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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf,len=788
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='13805v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='PR]  31 Jan 2023  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS  MORREY DRIFT  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We prove the unique weak solvability of stochastic differential equations with time-  inhomogeneous drift in essentially the largest (scaling-invariant) Morrey class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' with integra-  bility parameter q > 1 close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The constructed weak solutions constitute a Feller evolution  family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The proofs are based on a detailed Sobolev regularity theory of the corresponding  parabolic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We consider the problem of weak well-posedness of stochastic differential equation  Xt = x −  � t  0  b(r, Xr)dr +  √  2Bt,  x ∈ Rd,  (1)  where Bt is a Brownian motion in Rd, under minimal assumptions on the time-inhomogeneous  vector field b : R × Rd → Rd (drift), d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  This equation or, more generally, stochastic  equations additionally having variable, possibly discontinuous diffusion coefficients, arise e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' in  the problems of stochastic optimization and serve as a basis for many physical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' This  requires, generally speaking, dealing with irregular, locally unbounded drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' An illustrative  example is equation (1) with velocity field b obtained by solving 3D Navier-Stokes equations,  which models the motion of a small particle in a turbulent flow [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' One is thus led to the  problem of establishing weak and strong well-posedness of (1) under minimal assumptions on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The latter can also be stated as the problem of finding the most general integral characteristics  of b that determine whether (1) is weakly/strongly well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Let us give a brief outline of the literature on stochastic differential equations (SDEs) with  singular drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We will try to keep the chronological order, but will be somewhat loose with the  terminology by including in “well-posedness” the uniqueness results of different strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The “sub-critical” Ladyzhenskaya-Prodi-Serrin class  |b| ∈ Ll(R, Lp(Rd)),  p ≥ d, l ≥ 2,  d  p + 2  l < 1  (2)  was attained by Portenko [28] (weak solutions) and Krylov-Röckner [25] (strong solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' See  also Zhang [34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Between [28] and [25], Bass-Chen [3] proved existence and uniqueness  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' 60H10, 47D07 (primary), 35J75 (secondary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Stochastic differential equations, weak solutions, singular drifts, Morrey class, Feller  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The research of the author is supported by the NSERC (grant RGPIN-2017-05567).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  1  2  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  in law of weak solutions of (1) for b = b(x) in the Kato class of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The Kato class  contains {|b| ∈ Lp(Rd), p > d} as well as some vector fields with entries not even in L1+ε  loc (Rd),  ε > 0, however, it does not contain {|b| ∈ Ld(Rd)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Speaking of time-homogeneous drifts, of  course, the fact that p = d is the optimal exponent on the scale of Lebesgue spaces can be seen  from rescaling the parabolic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  In [4], Beck-Flandoli-Gubinelli-Maurelli developed an approach to proving strong well-posedness  of (1) with b in the critical Ladyzhenskaya-Prodi-Serrin class  |b| ∈ Ll(R, Lp(Rd)),  p ≥ d, l ≥ 2,  d  p + 2  l ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (3)  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' starting point x ∈ Rd via stochastic transport and stochastic continuity equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' They  also considered the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let  b(x) =  √  δ d − 2  2  1|x|<1|x|−2x,  (4)  so |b| just misses to be Ld(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' If δ > 4(  d  d−2)2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' the attraction to the origin by the drift is  large enough, then SDE (1) with the starting point x = 0 does not have a weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In [15],  Semënov and the author showed that the Feller generator ∆−b·∇ with “weakly form-bounded”  b = b(x), see (23) below, determines for every starting point x ∈ Rd a weak solution to (1)  that is, moreover, unique among weak solutions that can be constructed via approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' (To  the best of the author’s knowledge, this was the first result on weak well-posedness of (1) that  included both |b| ∈ Ld(Rd) and the model vector field (4) with δ small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' it also included the  elliptic Morrey class with q > 1 and the Kato class considered by Bass-Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=') Returning to  time-inhomogeneous drifts, we note that almost at the same time Wei-Lv-Wu [32], and later Nam  [27], obtained results on weak well-posedness of (1) for every x ∈ Rd for time-inhomogeneous  vector fields b that can be more singular than the ones in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Nevertheless, their results excluded  b = b(x) with |b| ∈ Ld(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In [21], Krylov proved strong well-posedness of (1), for every initial  point and |b| ∈ Ld(Rd), by developing an approach based on his and Veretennikov’s old criterion  for a weak solution to be a strong solution [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In [33], Xia-Xie-Zhang-Zhao established, among  other results, weak well-posedness of (1) for every initial point and b ∈ Cb(R, Ld(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Röckner-  Zhao [29] furthermore established weak well-posedness of (1), with any x ∈ Rd, for drifts in  L∞(R, Ld,w(Rd)), plus the drifts in the critical LPS class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Here Ld,w(Rd) denotes the weak Ld  class that contains vector fields in Ld(Rd), as well as more singular vector fields, such as (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In  [30], they obtained strong well-posedness of (1), for any starting point and b in the critical LPS  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In [13], the author and Madou established weak well-posedness of (1), for every starting  point and form-bounded drifts, see example 5) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' This class contains L∞(R, Ld,w(Rd)) as  well as some drifts that are not even in L∞(R, L2+ε(Rd)) for a given ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' By the way, in [18]  Semënov, Song and the author showed that the approach of [4] to the strong well-posedness of  (1) via the stochastic transport/continuity equations also works for form-bounded vector fields  b = b(x), although, again, one obtains strong well-posedness of (1) only for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  In a recent major advancement, Krylov [23] further verified the Veretennikov-Krylov criterion  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  3  for drifts in a large parabolic Morrey class:  b ∈ Eq,  q > d  2 + 1,  (5)  see definition below (ignoring here some differences with [23] in the definition of parabolic Morrey  class), thus establishing, for every starting point, strong well-posedness of (1) with drift in class  (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Class (5) contains L∞(R, Ld,w) as well as some drifts that are not in L∞(R, Lq+ε) for a  given ε > 0 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' example in [22, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let us note in passing that some steps in his proof,  such as gradient estimates, are, in fact, carried out for a larger class of form-bounded vector  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The above outline does not discuss many interesting results on distribution-valued drifts and  on the drifts satisfying additional assumptions on their structure, such as div b ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We also did  not discuss results on partial well-posedness of (1) with |b| ∈ Lp(Rd) in the supercritical regime  d  2 < p < d (in the sense of scaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In this regard, see Zhao [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  In the present paper we consider drifts in the Morrey class Eq with integrability parameter  q > 1 that can be chosen arbitrarily close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Denote  Cr(t, x) := {(s, y) ∈ Rd+1 | t ≤ s ≤ t + r2, |x − y| ≤ r}  and, given a vector field b : Rd+1 → Rd with components in Lq  loc(Rd+1), q ∈ [1, d + 2], set  ∥b∥E+  q :=  sup  r>0,z∈Rd+1 r  � 1  |Cr|  �  Cr(z)  |b(t, x)|qdtdx  � 1  q  and, reversing the direction of time in b,  ∥b∥E−  q :=  sup  r>0,z∈Rd+1 r  � 1  |Cr|  �  Cr(z)  |b(−t, x)|qdtdx  � 1  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We say that a vector field b belongs to the parabolic Campanato-Morrey (or, for  brevity, Morrey) class Eq if ∥b∥Eq := ∥b∥E+  q ∨ ∥b∥E−  q  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  One has  ∥b∥Eq ≤ ∥b∥Eq1 if q < q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  If above b = b(x), then one obtains the usual elliptic Morrey class Mq, that is, |b| ∈ Lq  loc(Rd)  and  ∥b∥Mq :=  sup  r>0,y∈Rd r  � 1  |Br|  �  Br(y)  |b(x)|qdx  � 1  q  < ∞,  where Br(y) is the closed ball of radius r centered at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Our result, stated briefly, is as follows (see Theorems 1-3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We will be using some  notations defined in the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let d ≥ 3, let b : Rd+1 → Rd be a vector field in the Morrey class Eq with q > 1 close  to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let p ∈]1, ∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' There exists a constant cd,p,q such that if ∥b∥Eq < cd,p,q, then the following  are true:  4  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  (i) There exists a unique weak solution to  (λ − ∂t − ∆ + b(t, x) · ∇)u = 0,  t < r,  u(r, ·) = g(·) ∈ Lp(Rd) ∩ L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The difference  u(t, ·) −  e−λ(r−t)  (4π(r − t))  d  2  �  Rd e− |·−y|2  4(r−t)g(y)dy  (t < r),  extended by 0 to t > r,  is in the parabolic Bessel potential space W1+ 1  p ,p(Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (ii) For p > d + 1, operators {P t,r}t<r defined by  P t,rg := u(t),  g ∈ C∞(Rd) ∩ L2(Rd)  are extended to a backward Feller evolution family on C∞(Rd) that determines, for every x ∈ Rd,  a weak solution to SDE (1) that is, moreover, unique in an appropriate class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Here are some examples of vector fields in Eq, q > 1:  1) The critical Ladyzhenskaya-Prodi-Serrin class  |b| ∈ Ll(R, Lp(Rd)),  p ≥ d, l ≥ 2,  d  p + 2  l ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  To prove the inclusion of this class into Eq it suffices to consider, by an elementary interpolation  argument, only the cases l = 2, p = ∞ and l = ∞, p = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In the former case the inclusion is  trivial, in the latter case the inclusion follows using Hölder’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  This example is strengthened in the next two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  2) The vector fields b with |b| ∈ L2,w(R, L∞(Rd)) are in Eq for 1 < q < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Indeed, by a well  known characterization of weak Lebesgue spaces, we have  r  � 1  |Cr|  �  Cr  |b|qdz  � 1  q  ≤ Cr  � 1  r2  � t+r2  t  |˜b|qds  � 1  q  ˜b(t) := ∥b(t, ·)∥L∞(Rd)  ≤ C∥˜b∥L2,w(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Hence, for example, a vector field b that satisfies  ∥b(t, ·)∥L∞(Rd) ≤ C  √  t,  t > 0  (and defined to be zero for t ≤ 0) is in Eq with 1 < q < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  3) Moreover, by the well known inclusion of the weak Lebesgue space Ld,w(Rd) in the elliptic  Morrey class,  |b| ∈ L∞(R, Ld,w(Rd))  ⇒  b ∈ Eq with 1 < q ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  4) For every ε > 0, one can find a vector field b ∈ Eq such that |b| is not in Lq+ε  loc (Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (In particular, selecting q > 1 sufficiently close to 1, we obtain vector fields b satisfying the  assumptions of the theorem, and that are not in L1+ε  loc (Rd+1) for a given ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=')  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  5  5) A vector field b is said to be form-bounded if |b| ∈ L2  loc(Rd+1) and for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' t ∈ R  ∥b(t, ·)ϕ∥2  L2(Rd) ≤ δ∥∇ϕ∥2  L2(Rd) + g(t)∥ϕ∥2  L2(Rd)  for all ϕ = ϕ(·) ∈ C∞  c (Rd), for some constant δ > 0 and a function g ∈ L1  loc(R) (written as  b ∈ Fδ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The constant δ is called a form-bound of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  This class itself contains 1) and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In particular, by Hardy’s inequality, the vector field (4)  is in Fδ with g = 0 (but not in any Fδ′ with δ′ < δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Note that if a form-bounded vector field b depends only on time, then b ∈ L2  loc(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' One can  compare this with example 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let us say a few more words about the form-bounded vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In the time-homogeneous  case b = b(x) the form-boundedness of b ensures, by the Lax-Milgram Theorem, that, whenever  δ < 1, the formal operator ∆ − b · ∇ has a realization as the generator of a quasi-contraction  semigroup in L2(Rd) which, moreover, is bounded as an operator W 1,2(Rd) → W −1,2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The  form-boundedness of b with δ < 1 is essentially the broadest condition on |b| that provides the  minimal “classical” theory of operator −∆ + b · ∇ in L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  By considering dilates and translates of a bump function, one can show that the class of  form-bounded vector fields Fδ is contained in E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' (Let us note, on the other hand, that the  time-homogeneous vector fields in Eq, q > 2 are form-bounded [7], see also [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' This remark  extends right away e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' to time-inhomogeneous vector fields b = b(t, ·) having uniformly bounded  in t elliptic Morrey norm ∥ · ∥Mq, q > 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' they are, obviously, contained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=') Note also that if  b is form-bounded with g = 0, then ∥b∥E2 = c  √  δ for appropriate constant c = cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The class of form-bounded vector fields is well known in the literature on regularity theory of  parabolic equations [19] (although, perhaps, less so than the Morrey class).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' As we mentioned  earlier, in the context of SDEs, condition b ∈ Fδ with δ < d−2 was used to develop a Sobolev  regularity theory of the corresponding parabolic equation and construct a Feller evolution family  [11] that determines, for every x ∈ Rd, a weak solution to (1) which is, moreover, unique in a  broad class of weak solutions [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In the present paper we deal with time-inhomogeneous vector fields that can be more  singular than the ones covered by the form-boundedness condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' These new vector fields are  situated in Eq − Es, 1 < q ≤ 2, s > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let us note that the distinction between Eq, 1 < q < 2  and smaller class Es, s > 2 is not only quantitative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Namely, consider a vector field b = b(x) in  Es, s > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Since it is form-bounded, the term b · ∇ in the parabolic equation can be handled  using quadratic inequality:  �  Rd(b · ∇u)udx ≤ γ  �  Rd |b|2u2dx + 1  4γ  �  Rd |∇u|2dx,  γ > 0,  (6)  where one estimates  �  Rd |b|2u2dx from above, using form-boundedness of b, by  �  Rd |∇u|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' This  elementary argument plays a key role e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' in the proof of gradient estimates in [4, 23, 11], needed  to prove well-posedness of the corresponding stochastic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' However, quadratic inequality  (6) is unavailable under our assumption b ∈ Eq, 1 < q < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  6  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The proofs in [13] use a Feller evolution family constructed in [11] using a parabolic variant  of the iteration procedure of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In this paper we pursue a simpler operator-theoretic approach,  replacing the iterations with the Duhamel series (written in “resolvent form”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' we  construct solution to inhomogeneous equation  (λ + ∂t − ∆ + b(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' x) · ∇)u = f ∈ Lp(Rd+1)  (which is essential to the rest of the paper) as  u := (λ + ∂t − ∆)−1f − (λ + ∂t − ∆)− 1  2− 1  2p Qp(1 + Tp)−1Rp(λ + ∂t − ∆)− 1  2p′ f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (7)  where p′ :=  p  p−1 and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' if b ∈ Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' q > 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' the operators  Rp = b  1  p · ∇(λ + ∂t − ∆)− 1  2 − 1  2p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Qp = (λ + ∂t − ∆)− 1  2p′ |b|  1  p′  are bounded on Lp(Rd+1) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' provided ∥b∥Eq is sufficiently small and λ is large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' the operator  Tp = RpQp has norm ∥Tp∥p→p < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' so the Duhamel series converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The regularizing factor  (λ+∂t−∆)−1/2−1/2p in (7) now yields the sought regularity of solution u provided that p is large  (> d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' A similar argument was used in [12] in the elliptic setting, where an even larger than  {b = b(x)} ∩ Eq, q > 1 class of time-homogeneous vector fields was treated (see (23) below);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' the  weak well-posedness of SDEs with such drifts was addressed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The proof of the gradient  estimates in [12], however, depends on the symmetry of the resolvents, and the transition from  elliptic estimates to the results for the parabolic equations required development of some new  approaches to constructing semigroups, using old ideas of Hille (pseudoresolvents) and Trotter,  see also [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In the present paper the proofs are much shorter since we work directly with the  parabolic operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' See also discussion after Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The necessity of the assumption “∥b∥Eq cannot be too large” follows from the aforementioned  counterexample to solvability of (1) with drift (4) when x = 0 and δ > 4(  d  d−2)2 (note that there  ∥b∥Eq = c  √  δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' It was recently proved in [17] that, given an arbitrary b ∈ Fδ with δ < 4, SDE  (1) has a weak solution for every starting point x ∈ Rd, so the above example and this result  become essentially sharp in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' (The fact that δ = 4 is critical can be seen from  multiplying the parabolic equation (11) corresponding to (1) by u|u|p−2, integrating by parts  and using quadratic inequality and form-boundedness as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The admissible p that give  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' an energy inequality turn out to be p >  2  2−  √  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=')  Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Set for 0 < α ≤ 2  (λ − ∂t − ∆)− α  2 h(t, x) :=  � ∞  t  �  Rd e−λ(s−t)  1  (4π(s − t))  d  2  1  (s − t)  2−α  2  e− |x−y|2  4(s−t) h(s, y)dsdy,  (8)  (λ + ∂t − ∆)− α  2 h(t, x) :=  � t  −∞  �  Rd e−λ(t−s)  1  (4π(t − s))  d  2  1  (t − s)  2−α  2  e− |x−y|2  4(t−s) h(s, y)dsdy,  (9)  where λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' By a standard result, if λ > 0, then these operators are bounded on Lp(Rd+1),  1 ≤ p ≤ ∞, with operator norm λ− α  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' If λ > 0, then (λ ± ∂t − ∆)−1 is the resolvent of a Markov  generator on Lp(Rd+1), 1 ≤ p < ∞, which we will denote by λ ± ∂t − ∆, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' (The  abuse of notation resulting from not indicating p should not cause any confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=') In particular,  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  7  one has well defined fractional powers (λ ± ∂t − ∆)  α  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We refer to articles [2, 9], among others,  regarding the properties of these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Denote by ⟨ , ⟩ the integration in d + 1 variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  ⟨h⟩ :=  �  Rd+1 hdz,  ⟨h, g⟩ := ⟨hg⟩  (all functions considered in this paper are real-valued).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  We denote by B(X, Y ) the space of bounded linear operators between Banach spaces X → Y ,  endowed with the operator norm ∥ · ∥X→Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Set B(X) := B(X, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Put  ∥h∥p := ⟨|h|p⟩  1  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Denote by ∥ · ∥p→q the (Lp(Rd+1), ∥ · ∥p) → (Lq(Rd+1), ∥ · ∥q) operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  We write T = s-X- limn Tn for T, Tn ∈ B(X) if Tf = limn Tnf in X for every f ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Let λ > 0 be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Set  Wα,p(Rd+1) := (λ + ∂t − ∆)− α  2 Lp(Rd+1)  endowed with the norm ∥h∥Wα,p := ∥(λ + ∂t − ∆)− α  2 h∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Denote by C∞(Rd) the space of continuous functions on Rd vanishing at infinity, endowed  with the sup-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Define in the same way C∞(Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  We fix T > 0 and put  DT := {(s, t) ∈ R2 | 0 ≤ s ≤ t ≤ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Recall that a family of positivity preserving L∞ contractions {Ut,s}(s,t)∈DT ⊂ B(C∞(Rd)) is  called a Feller evolution family (on C∞(Rd)) if Ut,rUr,s = Ut,s for all r ∈ [s, t], Us,s = Id and  Ur,s = s-C∞(Rd)- lim  t↓r Ut,s  for all s ≤ r < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Define the following regularization of b : Rd+1 → Rd, b ∈ Eq, q > 1:  bn := 1nb,  where 1n is the indicator of {|b| ≤ n} ⊂ Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (10)  We can additionally mollify bn to obtain a C∞ smooth approximation of b such that the Morrey  norm of the approximating vector field does not exceed (1 + ε)∥b∥Eq for any fixed ε > 0, as is  needed to turn a priori Sobolev regularity estimates for (11) into a posteriori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' However,  a regularization of b given by (10) will suffice (in particular, we will be able to apply Itô’s formula  to solutions of parabolic equations with drift bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Put b  1  p := b|b|−1+ 1  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Let E = Ep := ∪ε>0e−ε|b|Lp(Rd+1), a dense subspace of Lp(Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The author is grateful to Renming Song for some useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  8  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Main results  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We first develop a Sobolev regularity theory of the inhomogeneous parabolic equation  (λ + ∂t − ∆ + b(t, x) · ∇)u = f  on Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (11)  The next theorem is essential for the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Theorem 1 (Sobolev regularity theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let b = bs + bb, where  |bs| ∈ Eq for some q > 1 close to 1, and |bb| ∈ L∞(Rd+1)  (12)  (indices s and b stand for “singular” and “bounded”, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The following are true:  (i) For every p ∈]1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' ∞[ there exist constants cd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='q and λd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='q such that if  ∥bs∥Eq < cd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' for every λ ≥ λd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' solutions un ∈ Lp(Rd+1) to the approximating parabolic equations  (λ + ∂t − ∆ + bn · ∇)un = f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  f ∈ Lp(Rd+1)  converge in W1+ 1  p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p(Rd+1) to  u := (λ + ∂t − ∆)−1f − (λ + ∂t − ∆)− 1  2− 1  2p Qp(1 + Tp)−1Rp(λ + ∂t − ∆)− 1  2p′ f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (13)  where the operators  Rp = Rp(b) := b  1  p · ∇(λ + ∂t − ∆)− 1  2− 1  2p ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Qp = Qp(b) :=  �(λ + ∂t − ∆)− 1  2p′ |b|  1  p′ ↾ E  �clos  p→p  (14)  are bounded on Lp(Rd+1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' and the operator Tp := RpQp has norm  ∥Tp∥p→p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (15)  (ii) If above p > d+1, then, by (13) and by the parabolic Sobolev embedding, the convergence  is uniform on Rd+1 and u ∈ C∞(Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' If b is bounded, then Qp = (λ+∂t −∆)− 1  2p′ |b|  1  p′ , and so the RHS of (13) is simply  the Duhamel series representation for the solution to (11) in Lp(Rd+1) provided by the standard  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The constraint ∥bs∥Eq < cd,p,q is needed to ensure that ∥Tp∥p→p < 1, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' If e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' b ∈ C∞(R, Ld(Rd)) or b = b(x) is in Ld(Rd), then one can represent b = bs + bb with  ∥bs∥Eq arbitrarily small (by defining bb to be a cutoff of b such that the remaining part bs has  sufficiently small L∞(R, Ld(Rd)) norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Given a general b satisfying (12), it is natural to ask, in what sense u defined by (13) solves  the parabolic equation (11)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let us consider the case p = 2 and f ∈ L2(Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  9  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We say that a function u ∈ W  3  2 ,2 is a weak solution to (11) if the following identity  is satisfied:  ⟨(λ + ∂t − ∆)  3  4u, (λ + ∂t − ∆)  3  4 η⟩ + ⟨R2(b)(λ + ∂t − ∆)  3  4u, Q∗  2(b)(λ + ∂t − ∆)  3  4 η⟩  = ⟨f, (λ − ∂t − ∆)− 1  4 (λ + ∂t − ∆)  3  4 η⟩  (16)  for all η ∈ C∞  c (Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (Identity (16) is obtained by formally multiplying equation (11) by (λ−∂t−∆)− 1  4(λ+∂t−∆)  3  4 η  and integrating over Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Note that Q∗  2(b) = |b|  1  2 (λ �� ∂t − ∆)− 1  4 is in B(L2) by Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=')  Then u defined by (13) is the unique in W  3  2,2 weak solution to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' See Remark 4 for the  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Thus, in order to have weak well-posedness of (11) for drifts satisfying (12) for  q > 1 we have to shift the standard scale of Hilbert spaces W1,2 ⊂ L2(Rd+1) ⊂ W−1,2 to  W  3  2,2 ⊂ W  1  2 ,2 ⊂ W− 1  2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' If we were to consider instead of (11) a more general equation (λ + ∂t −  ∇ · a · ∇ + b · ∇)u = f with a uniformly elliptic discontinuous matrix a, then the second order  term would force us to work in the standard scale, and hence would require more restrictive  assumptions on b: the form-boundedness, see the beginning of the paper (on the scale of Morrey  spaces this will be (12) with q > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  An analogous result can be obtained for f ∈ Lp(Rd+1) with (16) modified according to (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Fix T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' For given n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' and 0 ≤ r < T, let vn ∈ Cb([r, T], C∞(Rd)) denote the  solution to the Cauchy problem  \uf8f1  \uf8f2  \uf8f3  (λ + ∂t − ∆ + bn(t, x) · ∇)vn = 0  (t, x) ∈]r, T] × Rd,  vn(r, ·) = g(·) ∈ C∞(Rd),  (17)  where bn’s are defined by (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' By a standard result, for every n, the operators  Ut,r  n g := vn(t),  0 ≤ r ≤ t ≤ T  constitute a Feller evolution family on C∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Let δs=r denote the delta-function in the time variable s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Put  (λ + ∂t − ∆)−1δs=rg(t, x) := 1t≥re−λ(t−r)(4π(t − r))− d  2  �  Rd e− |x−y|2  4(t−r) g(y)dy,  ∇(λ + ∂t − ∆)− 1  2 − 1  2p′ δs=rg := 1t≥re−λ(t−r)(t − r)− 1  2+ 1  2p′ (4π(t − r))− d  2  �  Rd ∇xe− |x−y|2  4(t−r) g(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Recall DT = {0 ≤ r ≤ t ≤ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Theorem 2 (C∞ theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Under the assumptions of Theorem 1, let ∥b∥Eq < cd,p,q for a p > d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Then the following are true:  (i) The limit  Ut,r := s-C∞(Rd)- lim  n Ut,r  n  uniformly in (r, t) ∈ DT  exists and determines a Feller evolution family on C∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  10  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  (ii) For every initial function g ∈ C∞(Rd) ∩ W 1,p(Rd), v(t) := Ut,rg, where (r, t) ∈ DT , has  representation  v = (λ + ∂t − ∆)−1δs=rg − (λ + ∂t − ∆)− 1  2− 1  2p Qp(1 + Tp)−1GpSpg,  (18)  where Gp = Gp(b) := b  1  p (λ + ∂t − ∆)− 1  2p ∈ B(Lp(Rd+1)) and Spg := ∇(λ + ∂t − ∆)− 1  2− 1  2p′ δs=rg  satisfies ∥Spg∥Lp(Rd+1) ≤ Cp,d∥∇g∥Lp(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (iii) As a consequence of (18) and the parabolic Sobolev embedding, we obtain  sup  (r,t)∈DT ,x∈Rd |v(t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' r)| ≤ C∥g∥W 1,p(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Recall that a probability measure Px, x ∈ Rd on (C([0, T], Rd), Bt = σ(ωr | 0 ≤ r ≤ t)),  where ωt is the coordinate process, is said to be a martingale solution to SDE  Xt = x −  � t  0  b(r, Xr)dr +  √  2Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (19)  if  1) Px[ω0 = x] = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  2) Ex  � r  0 |b(t, ωt)|dt < ∞, 0 < r ≤ T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  3) for every f ∈ C2  c (Rd) the process  r �→ f(ωr) − f(x) +  � r  0  (−∆f + b · ∇f)(t, ωt)dt  is a Br-martingale under Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  A martingale solution Px of (19) is said to be a weak solution if, upon completing Bt with  respect to Px (to, say, ˆBt), there exists a Brownian motion Bt on  �C([0, T], Rd), ˆBt, Px  � such that  ωr = x −  � r  0  b(t, ωt)dt +  √  2Br,  r ≥ 0  Px-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Put  P t,r(b) := UT−t,T−r(˜b),  ˜b(t, x) = b(T − t, x)  where 0 ≤ t ≤ r ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Under the assumptions of Theorem 2 the following are true:  (i) The backward Feller evolution family {P t,r}0≤t≤r≤T is conservative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' the density P t,r(x, ·)  satisfies  ⟨P t,r(x, ·)⟩ = 1  for all x ∈ Rd,  and determines probability measures Px, x ∈ Rd on (C([0, T], Rd), Bt), such that  Ex[f(ωr)] = P 0,rf(x),  0 ≤ r ≤ T,  f ∈ C∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (ii) For every x ∈ Rd, the probability measure Px is a weak solution to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  11  (iii) For every x ∈ Rd and f satisfying (12), given a p > d + 1 as in Theorem 2, there exists  a constant c such that for all h ∈ Cc(Rd+1)  Ex  � T  0  |f(r, ωr)h(r, ωr)|dr ≤ c∥1[0,T]|f|  1  p h∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (In particular, one can take f = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=')  (iv) Any martingale solution Qx to (19) that satisfies  EQx  � T  0  |b(r, ωr)h(r, ωr)|dr ≤ c∥1[0,T]|b|  1  p h∥p,  h ∈ Cc(Rd+1),  for some p > d + 1,  (20)  coincides with Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Key bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Here is the key bound used in the proofs of Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let |b| ∈ Eq for some q > 1 close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Then for every p ∈]1, ∞[ there exists a  constant cp,q such that  ∥|b|  1  p (± ∂t − ∆)− 1  2p ∥p→p ≤ cp,q∥b∥  1  p  Eq  In the time homogeneous case b = b(x) the estimate on ∥|b|  1  p (λ−∆)− 1  2p ∥Lp(Rd)→Lp(Rd) in terms  of the elliptic Morrey norm of |b| is due to [1, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Similar estimates in the parabolic  case were obtained in [24, proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' There the proof is carried out for a different set  of parameters than the one needed in this paper, so we included the details in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  As an immediate consequence of Proposition 1 we obtain the following  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let b = bs + bb, satisfy (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Then, for all λ > 0,  ∥Rp∥p→p ≤ Cd,p,q∥bs∥  1  p  Eq + cλ− 1  2p ∥bb∥  1  p  L∞(Rd+1)  (21)  ∥Qp∥p→p ≤ C′  p,q∥bs∥  1  p′  Eq + c′λ− 1  2p′ ∥bb∥  1  p′  L∞(Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (22)  The first estimate (21) follows from Proposition 1 using the boundedness of parabolic Riesz  transforms (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The second estimate (22) follows from Proposition 1 by duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Some remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Theorems 1, 2 are time inhomogeneous counterparts of the results in [11], Theorem  3 is a time inhomogeneous counterpart of the result in [15], where the authors treated vector  fields b : Rd → Rd that can be more singular than the ones considered in this paper, namely,  |b| ∈ L1  loc(Rd) and there exists δ > 0 such that, for some λ > 0,  ��|b|  1  2ϕ  ��  L2(Rd) ≤ δ  ��(λ − ∆)  1  4 ϕ  ��  L2(Rd),  ∀ ϕ ∈ C∞  c (Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (23)  These vector fields are called weakly form-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The fact that the time-homogeneous vector  fields in Eq, 1 < q ≤ 2 are weakly form-bounded follows from [1, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' A key difference  between [11] and the present paper is in the elliptic analogue of estimate (15), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  ∥ ˜Tp∥Lp(Rd)��Lp(Rd) < 1,  where ˜Tp := b  1  p · ∇(µ − ∆)−1|b|  1  p′ ,  (24)  12  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  needed to ensure the convergence of the Neumann series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' This estimate is proved in [11] directly,  without splitting ˜Tp into a product of operators b  1  p ·∇(µ−∆)− 1  2+ 1  2p and (µ−∆)− 1  2p′ |b|  1  p′ as we do  for Tp in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In fact, the previous two operators are not even bounded on Lp(Rd) under  the assumption (23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' to have the Lp(Rd) boundedness one has to replace the exponents − 1  2 + 1  2p,  − 1  2p′ by − 1  2 + 1  2p − ε, − 1  2p′ − ε for a ε > 0 (this shows, by the way, that the difference between  the elliptic Morrey class Mq with q > 1 and the larger class (23) is quite significant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' However,  the proof of (24) in [11] requires inequalities for symmetric Markov generators, not valid for the  parabolic operator ∂t − ∆ (although, it seems, one can address a parabolic counterpart of (23)  via an appropriate symmetrization of ∂t − ∆, which we plan to do in a subsequent paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Despite what was said in the introduction, [13, 11] and the present paper are not  directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Namely, the results in [13, 11] with b ∈ Fδ can be extended more or less  directly to the non-divergence form equation  � − a(t, x) · ∇2 + b(t, x) · ∇  �u = f,  u(s, ·) = g(·),  t > s,  (25)  and the corresponding Itô’s SDE having “form-bounded diffusion coefficients”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Namely, the  matrix a is assumed to be bounded, uniformly elliptic and have  ∇aij ∈ Fδij,  1 ≤ i, j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (26)  See [16] where this scheme was carried out in the time-homogeneous case b = b(x), a = a(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (Let us note that since ∇aij(x) are form-bounded, they are in the elliptic Morrey class with  q = 2, see the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Hence by Poincaré’s inequality such aij belong to the VMO class,  see details in [22, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=') However, a similar extension of the results of the present paper for  (1) with b ∈ Eq, 1 < q ≤ 2 requires more regular a, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Some corollaries of Theorem 1 and Theorem 2(ii)  The following results will be needed in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let vector fields b, f satisfy (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let bn be given by (10) and let us define fn  similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Then, under the assumptions on ∥bs∥Eq of Theorem 1, for every λ ≥ λd,p,q solutions  un ∈ Lp(Rd+1) to the approximating parabolic equations  (λ + ∂t − ∆ + bn · ∇)un = |fn|h,  h ∈ Cc(Rd+1)  converge in W1+ 1  p ,p(Rd+1) to  u := (λ + ∂t − ∆)−1|f|h − (λ + ∂t − ∆)− 1  2− 1  2p Qp(b)(1 + Tp(b))−1Rp(b)Qp(f)|f|  1  p h,  In particular, if p > d + 1, then the convergence is uniform on Rd+1 and  sup  Rd+1 |u| ≤ C∥f  1  p h∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  13  The reason we include h in Corollary 1 is because in general |f| is only in L1  loc(Rd+1), not in  L1(Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In the proof of Theorem 3 we will be applying Corollary 1 and Corollaries 2, 3 below  to fn equal to either bn or (with some abuse of notation) to bn − bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Under the assumptions and notation of Corollary 1, if p > d + 1, then, for  every λ ≥ λd,p,q, solutions vn ∈ Cb([r, T], C∞(Rd)) to the approximating inhomogeneous Cauchy  problems  (λ + ∂t − ∆ + bn · ∇)vn = 1[r,T]|fn|h  on ]r, T] × Rd,  vn(r, ·) = g(·) ∈ C∞(Rd) ∩ W 1,p(Rd),  where 0 ≤ r < T, converge uniformly on DT × Rd to  v := (λ + ∂t − ∆)−1�1|f|h + δs=rg  �  − (λ + ∂t − ∆)− 1  2− 1  2p Qp(b)(1 + Tp(b))−1�Rp(b)Qp(1f)|1f|  1  p h + Gp(b)Spg  �,  1 := 1[r,T],  sup  (r,t)∈DT ,x∈Rd |v(t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' r)| ≤ C1∥1f  1  p h∥p + C2∥g∥W 1,p(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  We also have the following weighted variant of Corollary 2, which we will record for bounded  vector fields bn and fn and λ = 0, as will be needed in the proof of Theorem 3 in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Let q > 1 (close to 1) be from the hypothesis on b in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Set  ρ(x) := (1 + l|x|2)−ν,  x ∈ Rd,  where ν > d  2p +  1  pq′ is fixed (so that ρ ∈ Lp(Rd) and (42) below holds) and l > 0 is to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  We have  |∇ρ| ≤ ν  √  lρ =: c1  √  lρ,  |∆ρ| ≤ 2ν(2ν + d + 2)lρ =: c2lρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (⋆)  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Under the assumptions of Corollary 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' if p > d + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' provided that the  constant l in the definition of ρ is chosen sufficiently small,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' solutions vn ∈ Cb([r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' C∞(Rd))  to the approximating inhomogeneous Cauchy problems  (∂t − ∆ + bn · ∇)vn = ±1[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='T]|fn|  on [r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' T] × Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  vn(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' ·) = g ∈ C∞(Rd) ∩ W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p(Rd),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  where 0 ≤ r < T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' satisfy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' for all t ∈]r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  sup  [r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='t]×Rd |ρvn| ≤ C1∥ρ1[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='t]|fn|  1  p ∥p + C2∥ρg∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p(Rd)  (27)  and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' putting ρy(x) := ρ(x − y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  sup  [r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='t]×Rd |vn| ≤ sup  y∈Zd(C1∥ρy1[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='t]|fn|  1  p ∥p + C2∥ρyg∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p(Rd))  (28)  ≤ ˜C1(t − r)γ∥f∥  1  p  Eq + ˜C2∥g∥W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='p  (29)  with constants C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' ˜C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' ˜C2 and γ > 0 independent of n and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  14  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  We prove Corollary 3 in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Proof of Proposition 1  It suffices to carry out the proof for bn defined by (10) and then use the Dominated Conver-  gence Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' So, without loss of generality, everywhere below b is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Below we follow  [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Set  Mβh(t, x) := sup  ρ>0  ρβ 1  |Cρ|  �  Cρ(t,x)  |h|dz,  0 ≤ β ≤ d − 2,  and define the maximal function  Mh(t, x) := sup  ρ>0  1  |Cρ|  �  Cρ(t,x)  |h|dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Also, define  ˆ  Mh(t, x) :=  sup  (t,x)∈C  1  |C|  �  C  |h|dz,  where the supremum is taken over all cylinders C = Cρ(z) ∋ (t, x), z ∈ Rd+1, ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Put Pα := (−∂t − ∆)− α  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We will need  Lemma 1 (see [24, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' For every β ∈]0, d + 2], 0 < α < β there exists C > 0 such  that, for all f ≥ 0,  Pαf ≤ C(Mβf)  α  β (Mf)1− α  β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Let us prove the first inequality:  |⟨|b|(P 1  p f)p⟩| ≤ cp  p,q∥b∥Eq∥f∥p  p,  f ∈ Lp(Rd+1)  (30)  (the proof of the second one is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Put u := P 1  p f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Then we estimate the LHS of (30) as  |⟨|b|(P 1  p f)p⟩| = |⟨|b||u|p−1, P 1  p f⟩|  ≤ |⟨P ∗  1  p (|b||u|p−1), f⟩| ≤ ∥P ∗  1  p (|b||u|p−1)∥p′∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (31)  Here P ∗  α = (∂t − ∆)− α  2 is the adjoint of the operator Pα in L2(Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' To obtain (30), we need  to bound the coefficient ∥P ∗  1  p (|b|up−1)∥p′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' To this end, we estimate pointwise  P ∗  1  p (|b||u|p−1) = P ∗  1  p (|b|  1  p +γ|b|  1  p′ −γ|u|p−1) ≤ P ∗  1  p (|b|1+γp)  1  p (P ∗  1  p (|b|1−γp′|u|p))  1  p′  for a small γ > 0 such that 1 + γp < q0 for some fixed q0 < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Hence  ∥P ∗  1  p (|b||u|p−1)∥p′  p′ ≤ ⟨|b|1−γp′|u|p, P 1  p [P ∗  1  p (|b|1+γp)]  1  p−1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (32)  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  15  1) By Lemma 1 with α = 1  p, β = 1 + γp (or rather its straightforward variant for P ∗  α),  P ∗  1  p (|b|1+γp) ≤ C∥b∥  1  p  E1+γp( ˆ  M|b|1+γp)1− 1  p  1  1+γp  ≤ C∥b∥  1  p  Eq0( ˆ  M|b|1+γp)1− 1  p  1  1+γp  At this point let us assume that  ˆ  M|b|1+γp ≤ C0|b|1+γp,  (33)  but will get rid of this assumption later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Then  P ∗  1  p (|b|1+γp) ≤ C2∥b∥  1  p  Eq0|b|1+γp− 1  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  2) After applying the last estimate in (32), it remains to estimate P 1  p (|b|  1  p +γp′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Selecting γ  even smaller, if needed, one may assume that 1  p + γp′ < q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' By Lemma 1,  P 1  p (|b|  1  p +γp′) ≤ C(M 1  p +γp′|b|  1  p +γp′)  1  p  1  1  p +γp′ (M|b|  1  p +γp′)  1− 1  p  1  1  p +γp′  ≤ C∥b∥  1  p  Eq0( ˆ  M|b|  1  p +γp′)  1− 1  p  1  1  p +γp′  In addition to (33), let us temporarily assume that  ˆ  M|b|  1  p +γp′ ≤ C0|b|  1  p +γp′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (34)  Then  P 1  p (|b|  1  p +γp′) ≤ C2∥b∥  1  p  Eq0|b|γp′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  3) Applying the results from 1), 2) in (32), we obtain  ∥P ∗  1  p (|b||u|p−1)∥p′ ≤ C3∥b∥  1  p  Eq0⟨|b||u|p⟩  1  p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Therefore, (31) yields  ⟨|b||u|p⟩ ≤ C3∥b∥  1  p  Eq0⟨|b||u|p⟩  1  p′ ∥f∥p  so  ⟨|b||u|p⟩  1  p ≤ C3∥b∥  1  p  Eq0∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (35)  4) Now one gets rid of the assumptions (33) and (34) at expense of replacing ∥b∥Eq0 in (35)  by ∥b∥Eq, where, recall, q0 < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' This, in turn, will give (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Fix q0 < q1 < q and define  ˜b := ( ˆ  M|b|q1)  1  q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Then ˜b ≥ |b| and ˜b satisfies  ˆ  M˜bq0 ≤ C0˜bq0  (36)  (see [8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='158]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Since 1+ γp < q0, 1  p + γp′ < q0, both inequalities (33) and (34) for ˜b follow from  (36), and so we have  ⟨|b||u|p⟩  1  p ≤ C3∥˜b∥  1  p  Eq0∥f∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  16  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  It remains to apply inequality ∥˜b∥Eq0 ≤ C∥b∥Eq, which was established in [24, proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Proof of Theorem 1  The fact that Qp, Rp are bounded on Lp(Rd+1), and the operator Tp = RpQp has norm  ∥Tp∥p→p < 1 provided that ∥bs∥Eq is sufficiently small and λ is sufficiently large, is an immediate  consequence of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Thus,  Θp(b) := (λ + ∂t − ∆)−1 − (λ + ∂t − ∆)− 1  2− 1  2p Qp(1 + Tp)−1Rp(λ + ∂t − ∆)− 1  2p′  is in B(Lp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Recall: bn := 1nb, where 1n is the indicator of {(t, x) ∈ Rd+1 | |b(t, x)| ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' If un ∈ Lp(Rd+1)  denotes the solution to (∂t − ∆ + bn · ∇)un = f, which exists by the classical theory, then  un = Θp(bn)f,  where Θp(bn)f coincides with the Duhamel series representation for un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Next, let us note that  Rp(bn) → Rp(b), Qp(bn) → Qp(b)  strongly in Lp(Rd+1)  (37)  as follows from (21), (22) and the Dominated Convergence Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Hence  un := Θp(bn) → u = Θp(b) in W1+ 1  p ,p(Rd+1),  (38)  as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In the comment after Theorem 1 we promised to prove the existence and uniqueness  of weak solution to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The argument is standard and goes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In the proof of Theorem  1 above take p = 2, so u ∈ W  3  2 ,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Multiplying (λ + ∂t − ∆ + bn · ∇)un = f, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' , by  ϕ = (λ − ∂t − ∆)− 1  4 (λ + ∂t − ∆)  3  4 η, η ∈ C∞  c (Rd+1) and integrating over Rd+1, we have  ⟨(λ + ∂t − ∆)  3  4un, (λ + ∂t − ∆)  3  4 η⟩+⟨R2(bn)(λ + ∂t − ∆)  3  4 un, Q∗  2(bn)(λ + ∂t − ∆)  3  4 η⟩  = ⟨f, (λ − ∂t − ∆)− 1  4 (λ + ∂t − ∆)  3  4 η⟩,  where, recall, Q∗  2(b) = |b|  1  2(λ − ∂t − ∆)− 1  4 ∈ B(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In view of (38),  ⟨(λ + ∂t − ∆)  3  4un, (λ + ∂t − ∆)  3  4 η⟩ → ⟨(λ + ∂t − ∆)  3  4u, (λ + ∂t − ∆)  3  4 η⟩  (n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Next,  ⟨R2(bn)(λ + ∂t − ∆)  3  4 un, Q∗  2(bn)(λ + ∂t − ∆)  3  4 η⟩  = ⟨R2(bn)(λ + ∂t − ∆)  3  4 (un − u), Q∗  2(bn)(λ + ∂t − ∆)  3  4η⟩  + ⟨R2(bn)(λ + ∂t − ∆)  3  4 u, (Q∗  2(bn) − Q∗  2(b))(λ + ∂t − ∆)  3  4η⟩  + ⟨R2(bn)(λ + ∂t − ∆)  3  4 u, Q∗  2(b)(λ + ∂t − ∆)  3  4η⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  17  By (38), Q∗  2(bn) → Q∗  2(b) strongly in L2(Rd+1) (by the same argument as in (37)) we get  that the first two terms in the RHS tend to 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' By (37), the last term tends to  ⟨R2(b)(λ + ∂t − ∆)  3  4u, Q∗  2(b)(λ + ∂t − ∆)  3  4η⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Hence u is a weak solution to (11) in the sense of  definition (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Let v ∈ W  3  2,2 be another weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Put  τ[v, η] := ⟨(λ + ∂t − ∆)  3  4 v, (λ + ∂t − ∆)  3  4 η⟩ + ⟨R2(b)(λ + ∂t − ∆)  3  4v, Q∗  2(b)(λ + ∂t − ∆)  3  4η⟩,  where η ∈ C∞  c (Rd+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We have  |⟨R2(b)(λ + ∂t − ∆)  3  4 v, Q∗  2(b)(λ + ∂t − ∆)  3  4 η⟩| ≤ c∥v∥W  3  2 ,2∥η∥W  3  2 ,2  where c < 1 by our assumption on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We extend τ[v, η] to η ∈ W  3  2,2 by continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Now we have  τ[v − u, η] = 0, where u is the weak solution constructed above, so it suffices to choose η = v − u  to arrive at 0 = τ[v − u, v − u] ≥ (1 − c)∥v∥2  W  3  2 ,2, hence v = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Proof of Theorem 2  Let us define Ut,rg := v(t) (t ≥ r), g ∈ C∞(Rd) ∩ W 1,p(Rd) where v(t) is given by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Since  Ut,r  n  are L∞ contractions, it suffices to prove (we consider convergence in (r, t) ∈ DT )  U = Cb(DT , C∞(Rd))- lim  n Ung,  (39)  and then extend operators Ut,r by continuity to g ∈ C∞(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The reproduction property of Ut,r  and the preservation of positivity will follow from the corresponding properties of Ut,r  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Proof of (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Put vn := Ut,r  n g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We have  vn = (λ + ∂t − ∆)−1δs=rg − (λ + ∂t − ∆)− 1  2− 1  2p Qp(bn)(1 + Tp(bn))−1Gp(bn)Spg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  This is the usual Duhamel series representation for vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We know from the proof of Theorem 1  that operators Qp(bn), Tp(bn), Gp(bn) are bounded on Lp(Rd+1) with operator norms indepen-  dent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' In turn, operator Sp satisfies  ∥Spg∥Lp(Rd+1) ≤ Cp,d∥∇g∥Lp(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (Indeed, taking for brevity r = 0, we have by definition  Spg(t, x) = 1t≥0e−λtt− 1  2+ 1  2p′ (4πt)− d  2  �  Rd ∇xe− |x−y|2  4t  g(y)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Hence  ∥Spg∥p  Lp(Rd+1) =  �  R  1t≥0∥Sp(t)∥p  Lp(Rd)dt ≤  � ∞  0  e−λptt(− 1  2+ 1  2p′ )pdt∥∇g∥p  Lp(Rd),  where (− 1  2 +  1  2p′ )p = − 1  2 (> −1 so the integral in time converges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=')  18  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  Clearly, (λ + ∂t − ∆)−1δs=rg ∈ Cb([r, ∞[, C∞(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Thus, to prove (39), it remains to note  that Qp(bn) → Qp(b), Tp(bn) → Tp(b) and Gp(bn) → Gp(b) strongly in Lp(Rd+1) (see proof of  Theorem 1), so that by the parabolic Sobolev embedding, since p > d + 1,  (λ + ∂t − ∆)− 1  2 − 1  2p Qp(bn)(1 + Tp(bn))−1Gp(bn)Spg  → (λ + ∂t − ∆)− 1  2− 1  2p Qp(b)(1 + Tp(b))−1Gp(b)Spg  in C∞(Rd+1) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The required convergence (39) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Proof of Corollary 3 (weighted bounds)  It will be convenient to carry out the proof for solutions vn to  (λ + ∂t − ∆ + bn · ∇)vn = ±1[r,T]|fn|,  on [r, T] × Rd,  vn(r, ·) = g ∈ C∞(Rd) ∩ W 1,p(Rd)  where λ ≥ λd,p,q > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Since we are working on a fixed finite time interval [r, T], this change will  amount to multiplying solution vn by a bounded function eλ(t−r) which, clearly, does not affect  the sought estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Put for brevity v = vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We will carry out the proof on the maximal interval [r, t], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' for  t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Also, without loss of generality, the RHS of the equation is 1[r,T]|fn| and the initial  function g ≥ 0, so v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We have for the weight ρ defined before the corollary,  λρv + ∂t(ρv) − ∆(ρv) + bn · ∇(ρv)  = ρ(λv + ∂tv − ∆v + bn · ∇v) − 2∇ρ · ∇v + (−∆ρ)v + bnv · ∇ρ  (v solves the parabolic equation above)  = ρ1[r,T]|fn| + K,  K := −2∇ρ · ∇v + (−∆ρ)v + bnv · ∇ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Let us rewrite the term K as follows:  K = −2  �∇ρ  ρ · ∇(ρv) − (∇ρ)2  ρ  v  �  + (−∆ρ)v + bnv · ∇ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Hence  λρv + ∂t(ρv) − ∆(ρv) + ˜bn · ∇(ρv) = ρ1[r,T]|fn| + ˜K,  (40)  where ˜bn := bn + 2∇ρ  ρ and  ˜K = 2(∇ρ)2  ρ  v + (−∆ρ)v + bnv · ∇ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Note that by (⋆) the term 2∇ρ  ρ  in ˜bn is a bounded vector field whose sup norm can be made  as small as needed by selecting l in the definition of ρ sufficiently small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' we will be selecting l  sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  19  By (⋆), the first two terms in ˜K are smaller than (c2  1 + c2)lρv, so they can be handled by  selecting λ ≥ λd,p,q + (c2  1 + c2)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Corollary 2 applied to (40) gives us  sup  [r,T]×Rd |ρv| ≤ C1∥ρ1[r,T]|fn|  1  p ∥p + C′  1  √  l∥|bn|  1  p ρv∥p + C2∥ρg∥W 1,p(Rd),  where, when we were treating the last term in ˜K, we used again (⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' However, now we have  to deal with ∥|bn|  1  p ρv∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' So, we will have to use instead a finer consequence of the solution  representation of Corollary 2:  1  2  sup  [r,T]×Rd |ρv| + C0  2 ∥(λ + ∂t − ∆)  1  2p ρv∥p  ≤ C1∥ρ1[r,T]|fn|  1  p ∥p + C′  1  √  l∥|bn|  1  p ρv∥p + C2∥ρg∥W 1,p(Rd)  (41)  for appropriate constant C0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' (Here we only need to justify that (λ + ∂t − ∆)− 1  2 − 1  2p′ δs=rρg is  in Lp(Rd+1) and its norm is bounded by ∥ρg∥Lp(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The proof of this fact repeats the argument  for operator Sp from the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=') We have  ∥|bn|  1  p ρv∥p ≤ ∥|bn|  1  p (λ + ∂t − ∆)− 1  2p ∥p→p∥(λ + ∂t − ∆)  1  2p ρv∥p  (we are applying Proposition 2)  ≤ Cλ∥(λ + ∂t − ∆)  1  2p ρv∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Applying the last bound in (41) with l chosen sufficiently small so that C0  2 − C′  1Cλ  √  l > 0, we  obtain (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Assertion (28) follows from (27) using translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Armed with (28), we now prove (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We obtain from (⋆) that supy∈Zd ∥ρyg∥W 1,p ≤ c0∥g∥W 1,p  for appropriate c0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' It remains to show that if |f| ∈ Eq, q > 1, then  sup  y∈Zd ∥ρy1[r,T]|f|  1  p ∥p ≤ c(T − r)γ∥f∥  1  p  Eq  (42)  for constants c = c(d, p, l) and γ = γ(q, p) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (Actually, f in Corollary 3 satisfies (12),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' f = f1 + f2 where |f1| ∈ Eq, q > 1 and f2 is bounded on Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The term f2 is dealt with using  the fact that ρ ∈ Lq(Rd+1), so we only need to apply (42) to |f1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=')  20  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  To prove (42),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' we estimate  ∥ρy1[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='T]|f|  1  p ∥p  p = ⟨ρp  y1[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='T]|f|⟩ ≤  ∞  �  k=0  (1 + lk2)−νp⟨1[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='T]|f|1Ck+1(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='y)−Ck(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='y)⟩  C0(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' y) := ∅  ≤  ∞  �  k=1  (1 + lk2)−νp⟨1[r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='T]|f|1Ck+1(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='y)⟩  ≤  ∞  �  k=1  (1 + lk2)−νp|Ck+1(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' y) ∩ ([r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' T] × Rd)|  1  q′ |Ck+1|  1  q  k + 1 (k + 1)  �  1  |Ck+1|⟨|f|q1Ck+1(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='y)⟩  � 1  q  ≤ c0(d)  ∞  �  k=1  (1 + lk2)−νp|T − r|  1  q′ (k + 1)  d  q′ |Ck+1|  1  q  k + 1 ∥f∥Eq  ≤ c1(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' l)|T − r|  1  q′  ∞  �  k=1  k−2νpk  d  q′ k  d+2  q −1∥f∥Eq =: cp|T − r|  1  q′ ∥f∥Eq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  where c = c(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' l) < ∞ since,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' by our choice of ν in (⋆),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' −2νp + d  q′ + d+2  q  − 1 < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  □  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Proof of Theorem 3  Below we follow an argument from [13] but use different embeddings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' the ones established  in Corollaries 1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  The first assertion (i), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' for all x ∈ Rd, 0 ≤ t ≤ r ≤ T,  ⟨P t,r(x, ·)⟩ = 1,  follows from from (27) with f = 0 and Theorem 2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Namely, we first show for a fixed x, using  weight ρ, that for every ε > 0 there exists R > 0 such that ⟨P t,r  m (x, ·)1Rd−B(0,R)(·)⟩ < ε for all  m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' , and so ⟨P t,r  m (x, ·)1B(0,R)(·)⟩ ≥ 1−ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Passing to the limit in m and then in R → ∞,  we obtain ⟨P t,r(x, ·)⟩ ≥ 1 − ε, which yields the required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (ii) For every n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' , let Xn  t = Xn  t,x denote the strong solution to the approximating  SDE  Xn  t = x −  � t  0  bn(s, Xn  s )ds +  √  2Bt,  x ∈ Rd,  on a complete probability space (Ω, Ft, P), where {bn} are given by (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Step 1: There exists a constant C > 0 independent of n, k such that  sup  n  sup  x∈Rd E  � r  s  |bk(t, Xn  t,x)|dt ≤ CF(r − s)  (43)  for 0 ≤ s ≤ r ≤ T, where  F(h) := hγ,  constants C and γ > 0 (from Corollary 3) are independent of n and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Indeed, let v = vn,k be  the solution to the terminal-value problem  (∂t + ∆ − bn · ∇)v = −|bk|,  v(r, ·) = 0,  t ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  21  By Itô’s formula,  v(r, Xn  r ) = v(s, Xn  s ) +  � r  s  (∂tv + ∆v − bn · ∇v)(t, Xn  t )dt +  √  2  � r  s  ∇v(t, Xn  t )dBt,  hence  0 = v(s, Xn  s ) −  � r  s  |bk(t, Xn  t )|dt +  √  2  � r  s  ∇v(t, Xn  t )dBt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Taking expectation, we obtain  E  � r  s  |bn(t, Xn  t )|dt = Ev(s, Xn  s ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Since Ev(s, Xn  s ) ≤ ∥v(s, ·)∥L∞(Rd), we obtain from Corollary 3 with fk = bk and initial data  g = 0  E  � r  s  |bk(t, Xn  t )|dt ≤ C(r − s)γ  with constants C and γ > 0 independent of k, n ⇒ (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  By a standard result (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' [10, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' 2]), given a conservative backward Feller evolution  family, there exist probability measures Px (x ∈ Rd) on (D([0, T], Rd), B′  t = σ(ωr | 0 ≤ r ≤ t)),  where D([0, T], Rd) is the space of right-continuous functions having left limits, and ωt is the  coordinate process, such that  Ex[f(ωr)] = P 0,rf(x),  0 ≤ r ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Here and below, Ex := EPx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Also, put {Pn  x := (PXn)−1}∞  n=1 and set En  x := EPnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Step 2: Ex[  � r  0 |b(t, ωt)|dt] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Indeed, by Step 1, supn supx∈Rd En  x  � r  s |bk(t, ωt)|dt ≤ CF(r − s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Hence by the convergence  result in Corollary 2 (with f := 1B(0,k)bk)  Ex[  � r  s  |1B(0,k)bk(t, ωt)|dt] ≤ CF(r − s) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  It remains to apply Fatou’s Lemma in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Step 3: For every f ∈ C2  c (Rd), the process  Mf  r := f(ωr) − f(x) +  � r  0  (−∆f + b · ∇f)(t, ωt)dt  (44)  is a B′  r-martingale under Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Indeed, let us note first that  Em  x [f(ωr)] → Ex[f(ωr)],  Em  x [  � r  0  (−∆f)(ωt)dt] → Ex[  � r  0  (−∆f)(ωt)dt]  (m → ∞),  (⋆)  as follows from the convergence result in Theorem 2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Next, we note that  Em  x  � r  0  (bm · ∇f)(t, ωt)dt → Ex  � r  0  (b · ∇f)(t, ωt)dt  (m → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (⋆⋆)  22  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  The latter follows from  Em  x  ����  � r  0  �(bm − bn) · ∇f  �(t, ωt)dt η(ω)  ���� ≤ C∥η∥∞∥1[0,r]|bm − bn|  1  p |∇f|∥p  (a)  as m, n → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Em  x  � � r  0  (bn · ∇f)(t, ωt)dt · η(ω)  �  → Ex  � � r  0  (bn · ∇f)(t, ωt)dt · η(ω)  �  (b)  as m → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Ex  ����  � r  0  �(b − bn) · ∇f  �(t, ωt)dt η(ω)  ���� ≤ C∥η∥∞∥1[0,r]|b − bn|  1  p |∇f|∥p → 0  (c)  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The proof of the inequality in (a) follows the proof of (43) but uses Corollary 2  with f = bn − bm and g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The convergence in (a) follows from the fact that bn − bm → 0 in  [L1  loc(Rd+1)]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Assertion (b) follows from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' The proof of (c) is similar to the proof of  (a) except that we pass to the limit in m and then in k using Fatou’s Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Now, since  Mf  r,m := f(ωr) − f(x) +  � r  0  (−∆f + bm · ∇f)(t, ωt)dt  is a B′  r-martingale under Pm  x ,  x �→ Em  x [f(ωr)] − f(x) + Em  x  � r  0  (−∆f + bm · ∇f)(t, ωt)dt  is identically zero on Rd,  and so by (⋆), (⋆⋆)  x �→ Ex[f(ωr)] − f(x) + Ex  � r  0  (−∆f + b · ∇f)(t, ωt)dt  is identically zero in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Since {Px}x∈Rd are determined by a Feller evolution family, and thus constitute a Markov  process, we can conclude (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' the proof of [20, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='2]) that Mf  r is a B′  r-martingale  under Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Step 4: {Px}x∈Rd are concentrated on (C([0, T], Rd), Bt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  By Step 3, ωt is a semimartingale under Px, so Itô’s formula yields, for every g ∈ C∞  c (Rd),  that  g(ωt) − g(x) =  �  s≤t  �g(ωs) − g(ωs−)  � + St,  (45)  where St is defined in terms of some integrals and sums of (∂xig)(ωs−) and (∂xi∂xjg)(ωs−) in s,  see [5, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' 2] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Now, let A, B be arbitrary compact sets in Rd such that dist(A, B) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Fix g ∈ C∞  c (Rd) that separates A, B, say, g = 0 on A, g = 1 on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Set  Mg  t := g(ωt) − g(x) +  � t  0  (−∆g + b · ∇g)(ωs)ds,  Kg  t :=  � t  0  1A(ωs−)dMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  PARABOLIC EQUATIONS AND SDES WITH TIME-INHOMOGENEOUS MORREY DRIFT  23  In view of (44) and (45), when evaluating Kg  t one needs to integrate 1A(ωs−) with respect to  St, however, one obtains zero since (∂xig)(ωs−) = (∂xi∂xjg)(ωs−) = 0 if ωs− ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Thus,  Kg  t =  �  s≤t  1A (ωs−) g(ωs) +  � t  0  1A(ωs−)  �−∆g + b · ∇g  �(ωs)ds  =  �  s≤t  1A (ωs−) g(ωs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Since Mg  t is a martingale, so is Kg  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Thus, Ex  ��  s≤t 1A(ωs−)g(ωs)  � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Using the Dominated  Convergence Theorem, we further obtain Ex  ��  s≤t 1A(ωs−)1B(ωs)  � = 0, which yields the re-  quired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' (By the way, this construction, in a more general form, can be used to control the jumps  of stable process perturbed by a drift, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=')  We denote the restriction of Px from (D([0, T], Rd), B′  t) to (C([0, T], Rd), Bt) again by Px, and  thus obtain that for every x ∈ Rd and all f ∈ C2  c (Rd)  Mf  r = f(ωr) − f(x) +  � r  0  (−∆f + b · ∇f)(t, ωt)dt,  ω ∈ C([0, T], Rd),  is a Br-martingale under Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Thus, Px is a Br-martingale solution to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' (Alternatively, we could have used a tightness  argument, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=')  To show that Px is a weak solution it suffices to show that Mf  r is also a martingale for  f(x) = xi and f(x) = xixj, which is done by following closely [15, proof of Lemma 6] and  employing weight ρ and (27) in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (iii) This follows from Corollary 2, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' proof of (ii) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  (iv) Suppose that there exist P1  x, P2  x, two martingale solutions to (19), that satisfy  Ei  x  � T  0  |b(r, ωt)h(t, ωt)|dt ≤ c∥1[0,T]b  1  p h∥p,  h ∈ Cc(Rd+1)  (46)  with constant c independent of h (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Here and below, E1  x := EP1x, E2  x := EP2x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We will  show that for every F ∈ Cc(Rd+1) we have  E1  x[  � T  0  F(t, ωt)dt] = E2  x[  � T  0  F(t, ωt)dt],  (47)  which implies P1  x = P2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Proof of (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Let un ∈ C([0, T], C∞(Rd)) be the solution to  (∂t − ∆ + bn · ∇)un = F,  un(T, ·) = 0,  (48)  where, recall, bn = 1nb, and 1n is the indicator of {|b| ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Set τR := inf{t ≥ 0 | |ωt| ≥ R},  R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' By Itô’s formula  Ei  xun(T ∧ τR, ωT∧τR) = un(0, x) + Ei  x  � T∧τR  0  F(t, ωt)dt  + Ei  x  � T∧τR  0  �(b − bn) · ∇un  �(t, ωt)dt  (49)  24  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' KINZEBULATOV  (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' We have  ����Ei  x  � T∧τR  0  �(b − bn) · ∇un  �(t, ωt)dt  ���� ≤ Ei  x  � T∧τR  0  �|b|(1 − 1n)|∇un|  �(t, ωt)dt  (we are applying (46))  ≤ c∥1[0,T]×B(0,R)|b|  1  p (1 − 1n)|∇un|∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  At this point we note that ˜un(t) := eλ(T−t)un(t) satisfies  (λ + ∂t + ∆ + bn · ∇)un = 1[0,T]eλ(T−t)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Hence we can apply to |b|  1  p |∇un| the solution representation of Corollary 2 (after reversing time  and taking there g = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  Using |b| ≥ |bn|, we then obtain an independent on n Lp(Rd+1)  majorant on |b|  1  p |∇un|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Therefore, since 1 − 1n → 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' on Rd+1 as n → ∞, we have  Ei  x  � T∧τR  0  �(b − bn) · ∇un  �(t, ωt)dt → 0  (n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  We are left to note, using again Corollary 2, that solutions un converge to a function u ∈  C([0, T], C∞(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Therefore, we can pass to the limit in (49), first in n and then in R → ∞, to  obtain  0 = u(0, x) + Ei  x  � T  0  F(t, ωt)dt  i = 1, 2,  which gives (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content='  □  References  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
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+page_content=' Gubinelli and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Maurelli, Stochastic ODEs and stochastic linear PDEs with  critical drift:  regularity, duality and uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' Electr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFST4oBgHgl3EQfdDhQ/content/2301.13805v1.pdf'}
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