 − ε ) n ( r ) , which is impossible. This proves (4.9).Next we apply Theorem 3.10 and get that u ( t , x ) ≥ π n ( r )2 −
10 N.V. KRYLOV if | x − x | ≤ ( R − r )4 − , where π = π ( d, δ, p , R, ¯ R, ¯ b ¯ R , γ ) >
0. After thatit only remains to apply Theorem 2.1 to conclude that for | x | ≤ R we have u (0 , x ) ≥ π n ( r ) N − (cid:16) R − r (cid:17) ν = 2 − ν − π N − u (4 , . The theorem is proved.By using Lemma 4.3 and Theorem 6.7 one derives in three lines the fol-lowing analog of Theorem 6.5 of [10].
Theorem 4.5.
Assume that (3.14) holds with ν = 0 . Let R ∈ (0 , ¯ R/ and let g be a Borel bounded function on ¯ C R and f ∈ L p,q ( C R ) . For ( t, x ) ∈ C R define u ( t, x ) = E t,x Z τ R f ( t + s, x s ) ds + E t,x g ( t + τ R , x τ R ) , (4.10) where τ R is the first exit time of ( t + s, x s ) from C R . Then there exists aconstant N , which depends only on δ, d, R, ¯ R, p, p , and ¯ b ∞ , such that (cid:12)(cid:12) u ( z ) − u ( z ) (cid:12)(cid:12) ≤ N (cid:0) R − α ρ α ( z , z ) sup ¯ C R | g | + R (2 d − d ) /p k f k L p,q ( C R ) (cid:1) (4.11) for z , z ∈ C R , α ∈ (0 , α ] , and α is taken from Lemma 4.3. By playing with R for fixed z , z ∈ C R as in the proof of Theorem 6.5of [10] we get the following. Theorem 4.6.
Under the conditions and notation from Theorem 4.5 thereexists a constant N , which depends only on δ, d, R, ¯ R, p, p , and ¯ b ¯ R , suchthat (cid:12)(cid:12) u ( z ) − u ( z ) (cid:12)(cid:12) ≤ N R − β ρ β ( z , z ) (cid:0) sup ¯ C R | u | + R (2 d − d ) /p k f k L p,q ( C R ) (cid:1) (4.12) for z , z ∈ C R , where β = α (2 d − d ) α p + 2 d − d . As a standard consequence of just continuity of u we have the following. Theorem 4.7.
The process X = { ( t t , x t ) , ∞ , N t + , P t,x ) is strong Markov. Applications
Here we suppose that Assumption 4.1 is satisfied and set a = σ , Lu ( t, x ) = (1 / a ij ( t, x ) D ij u ( t, x ) + b i ( t, x ) D i u ( t, x ) . IFFUSION PROCESSES WITH DRIFT IN L d +1 Theorem 5.1.
Let R ∈ (0 , ¯ R/ and assume that (3.14) holds with ν = 0 , p < ∞ , q < ∞ and that we are given a function u ∈ W , p,q, loc ( C R ) ∩ C ( ¯ C R ) .Then for − f = ∂ t u + Lu we have (cid:12)(cid:12) u ( z ) − u ( z ) (cid:12)(cid:12) ≤ N R − β ρ β ( z , z ) (cid:0) sup ¯ C R | u | + R (2 d − d ) /p k f k L p,q ( C R ) (cid:1) (5.1) for z , z ∈ C R , where N and β are taken from Theorem 4.6. Proof. Approximating C R by C R − ε we see that we may assume that u ∈ W , p,q ( C R ) ∩ C ( ¯ C R ). This gives us the opportunity to replace L in thedefinition of f with L n := I | b |≥ n ∆ + I | b | Let R ∈ (0 , ¯ R ] and assume that (3.14) holds with ν = 0 , p < ∞ , q < ∞ . Let u ∈ W , p,q, loc ( C R ,R ) ∩ C ( ¯ C R ,R ) be such that u > on ∂ ′ C R ,R . Then there exists a constant N , which depends only on δ, d, R, ¯ R, p, p , and ¯ b ¯ R , such that u ( R , ≤ N u (0 , x ) + N R (2 d − d ) /p k f k L p,q ( C R ,R ) whenever | x | ≤ R/ , where − f = ∂ t u + Lu . In particular, if ∂ t u + Lu = 0 in C R ,R (a.e.), then (Harnack’s inequality) u ( R , ≤ N u (0 , x ) . Proof. As in the proof of Theorem 5.1, the general case is reduced to theone in which b is bounded and u ∈ W , p,q ( C R ,R ) ∩ C ( ¯ C R ,R ). In that case,as in the proof of Theorem 5.1, by Itˆo’s formula for the Markov process fromSection 4 u ( t, x ) = h ( t, x ) + F ( t, x ) , (5.2)where h ( t, x ) = E t,x u ( t + τ, x τ ) ≥ , F ( t, x ) = E t,x Z τ f ( t + s, x s ) ds, and τ is the first exit time of ( t + s, x s ) from C R ,R . By Theorem 4.4, h ( R , ≤ N h (0 , x ) and it only remains to use Theorem 6.7 to estimate F .The theorem is proved.Here is a generalization of the Fanghua Lin estimate for operators withsummable drift which is one of the main tools in the Sobolev space theoryof fully nonlinear parabolic equations (see, for instance, [8]). Theorem 5.3. Let R ∈ (0 , ¯ R ] , p, q satisfy (3.14) with ν = 0 , p < ∞ , q < ∞ .Let u ∈ W , p,q, loc ( C R ) ∩ C ( ¯ C R ) , and c ∈ L p,q ( C R ) , c ≥ . Then (cid:16) – Z C R (cid:0) | D u | + ( | b | + R − ) | Du | (cid:1) / (2 µ ) dxdt (cid:17) µ ≤ N R − d/p − /q k f k L p,q ( C R ) + N R − sup ∂ ′ C R | u | , (5.3) where − f = ∂ t u + Lu − cu , µ is taken from Corollary 3.7 with κ = 1 / and N depends only on d, δ, R, ¯ R, p, p , R − d/p − /q k c k L p,q ( C R ) , ¯ b R , ¯ b ¯ R , and thefunction ¯ N ( d, p , · ) (see (6.2) ). Proof. On the account of moving R , we may assume that u ∈ W , p,q ( C R ).After that we observe that in light of Theorem 6.9 k ∂ t u + Lu k L p,q ( C R ) ≤ k ∂ t u + Lu − cu k L p,q ( C R ) + k c k L p,q ( C R ) sup C R | u |≤ (cid:0) N R − d/p − /q k c k L p,q ( C R ) (cid:1) k ∂ t u + Lu − cu k L p,q ( C R ) + k c k L p,q ( C R ) sup ∂ ′ C R | u | and reduce the case of general c to the one with c ≡ 0. As a few timesbefore we may assume that b is bounded and then using approximationswe see that assuming that u ∈ C , ( ¯ C R ) and that the coefficients a ij areinfinitely differentiable do not restrict generality. In that case introduce L ′ ( t, x ) = L ( t, x ) for t ≥ L ′ ( t, x ) = L ( − t, x ) for t < v as a unique W , d +1 ( C R ,R ( − R , ∂ t v + L ′ v = − f I C R with boundary condition v = u on { t ≥ } ∩ ∂ ′ C R ,R ( − R , 0) and v ( t, x ) = u ( − t, x ) on { t ≤ } ∩ ∂ ′ C R ,R ( − R , v = u in C R and by Theorem 6.9 in C R ,R ( − R , 0) we have | v | ≤ N R − d/p − /q k f k L p,q ( C R ) + sup ∂ ′ C R | u | . (5.4) IFFUSION PROCESSES WITH DRIFT IN L d +1 Next, it is easy to see that for sufficiently small ε > 0, depending only on δ and the function ¯ N ( d, p , · ), we have that ˆ a := a − εI C R ( D ij u ) / | D u | ∈ S ( δ − ε ) / and ¯ N ( d, p , ( δ − ε ) / )¯ b R < b = b − εI C ¯ R ( | b | +1) Du/ | Du | any ρ > t, x ) ∈ R d +1 we have k ˆ b k L p ,q ( C ρ ( t,x )) ≤ (1 + ε ) k b k L p ,q ( C ρ ( t,x )) + εN ( d )( ρ ∧ ¯ R ) . It follows that k ˆ b k q L p ,q ( C ρ ( t,x )) ≤ (cid:16) (1 + ε )¯ b /q ρ + N ε (cid:17) q ρ, where N depends only on d, p , and ¯ R . It is seen that for a ε > 0, dependingonly on d, δ, p , ¯ R , and the function ¯ N ( d, p , · ), not only (5.5) is satisfied butalso ¯ N ( d, p , ( δ − ε ) / ) (cid:16) (1 + ε )¯ b /q R + N ε (cid:17) q < . Therefore the above theory is applicable to the operatorˆ L = (1 / a ij D ij + ˆ b i D i . Then set ˆ σ = ˆ a / and consider the diffusion process ( − R + t, x t ), t ≥ − R , 0) with diffusion matrix ˆ σ and drift ˆ b . By Itˆo’s formula v ( − R , 0) = − E Z τ ( ∂ t v + ˆ Lv )( − R + t, x t ) dt + Ev ( − R + τ, x τ ) , where τ is the first exit time of ( − R + t, x t ) from C R ,R ( − R , | Ev ( − R + τ, x τ ) | ≤ sup ∂ ′ C R | u | . Furthermore, on C R we have ∂ t v + ˆ Lv = ∂ t u + ˆ Lu = − f − ε | D u | − ε ( | b | + 1) | Du | . In the remaining part of C R ,R ( − R , 0) we have ∂ t v + ¯ Lv = 0. It followsthat E Z τ I C ε (cid:0) | D u | + ( | b | + 1) | Du | (cid:1) ( − R + t, x t ) dt ≤ v ( − R , − E Z τ I C R f ( − R + t, x t ) dt + sup ∂ ′ C R | u | . After that it only remains to recall (5.4) and apply Theorem 6.9 and Corol-lary 3.7. The theorem is proved. Appendix Here we present without proofs some results from [12] frequently used inthe main text.Set τ ′ R ( x ) = inf { t ≥ x + x t B R } , γ R ( x ) = inf { t ≥ x + x t ∈ ¯ B R } . (6.1) Theorem 6.1 (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 ≤ , (6.2) then for | x | ≤ RP ( τ R ( x ) = R ) ≤ − ¯ ξ, P ( τ R = R ) ≥ ¯ ξ. (6.3) Moreover for n = 1 , , ... and | x | ≤ RP ( τ ′ R ( x ) ≥ nR ) = P ( τ nR ,R ( x ) = nR ) ≤ (1 − ¯ ξ ) n , (6.4) so that Eτ ′ R ( x ) ≤ N ( d, δ ) R .Furthermore, for any x ∈ ¯ B R/ P ( τ ′ R ( x ) > γ R/ ( x )) ≥ ¯ ξ. (6.5) Theorem 6.2 (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, (6.6) where ˙ λ = λ min(1 , λ/λ ) , λ = R − . 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) . (6.7) Theorem 6.3 (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 ) . Thenthere exist T , T , depending only on ¯ ξ , such that < T < T < ∞ andthe probability π that x + x t will reach ¯ B R/ ( y ) before exiting from S R ( x, y ) and this will happen on the time interval [ nT R , nT R ] is greater than π n ,where n = j | x − y | + R R k and π = ¯ ξ/ . Theorem 6.4 (Theorem 4.3) . There exists d ∈ (1 , d ) , depending only on δ , d , R , p , such that 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 ) . IFFUSION PROCESSES WITH DRIFT IN L d +1 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 − . Theorem 6.5 (Theorem 4.8) . Suppose p, q ∈ [1 , ∞ ] , ν := 1 − d p − q ≥ . (6.8) 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+ ) , (6.9) 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 6.6 (Theorem 4.9) . Assume that (6.8) 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+ ) , (6.10) 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+ ) , (6.11) where N = N ( δ, d, R, p, q, p , ¯ b ∞ ) . Theorem 6.7 (Theorem 4.10) . Assume that (6.8) holds with ν = 0 . Thenfor 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 ) , (6.12) where N = N ( δ, d, R, p, p , ¯ b ¯ R , ¯ R ) . Theorem 6.8 (Theorem 4.11) . Assume that (6.8) 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 (6.13) and the stochastic integral above is a square-integrable martingale. Theorem 6.9 (Theorem 5.1) . Let < R ≤ ¯ R , domain Q ⊂ C R , andassume that (3.14) holds with ν = 0 , p < ∞ , q < ∞ , and that we are givena function u ∈ W , p,q, loc ( Q ) ∩ C ( ¯ Q ) . Take a function c ≥ on Q . 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