Eldan's stochastic localization and tubular neighborhoods of complex-analytic sets
aa r X i v : . [ m a t h . M G ] J un Eldan’s stochastic localization and tubularneighborhoods of complex-analytic sets
Bo’az Klartag
Abstract
Let f : C n → C k be a holomorphic function and set Z = f − (0) . Assume that Z isnon-empty. We prove that for any r > , γ n ( Z + r ) ≥ γ n ( E + r ) , where Z + r is the Euclidean r -neighborhood of Z , where γ n is the standard Gaussianmeasure in C n , and where E ⊆ C n is an ( n − k ) -dimensional, affine, complex subspacewhose distance from the origin is the same as the distance of Z from the origin. Eldan’s stochastic localization is a rather recent analytic technique that has emerged fromconvex geometry. In a nutshell, the idea is to construct a time-parameterized family ofdecompositions of a given probability measure on R n into a mixture of increasingly curvedmeasures, in the sense that their densities become more log-concave with time.The technique was used by Eldan [7] and by Lee and Vempala [14] in their works onthe KLS conjecture, by Eldan in his works on noise sensitivity [8] and on Skorokhod em-beddings [9], by Eldan and Lehec [10] in connection with Ball’s thin-shell problem andBourgain’s slicing conjecture, and by Ding, Eldan and Zhai [6, Theorem 1.6] for studyingthe concentration of a supremum of a Gaussian process.In this paper we apply this technique to the study of complex-analytic subvarieties of C n . Write γ n for the standard Gaussian probability measure in C n , whose density is z (2 π ) − n exp( −| z | / , where | z | = qP j | z j | for z = ( z , . . . , z n ) ∈ C n . For a non-emptyset S ⊆ C n we write d (0 , S ) = inf z ∈ S | z | for the Euclidean distance from the origin to the set S . Theorem 1.1.
Let ≤ k ≤ n , let f : C n → C k be a holomorphic map and set Z = f − (0) .Assume that Z is non-empty. Let E ⊆ C n be an ( n − k ) -dimensional, affine, complexsubspace with d (0 , E ) = d (0 , Z ) . Then, γ n ( Z + r ) ≥ γ n ( E + r ) for all r > , (1) where Z + r = { z + w ; z ∈ Z, w ∈ C n , | w | ≤ r } . ntuitively, a consequence of Theorem 1.1 is that a fiber of a holomorphic map cannotescape from the origin faster than a flat that has the same distance from the origin. The limitcase r → in (1) implies that Z Z e −| z | / (2 π ) n ≥ Z E e −| z | / (2 π ) n = e − d (0 ,Z ) / (2 π ) k , where both integrals are carried out with respect to the Hausdorff measure of real dimension n − k ) . By specializing Theorem 1.1 to the case where ∈ Z we obtain the following: Corollary 1.2.
Let ≤ k ≤ n and let f : C n → C k be a holomorphic map with f (0) = 0 .Set Z = f − (0) . Then, γ n ( Z + r ) ≥ γ n ( C n − k + r ) for all r > , (2) where we think of C n − k as the subspace of C n of all vectors whose last k coordinates vanish. We view Corollary 1.2 in the context of Gromov’s Gaussian waist inequality [11]. If thefunction f in Corollary 1.2 is assumed to be merely continuous, then Gromov showed thatthere exists some t ∈ C k such that the set Z = f − ( t ) satisfies (2). From Corollary 1.2 wethus gather that when f : C n → C k is not only continuous but also holomorphic, it is alwayspossible to select t = f (0) in the Gaussian waist inequality.Of course, the complex structure is irrelevant for Gromov’s Gaussian waist inequality,which is usually formulated for continuous maps f : R n → R k . With the exception of thecase k = 1 , very little is known about the special value t for which the fiber Z = f − ( t ) islarge. A random choice of t is worthless in general, as explained by Alpert and Guth [2]. Theidea that t = f (0) should be suitable for a holomorphic f : C n → C k is rather natural, inview of the following statement (see Lelong [15], Rutishauser [16] or e.g. Chirka [5, Section15.1]): Under the assumptions of Corollary 1.2, the function r V ol n − k ) ( Z ∩ rB n ) V ol n − k ) ( rB n − k ) ) is a non-decreasing function of r > , whose limit as r → + is at least one. Here B n = { z ∈ C n ; | z | ≤ } and V ol n − k ) is the n − k ) -dimensional Hausdorff measure. Weremark that this statement may be used to show that lim inf r → + h γ n ( Z + r ) /γ n ( C n − k + r ) i ≥ . We thus obtain a rather direct proof of a limit case of Corollary 1.2. A corresponding limitcase of Theorem 1.1 may be deduced from the results of Brendle and Hung [4] and Alexan-der, Hoffman and Osserman [1], even in the more general setting of a minimal submanifold.Still, the only proof of Theorem 1.1 or Corollary 1.2 of which we are aware is quite indi-rect and it involves Eldan’s stochastic localization. It would be interesting to find a proof ofTheorem 1.1 which does not involve stochastic processes. e proceed with a non-Euclidean waist inequality for scalar holomorphic functions f : C n → C that follows from Theorem 1.1. By a convex body we mean a compact, convex setwith a non-empty interior. We say that a convex body K ⊆ C n is circled if e it K = K for all t ∈ R . Here, λK = { λz ; z ∈ K } , A + B = { a + b ; a ∈ A, b ∈ B } and x + A = { x } + A .Equivalently, a subset K ⊆ C n is a circled convex body if and only if K is the unit ball of acomplex norm on C n . Thus the following corollary is concerned with tubular neighborhoodswith respect to an arbitrary complex norm on C n . Corollary 1.3.
Let K ⊆ C n be a circled convex body. Then there exists a complex, ( n − -dimensional, linear subspace H ⊆ C n with the following property: Let f : C n → C beholomorphic, set Z = f − (0) and assume that Z is non-empty. Then for any r > , γ n ( Z + rK ) ≥ γ n ( H + rK ) , where H ⊆ C n is a translate of H with d (0 , H ) = d (0 , Z ) . Let us now discuss the proof of Theorem 1.1 in the case where ∈ Z . By approximation,it suffices to consider the case where is a regular value of f , thus Z = f − (0) ⊆ C n isan ( n − k ) -dimensional complex submanifold of C n . We fix a probability space (Ω , F , P ) ,which for concreteness is set to be the unit interval Ω = [0 , equipped with the Lebesguemeasure. With P -almost any ω ∈ Ω we will associate a Gaussian measure µ ω on C n suchthat γ n ( A ) = Z Ω µ ω ( A ) d P ( ω ) = E ω µ ω ( A ) for any measurable set A ⊆ C n . (3)The Gaussian probability measure µ ω has several important properties: It is almost-surelycentered at a certain point of Z , it is supported on a k -dimensional affine subspace of C n , andit is more curved than the standard Gaussian measure in C n . The measure decomposition in(3), which is formulated precisely in Theorem 3.3 below, is the main technical result in thispaper.The decomposition (3) is obtained as the limit t → ∞ of a stochastic process of de-compositions parameterized by time. Recall that a stochastic process is a family of randomvariables ( X t ) t ≥ , i.e., a family of functions defined on Ω . All of our stochastic processesare continuous, meaning that for almost any ω ∈ Ω , the map t X t ( ω ) is continuous in [0 , ∞ ) . From now on in this paper, we will supress the dependence on the sample point ω ∈ Ω , and replace integrals with respect to P by the expectation sign E .Let us briefly explain the relation of holomorphic functions to stochastic processes andin particular to Itˆo martingales. Recall the well-known fact, that f ( W t ) is a local martin-gale whenever f is harmonic and ( W t ) t ≥ is a Brownian motion. When f : C n → C isholomorphic (pluriharmonic is sufficient), the stochastic process f ( X t ) is a local martingalewhenever dX t = Σ t dW t , where Σ t ∈ C n × n is an arbitrary adapted process. This provides much more flexibility,which allows us to control the center of the Gaussian measure µ t and confine it to the mani-fold Z . The details are below. or z, w ∈ C n we write z · w = P j z j w j . Note that in our notation the vector w is notconjugated. We use | A | to denote the Hilbert-Schmidt norm of a matrix A ∈ C n × n , and Id isthe identity matrix. For Hermitian matrices A, B ∈ C n × n , we write A ≤ B if the difference B − A is positive semi-definite, while A > means that A is positive-definite. We write log for the natural logarithm. By a smooth function we mean C ∞ -smooth. Acknowledgements.
I would like to thank Bo Berndtsson, Ronen Eldan, Sasha Logunovand Sasha Sodin for interesting related discussions. Thanks also to the anonymous refereefor comments simplifying the proof of Lemma 2.3. Supported by a grant from the EuropeanResearch Council (ERC).
Denote by M + n ( C ) the collection of all Hermitian n × n matrices that are positive-definite.Write Q n for the space of all quadratic polynomials P : C n → R of the form p ( z ) = ( z − a ) ∗ B ( z − a )2 − log det( B ) with B ∈ M + n ( C ) , a ∈ C n . (4)Here z = ( z , . . . , z n ) ∈ C n is actually viewed as a column vector, and z ∗ is the row vectorwith entries z , . . . , z n . The representation (4) of p ∈ Q n clearly determines the vector a p = a and the positive-definite Hermitian matrix B p = B ( p ) = B . Note that for any p ∈ Q n , Z C n e − p ( z ) dλ ( z ) = det( B ) Z C n e − ( z − a ) ∗ B ( z − a ) / dλ ( z ) = Z C n e −| w | / dλ ( w ) = (2 π ) n , (5)where λ is the Lebesgue measure in C n .Let ( W t ) t ≥ be a standard Brownian motion in C n with W = 0 which is defined onour probability space (Ω , F , P ) . This means that Re( W t ) and Im( W t ) are two independentstandard Brownian motions in R n , so in particular E | W t | = 2 nt for all t ≥ . Recall thatthe quadratic variation of two continuous functions ϕ, ψ : [0 , t ] → C is [ ϕ, ψ ] t = lim ε ( P ) → N P X i =1 ( ϕ ( t i ) − ϕ ( t i − )) · ( ψ ( t i ) − ψ ( t i − )) whenever the limit exists, where P = { t < t < . . . < t N P = t } is a partition of [0 , t ] into N P intervals and ε ( P ) = max ≤ i ≤ N P | t i − t i − | is the mesh of the partition. Note thatthe quadratic variation of the Brownian motion is a deterministic function, namely [ W j , W k ] t = [ W j , W k ] t = 0 , [ W j , W k ] t = 2 tδ jk ( t > (6)where W t = ( W t , . . . , W tn ) ∈ C n and δ jk is Kronecker’s delta. The following propositionclaims the existence a certain stochastic process, adapted to the filtration induced by theBrownian motion and attaining values in the space Q n . Proposition 2.1 (“existence of the Eldan process”) . Assume that
Σ : Q n → C n × n is asmooth matrix-valued map with | B / p Σ( p ) | ≤ for all p ∈ Q n . Then there exists a Q n -valued adapted stochastic process ( p t ) t ≥ such that the following hold: i) p ( z ) = | z | / for all z ∈ C n .(ii) For any measurable set S ⊆ C n and for any t > , Z S e − p ( z ) dλ ( z ) = E Z S e − p t ( z ) dλ ( z ) . (iii) Denote a t = a p t , B t = B p t and Σ t = Σ( p t ) . Then the following system of stochasticdifferential equations hold true in t ∈ (0 , + ∞ ) : da t = Σ t dW t and B t = B t Σ t Σ ∗ t B t dt. (7) Proof.
Equation (7) is a stochastic differential equation with smooth coefficients in a and B . According to the standard theory (e.g., Kallenberg [12, Section 21]) this equation has aunique strong solution for all ≤ t < T with the initial condition B = Id, a = 0 , wherethe random variable T ∈ [0 , + ∞ ] is a stopping time which is almost-surely non-zero. Thestopping time T is the explosion time of the process, in the sense that T = sup k ≥ T k where T k = inf { t ≥ | a t | + | B t | ≥ k } ( k = 1 , , . . . ) . (8)Let us show that P ( T = + ∞ ) = 1 . We first explain why B t cannot blow up in finite time.In fact, we will show that B t is bounded by a deterministic function of t . For k ≥ and t > we abbreviate t k = min { t, T k } . We know that B = Id and by (7), B t k = Id + Z t k B s Σ s Σ ∗ s B s ds ( t > . (9)In particular, B t k ≥ Id for all t > . Since | B / t Σ t | ≤ , we deduce from (9) that Tr[ B t k ] = n + Z t k Tr[ B / s ( B / s Σ s Σ ∗ s B / s ) B / s ] ds ≤ n + Z t k Tr[ B s ] ds, where Tr[ A ] is the trace of the matrix A ∈ C n × n . Thus the Gr¨onwall inequality (e.g., [12,Lemma 21.4]) implies that Tr[ B t k ] ≤ ne t for all t > and k ≥ . Since B t k ≥ Id , thebound on the trace shows that B t cannot blow up in finite time. Next, the process ( a t ) t ≥ is not necessarily bounded by a deterministic function, yet for any s ∈ (0 , t k ) we have that | Σ s | = Tr[Σ s Σ ∗ s ] ≤ Tr[ B s Σ s Σ ∗ s ] = | B / s Σ s | ≤ . Hence, by the Itˆo isometry, E | a t k | = 2 E Z t k | Σ s | ds ≤ t k ≤ t. Now Doob’s martingale inequality (e.g., [12, Proposition 7.16]) shows that P (sup ≤ s ≤ t k | a s | >k/ −→ as k → ∞ , for any fixed t > . Consequently, lim k →∞ P ( T k < t ) = 0 for all t > . It follows that T = + ∞ with probability one. We have thus shown that the Q n -valuedstochastic process ( p t ) t ≥ is well-defined for all t > , and moreover (i) and (iii) hold true.It remains to prove (ii). We begin by deducing from (7) that d log det( B t ) dt = Tr (cid:20) B − t dB t dt (cid:21) = Tr[Σ t Σ ∗ t B t ] = Tr[Σ ∗ t B t Σ t ] . (10) ix z ∈ C n . Note that as | B t | ≤ ne t and | Σ t | ≤ , for any fixed t > , E Z t ( z − a s ) ∗ B s Σ s Σ ∗ s B s ( z − a s ) ds ≤ n e t · E Z t | z − a s | ds < ∞ . (11)Hence the process L t = L t ( z ) = Re nR t ( z − a s ) ∗ B s Σ s dW s o is a martingale whosequadratic variation process satisfies d [ L, L ] t = ( z − a t ) ∗ B t Σ t Σ ∗ t B t ( z − a t ) dt . We claimthat dp t ( z ) = − dL t + d [ L, L ] t / t > . (12)Indeed, for j = 1 , . . . , n write Σ jt for the j th row of the matrix Σ t . Then by (6) and (7) thequadratic variation process [ a j , a k ] t satisfies d [ a j , a k ] t = 2Σ jt (Σ kt ) ∗ dt. (13)Note that [ a j , a k ] t = [ a j , a k ] t = 0 , and also that B t is a process of finite variation. By Itˆo’sformula, for any fixed z ∈ C n , dp t ( z ) = ( z − a t ) ∗ dB t ( z − a t ) − ( da t ) ∗ B t ( z − a t ) − ( z − a t ) ∗ B t da t − d log det( B t ) + 2Tr[Σ ∗ t B t Σ t ] dt. (14)Now (12) follows from (7), (10) and (14). From (12) and the Itˆo formula, de − p t ( z ) = e − p t ( z ) · dL t . Hence ( e − p t ( z ) ) t ≥ is a local martingale. Since e − p t ( z ) ≤ det( B t ) ≤ n n e tn , the non-negative process ( e − p t ( z ) ) t ≥ is in fact a martingale. By the martingale property, E e − p t ( z ) = e − p ( z ) for all t and z . In order to prove (ii), observe that for any t > , E Z S e − p t ( z ) dλ ( z ) = Z S E e − p t ( z ) dλ ( z ) = Z S e − p ( z ) dλ ( z ) < ∞ . The application of Fubini’s theorem in the space C n × Ω , where Ω is the probability spaceon which the Brownian motion ( W t ) t ≥ is defined, is legitimate as the integrand is non-negative. This completes the proof of the proposition.Our next lemma shows that with probability one, the center a t of the probability density (2 π ) − n e − p t ( z ) does not escape to infinity as t → ∞ , and in fact almost-surely it has a finitelimit in C n . Lemma 2.2.
We work under the notation and assumptions of Proposition 2.1. Define a ∞ := lim t →∞ a t and A ∞ := lim t →∞ B − t . Then these limits are well-defined almost-everywhere in our probability space (Ω , F , P ) .Moreover, the random point a ∞ ∈ C n satisfies E | a ∞ | ≤ n while the random Hermitianmatrix A ∞ satisfies ≤ A ∞ ≤ · Id almost-surely. roof. According to (7), dB − t dt = − B − t dB t dt B − t = − B − t ( B t Σ t Σ ∗ t B t ) B − t = − Σ t Σ ∗ t . (15)Recall also that B = Id with B t ≥ Id for all t . Therefore, for any t > , Z t Σ s Σ ∗ s ds = − Z t dB − s ds ds = Id − B − t ≤ Id. (16)Since da t = Σ t dW t , we deduce from (16) that the quadratic variation processes of themartingales Re[ a jt ] and Im[ a jt ] are bounded by n for all t and j . Hence, according toDoob’s margingale convergence theorem (e.g., [12, Chapter 7]), a ∞ = lim t →∞ a t (17)is a well-defined random vector in C n , where the convergence in (17) is both almost-everywherein Ω and in the sense of L . Moreover, by (16), E | a ∞ | = 2 E Z ∞ | Σ s | ds ≤ n. Finally, the positive-definite Hermitian matrix B − t satisfies B − = Id and dB − t /dt ≤ ,according to (15). Thus the limit A ∞ = 2 · lim t →∞ B − t exists almost-surely, and it satisfies ≤ A ∞ ≤ · Id .Recall that A ∈ C n × n is an orthogonal projection if A = A = A ∗ . Write G n,k for theGrassmannian of all k -dimensional complex subspaces of C n . For a subspace E ∈ G n,k wedenote by π E the orthogonal projection matrix whose kernel equals E . Given a Hermitianmatrix A ∈ C n × n we denote its eigenvalues by λ ( A ) ≤ . . . ≤ λ n ( A ) . We write M + n ( C ) for the set of all Hermitian n × n matrices that are positive semi-definite. Proposition 2.1states that B t = Id + Z t B s Σ s Σ ∗ s B s ds ( t ≥ . (18)Since t B t Σ t Σ ∗ t B t is continuous, then with probability one, the map t B t is continu-ously differentiable in (0 , ∞ ) . Set M t = B / t Σ t Σ ∗ t B / t . Later on we will select a suitablemap Σ : Q n → C n × n such that with probability one, B t and M t will satisfy the assumptionsof the following standard, non-probabilistic lemma. Lemma 2.3 (“eigenvalue growth”) . Let ≤ k ≤ n − , ε > and let B, M : [0 , ∞ ) → M + n ( C ) be two functions, with M continuous and B continuously differentiable. Assumethat for all t ≥ , λ k +1 ( M t ) ≥ ε and dB t dt = B / t M t B / t (19) where M t = M ( t ) and B t = B ( t ) . Assume also that B = Id . Then for all t > , λ k +1 ( B t ) ≥ εtk + 1 . (20) roof. We recall the min-max characterization of the eigenvalues of a Hermitian n × n matrix A : λ k +1 ( A ) = min E ∈ G n,k +1 max = v ∈ E v ∗ Av | v | . (21)Set ℓ = n − ( k + 1) . For any F ∈ G n,ℓ and A ∈ M + n ( C ) , Tr[ Aπ F ] ≥ max = v ∈ F ⊥ v ∗ Av | v | ≥ min E ∈ G n,k +1 max = v ∈ E v ∗ Av | v | = λ k +1 ( A ) , (22)where F ⊥ = { z ∈ C n ; ∀ w ∈ F, w ∗ z = 0 } is the orthogonal complement. It also followsfrom (21) that for any A, C ∈ M + n ( C ) with C ≥ Id , λ k +1 ( C ∗ AC ) = min E ∈ G n,k +1 max = v ∈ E ( Cv ) ∗ A ( Cv ) | v | (23) ≥ min E ∈ G n,k +1 max = v ∈ E ( Cv ) ∗ A ( Cv ) | Cv | = min F ∈ G n,k +1 max = w ∈ F w ∗ Aw | w | = λ k +1 ( A ) . Note that (19) implies the inequality B t ≥ B = Id for all t ≥ . Consequently, we deducefrom (22) and (23) that for any t ≥ and F ∈ G n,ℓ , ddt Tr[ B t π F ] = Tr[ B / t M t B / t π F ] ≥ λ k +1 ( B / t M t B / t ) ≥ λ k +1 ( M t ) ≥ ε. (24)We integrate (24) and obtain that for t > and F ∈ G n,ℓ , Tr[ B t π F ] ≥ Tr[ B π F ] + εt = k + 1 + εt. (25)In particular, (25) holds true for the subspace F = F t that is spanned by the eigenvectors of B t that corresponds to the eigenvalues λ k +2 ( B t ) , . . . , λ n ( B t ) . We conclude from (25) that k + 1 + εt ≤ Tr[ B t π F t ] = Tr[ π F t B t π F t ] = k +1 X j =1 λ j ( B t ) ≤ ( k + 1) λ k +1 ( B t ) , completing the proof of (20).It is certainly possible to perform a more accurate analysis and improve the estimateof Lemma 2.3, yet it is not needed here. Once we know that λ k +1 ( B t ) is large, the density (2 π ) − n e − p t ( z ) is going to be concentrated on a small neighborhood of a k -dimensional affinesubspace in C n .A complex Gaussian probability measure in C n , or a complex Gaussian in short, is aprobability measure µ supported in a complex, affine subspace E ⊆ C n with density in E that is proportional to z e − ( z − a ) ∗ B ( z − a ) / ( z ∈ E ) (26)for some a ∈ E and a positive-definite, Hermitian operator B . We say that µ is more curvedthan the standard Gaussian if B ≥ Id. rite G n for the collection of all complex Gaussians in C n equipped with the topology ofweak convegrence, i.e., µ m −→ µ if ∀ ϕ ∈ C ( C n ) , Z C n ϕdµ m m →∞ −→ Z C n ϕdµ. Here, C ( C n ) is the collection of all bounded, real-valued, continuous functions on C n . Notethat for any quadratic polynomial p ∈ Q n , we deduce from (5) that the measure with density (2 π ) − n e − p belongs to G n . By a random complex Gaussian we mean a random variableattaining values in G n . For µ ∈ G n and j, k = 1 , . . . , n we denote a µ := Z C n zdµ, A j ¯ kµ = Z C n z j z k dµ ( z ) − Z C n z j dµ ( z ) Z C n z k dµ ( z ) . Thus a µ ∈ C n is the center of µ while A µ = ( A j ¯ kµ ) j,k =1 ,...,n ∈ M + n ( C ) is the complexcovariance matrix. Clearly µ ∈ G n is determined by a µ and A µ , and in fact under therepresentation (26) we have A µ = 2 B − in the case where E = C n . A standard argument shows that if µ , µ , . . . ∈ G n satisfy a µ m m →∞ −→ a ∈ C n , A µ m m →∞ −→ A ∈ M + n ( C ) , then µ m −→ µ where µ is a complex Gaussian in C n with a µ = a and A µ = A . Proposition 2.4.
We work under the notation and assumptions of Proposition 2.1. Let ε > and assume that with probability one, ∀ t ≥ , λ k +1 ( M t ) ≥ ε, (27) where M t = B / t Σ t Σ ∗ t B / t . Denote by µ t the measure on C n whose density is z (2 π ) − n e − p t ( z ) .Then µ t ∈ G n for all t ≥ and µ ∞ := lim t →∞ µ t is a well-defined random complexGaussian. Moreover, µ ∞ is supported in a k -dimensional, complex, affine subspace, it ismore curved than the standard Gaussian measure and ∀ ϕ ∈ L ( γ n ) , Z C n ϕdγ n = E Z C n ϕdµ ∞ , (28) where L ( γ n ) is the collection of all γ n -integrable, real valued functions on C n .Proof. Since the quadratic polynomial p t belongs to Q n , the measure µ t belongs to G n forall t , according to (5). Moreover, by Lemma 2.2 we know that with probability one, a t = a µ t t →∞ −→ a ∞ , A µ t = 2 B − t t →∞ −→ A ∞ , where a ∞ ∈ C n and A ∞ ∈ M + n ( C ) are random variables with A ∞ ≤ Id . Denote by µ ∞ ∈ G n the random complex Gaussian with center a ∞ and complex covariance matrix A ∞ . Then almost-surely, µ t t →∞ −→ µ ∞ . (29) ince A ∞ ≤ Id , with probability one the complex Gaussian µ ∞ is more curved than thestandard Gaussian measure in C n . For any ϕ ∈ C ( C n ) , it follows from Proposition 2.1(ii)that Z C n ϕdγ n = (2 π ) − n E Z C n ϕ ( z ) e − p t ( z ) dz = E Z C n ϕdµ t t →∞ −→ E Z C n ϕdµ ∞ , where we used (29) as well as the bounded convergence theorem in (Ω , F , P ) in the lastpassage. This proves (28) in the case where ϕ ∈ C ( C n ) . By approximation from below, wesee that (28) holds true for all non-negative, γ n -integrable functions. This implies (28) is thegeneral case. Next, thanks to our assumption (27) and to conclusion (iii) of Proposition 2.1,we may apply Lemma 2.3 and obtain that λ k +1 ( B t ) ≥ εk + 1 · t ( t ≥ . Therefore, as A µ t = 2 B − t , λ n − k ( A ∞ ) = lim t →∞ λ n − k ( A µ t ) = lim t →∞ λ k +1 ( B t ) = 0 . Thus A ∞ is a matrix whose rank is at most k , and with probability one, the random complexGaussian µ ∞ is supported in a k -dimensional complex, affine subspace of C n . We prove the following:
Proposition 3.1.
Let ≤ k ≤ n , let Z ⊆ C n be a closed set and let U ⊆ C n be an open setcontaining the origin. Assume that f : U → C k is holomorphic, with f (0) = 0 , such that Z = { z ∈ U ; f ( z ) = 0 } . Assume also that is a regular value of f .Then there exists a Q n -valued adapted stochastic process ( p t ) t ≥ , satisfying (i), (ii) and(iii) from Proposition 2.1, and moreover the following holds with probability one: For all t > we have a t ∈ Z and λ k +1 ( M t ) ≥ c n , where c n = 1 /n and where as before M t = B / t Σ t Σ ∗ t B / t . The proof requires some preparation. Fix
Z, U ⊆ C n and f = ( f , . . . , f k ) : C n → C k as in Proposition 3.1. The value is a regular value of f , hence for any z ∈ Z , the Jacobianmatrix ( ∂f j /∂z ℓ ) ≤ j ≤ k, ≤ ℓ ≤ n (30)has rank k . By continuity, there exists an open set ˜ U ⊆ U containing Z such that theJacobian matrix in (30) has rank k throughout ˜ U . We may replace U by the smaller set ˜ U and assume from now on that the Jacobian matrix in (30) has rank k throughout U . For aholomorphic function g : C n → C we write ∇ g = ( ∂g/∂z , . . . , ∂g/∂z n ) for the holomorphic gradient, viewed as a row vector. emma 3.2. There exists a smooth map H : U → G n,k such that ( ∇ f j ( z )) ∗ ∈ H ( z ) ( z ∈ U, j = 1 , . . . , k ) . (31) Proof.
For z ∈ U write H ( z ) for the complex subspace spanned by the vectors ( ∇ f ( z )) ∗ , . . . , ( ∇ f k ( z )) ∗ ∈ C n . Then H ( z ) is a k -dimensional subspace for any z ∈ U , since the Jacobian matrix in (30) hasrank k . Clearly the subspace H ( z ) ∈ G n,k varies smoothly with z ∈ U .Pick a smooth function θ : C n → [0 , that is supported on U with θ ( z ) = 1 for all z ∈ Z. (32)For a quadratic polynomial p ∈ Q n with a p ∈ U we set ˜ H ( p ) = B ( p ) − / · H ( a p ) ∈ G n,k , where H : U → G n,k is as in Lemma 3.2. Thus, we “rotate” the subspace H ( a p ) using theoperator B ( p ) − / in order to obtain the subspace ˜ H ( p ) . For a quadratic polynomial p ∈ Q n we define Σ( p ) := θ ( a p ) √ n · B ( p ) − / · π ˜ H ( p ) , (33)where we recall that π ˜ H ( p ) is the orthogonal projection matrix whose kernel is ˜ H ( p ) . Then Σ( p ) varies smoothly with p ∈ Q n , and moreover always | B / p · Σ( p ) | ≤ . (34)It follows from Lemma 3.2 that if a p ∈ U and j = 1 , . . . , k , then the vector ( ∇ f j ( a p )) ∗ belongs to the kernel of π H ( a p ) . Therefore B ( p ) − / ( ∇ f j ( a p )) ∗ belongs to the kernel of π ˜ H ( p ) . Consequently, for all p ∈ Q n , if a p ∈ U then ∇ f j ( a p ) Σ( p ) = 0 for j = 1 , . . . , k. (35)Thanks to (34) we may apply Proposition 2.1, and conclude the existence of the Q n -valuedstochastic process ( p t ) t ≥ . Recall that we abbreviate a t = a p t , B t = B p t and Σ t = Σ( p t ) . Proof of Proposition 3.1.
Consider the following stopping time: T = inf { t > a t U } . Since a = 0 ∈ U and U is open, the stopping time T is almost-surely positive. We claimthat a t ∈ Z for all t ∈ [0 , T ) . Indeed, recall that da t = Σ t dW t . Therefore the quadraticvariation [ a jt , a ℓt ] vanishes for all j, ℓ = 1 , . . . , n . This means that when applying the Itˆo ormula for df j ( a t ) , there will be no quadratic variation term as f j is holomorphic. Thus, for j = 1 , . . . , k and < t < T , df j ( a t ) = ∇ f j ( a t ) · da t = ∇ f j ( a t ) · Σ t dW t = ∇ f j ( a p t )Σ( p t ) dW t = 0 , (36)according to (35). For t > abbreviate t ∧ T = min { t, T } . It follows from (36) that f j ( a t ∧ T ) = f j ( a ) = f j (0) = 0 for all t ≥ , j = 1 , . . . , k. (37)Since Z = { z ∈ U ; ∀ j f j ( z ) = 0 } , from (37) we deduce that a t ∈ Z for all t ∈ [0 , T ) . (38)Consider the stopping time S = inf { t > a t Z } . From (38) we learn that S ≥ T with probability one. However, since Z is a closed set that iscontained in the open set U , the continuity of the process a t implies that S < T whenever T is finite. Therefore T = S = + ∞ almost-surely, and (38) shows that a t ∈ Z for all t > .In particular, for any t > , by (32), θ ( a t ) = 1 . Consequently, for all t > , B / t Σ t = θ ( a t ) √ n π ˜ H ( p t ) = π ˜ H ( p t ) √ n ( t > , (39)where ˜ H ( p t ) is a certain complex subspace of dimension k . Since M t = B / t Σ t Σ ∗ t B / t ,we understand from (39) that M t ∈ M + n ( C ) and λ k +1 ( M t ) = 1 /n for all t . This completesthe proof of the proposition.The decomposition of the Gaussian measure γ n alluded to in the Introduction is describedin the following theorem: Theorem 3.3.
Let ≤ k ≤ n , let Z ⊆ C n be a closed set and let U ⊆ C n be an open setcontaining the origin. Assume that f : U → C k is holomorphic, with f (0) = 0 , such that Z = { z ∈ U ; f ( z ) = 0 } and such that is a regular value of f . Then there exists a randomcomplex Gaussian probability measure µ ∈ G n with the following properties:(i) With probability one, µ is more curved than the standard Gaussian measure and it issupported in a k -dimensional complex, affine subspace of C n .(ii) For any ϕ ∈ L ( γ n ) , Z C n ϕdγ n = E Z C n ϕdµ. (iii) With probability one, the center a µ satisfies a µ ∈ Z . roof. In view of Proposition 3.1 we may apply Proposition 2.4 and set µ := µ ∞ . Nowproperties (i) and (ii) follow from Proposition 2.4 while (iii) follows from Proposition 3.1 as a µ = a ∞ = lim t →∞ a t ∈ Z since a t ∈ Z for all t > and Z is closed. Remark 3.4.
The point a µ = a ∞ is a random vector supported in Z . By setting ν ( A ) = P ( a µ ∈ A ) for A ⊆ C n we thus obtain a certain probability measure ν supported on Z .Does this probability measure ν associated with Z have any significance? In the case where Z is a linear subspace, the measure ν is a standard Gaussian in Z . Remark 3.5.
The stochastic process ( a t ) t ≥ has three important properties: The point a t belongs to Z at all times, this process is a martingale with da t = Σ t dW t , and furthermore Z ∞ Σ ∗ t Σ t dt = Id − lim t →∞ B − t = Id − P for some positive-definite matrix P whose rank is the codimension of Z . These three prop-erties suffice for obtaining the conclusion of Theorem 3.3.What is the rˆole of the complex structure in our proof? Can we construct a process withthese three properties given a submanifold Z ⊆ R n ? This question poses a challenge, evenin the case where Z is a minimal surface. A na¨ıve approach towards the real case could beto replace the definition (33) of Σ( p ) by something of the form Σ( p ) := θ ( a p ) √ n · B ( p ) − / · π ˜ H ( p ) · S, for a certain matrix S . The matrix S needs to satisfy two requirements: First, the Itˆo termshould vanish in order to ensure that a t ∈ Z . This is equivalent to the requirement that Tr h SS ∗ π ˜ H ( p ) B ( p ) − / ∇ f j B ( p ) − / π ˜ H ( p ) i = 0 ( j = 1 , . . . , k ) , (40)where ∇ f j is the Hessian of f j : R n → R . Note that this requirement automatically holds inthe complex case. Second, some regularity is needed, maybe of the form cId ≤ SS ∗ ≤ CId for some constants c, C depending on Z . The requirement (40) amounts to linear constraintson the matrix SS ∗ . However, as B ( p t ) − / is likely to be an almost-degenerate matrix forlarge t , it is not entirely clear how to satisfy these linear constraints while keeping S and S − bounded. For v ∈ C k and R > write D ( v, R ) = { z ∈ C k ; | z − v | ≤ R } . We claim that for any z , z ∈ C k , | z | ≤ | z | = ⇒ γ k ( D ( z , R )) ≥ γ k ( D ( z , R )) . (41) ndeed, consider the function ρ R ( z ) = γ k ( D ( z, R )) , defined for z ∈ C k . This function isclearly a radial function in C k . By the Pr´ekopa-Leindler inequality, the function ρ R is log-concave since it is the convolution of the Gaussian measure with the characteristic functionof a Euclidean ball (see, e.g., [3, Corollary 1.4.2]). Since ρ R is even and log-concave, itis necessarily non-increasing on any ray emanating from the origin, proving (41). We shallneed the following lemma: Lemma 4.1.
Let µ be a complex Gaussian measure in C k that is more curved than thestandard Gaussian γ k . Then for any v ∈ C k and R > , Z D ( a µ ,R ) e Re( v ∗ z ) dµ ( z ) ≥ γ k ( D ( v, R )) · Z C k e Re( v ∗ z ) dµ ( z ) . (42) Proof.
Translating, we may assume that a µ = 0 . Let ν be the probability measure on C n whose density with respect to µ is proportional to the function z e Re( v ∗ z ) . Our goal is toprove that ν ( D (0 , R )) ≥ γ k ( D ( v, R )) . (43)Assume first that the support of µ spans the entire space C k . Then for some B ∈ M + n ( C ) ,the density of µ with respect to the Lebesgue measure is dµdλ ( z ) = (2 π ) − n det( B ) · e − z ∗ Bz/ ( z ∈ C n ) . Since µ is more curved than the standard Gaussian, necessarily B ≥ Id . Set u = B − v .Note that dνdλ ( z ) = (2 π ) − n det( B ) · e − ( z − u ) ∗ B ( z − u ) / . Hence ν is the push-forward of the standard Gaussian measure γ k under the affine map T ( z ) = u + B − / z = B − v + B − / z ( z ∈ C n ) . Note that T − ( z ) = − B − / v + B / z . Since B / ≥ Id , then B / ( D (0 , R )) ⊇ D (0 , R ) ,and hence T − ( D (0 , R )) = − B − / v + B / ( D (0 , R )) ⊇ − B − / v + D (0 , R ) = D ( − B − / v, R ) . Since B − / ≤ Id we know that | B − / v | ≤ | v | . We may now use (41) to obtain ν ( D (0 , R )) = γ k ( T − ( D (0 , R ))) ≥ γ k ( D ( − B − / v, R )) ≥ γ k ( D ( v, R )) , and (43) is proven. This completes the proof of (42) in the case where the support of µ spans C k . The general case follows by an approximation argument (e.g., convolve µ with a smallGaussian). Proposition 4.2.
Let ≤ k ≤ n , let Z ⊆ C n be a non-empty closed set and let U ⊆ C n bean open set. Suppose that f : U → C k is holomorphic and that Z = { z ∈ U ; f ( z ) =0 } . Assume also that is a regular value of f . Then for any R ≥ and v ∈ C k with | v | ≥ d (0 , Z ) , γ n ( Z + R ) ≥ γ k ( D ( v, R )) . (44) roof. The set Z is closed and non-empty, hence there exists z ∈ Z with | z | = d (0 , Z ) .Set f ( z ) = f ( z + z ) and Z = Z − z , U = U − z . Thus Z = { z ∈ U ; f ( z ) = 0 } , theset Z contains the origin, and is a regular value of f . We may therefore apply Theorem 3.3and conclude that there exists a certain random complex Gaussian µ with Z C n ϕdγ n = E Z C n ϕdµ for all ϕ ∈ L ( γ n )) . (45)In particular, it follows from (45) that Z Z + R e − Re( z ∗ z ) dγ n ( z ) = E Z Z + R e − Re( z ∗ z ) dµ ( z ) ≥ E Z D ( a µ ,R ) e − Re( z ∗ z ) dµ ( z ) , (46)where we used the fact that P ( a µ ∈ Z ) = 1 in the last passage, as follows from Theorem3.3(iii). According to Theorem 3.3(i), with probability one the complex Gaussian µ is sup-ported in a k -dimensional, complex affine subspace, and it is more curved than the standardGaussian. By Lemma 4.1 and (41), E Z D ( a µ ,R ) e − Re( z ∗ z ) dµ ( z ) ≥ γ k ( D ( v, R )) · E Z C n e − Re( z ∗ z ) dµ ( z ) , (47)where we used that | v | ≥ | z | . Let us now use (45) with ϕ ( z ) = e − Re( z ∗ z ) . From (46) and(47) we conclude that Z Z + R e − Re( z ∗ z ) dγ n ( z ) ≥ γ k ( D ( v, R )) · Z C n e − Re( z ∗ z ) dγ n ( z ) . (48)We now change variables w = z + z in the integrals in (48). We obtain that Z Z + R e − Re( z ∗ w ) −| w − z | / dλ ( w ) ≥ γ k ( D ( v, R )) · Z C n e − Re( z ∗ w ) −| w − z | / dλ ( w ) , or equivalently, Z Z + R e −| w | / dλ ( w ) ≥ γ k ( D ( v, R )) · Z C n e −| w | / dλ ( w ) . The desired inequality (44) is thus proven.The conclusion of Proposition 4.2 is stable under upper Hausdorff limits in C n . We saythat a closed set Z ⊆ C n contains the upper Hausdorff limit of a sequence of closed sets Z , Z , . . . ⊆ C n if for any R, δ > there exists N ≥ with ∀ m ≥ N, D (0 , R ) ∩ Z m ⊆ Z + δ. (49) Lemma 4.3.
Let ≤ k ≤ n − and let K ⊆ C n be a convex body. Assume that the closedset Z ⊆ C n contains the upper Hausdorff limit of a sequence Z , Z , . . . of closed sets in C n . Then, γ n ( Z + K ) ≥ lim sup m →∞ γ n ( Z m + K ) . roof. Set L = lim sup m →∞ γ n ( Z m + K ) . It suffices to prove that for any fixed ε > , γ n ( Z + K ) ≥ L − ε. (50)Denote r = sup z ∈ K | z | < ∞ . In order to prove (50) we pick a large number R > r suchthat γ n ( C n \ D (0 , R − r )) ≤ ε. (51)Then for any δ > , according to (49), γ n ([ Z + δ ] + K ) ≥ lim sup m →∞ γ n ([ D (0 , R ) ∩ Z m ] + K ) ≥ − ε + lim sup m →∞ γ n ( Z m + K ) = L − ε. The set Z + K is closed. Since γ n is a probability measure, γ n ( Z + K ) = γ n \ δ> [ Z + K + δ ] ! = lim δ → + γ n ([ Z + δ ] + K ) ≥ L − ε, completing the proof of (50).Assume that the sequence of continuous functions f , f , . . . : C n → C k converges to alimit function f : C n → C k uniformly on compacts in C n . Then Z = f − (0) contains theupper Hausdorff limit of the sequence ( Z m ) m ≥ , where Z m = f − m (0) . For a short argumentsee, e.g., the proof of Theorem 4.2 in [13]. Proof of Theorem 1.1.
There exists a point p = ( p , . . . , p n ) ∈ Z with | p | = d (0 , Z ) . For δ > denote g δ ( z ) = f ( z ) + δ ( z − p , . . . , z k − p k ) ∈ C k , ( z ∈ C n ) . For all but finitely many values of δ , the point p is a regular point of the smooth map g δ : C n → C k . Pick a sequence δ m ց such that p is a always a regular point of g δ m . Thenfor each m , the image g δ m ( D ( p, /m )) contains an open neighborhood of the origin in C k .By Sard’s theorem, for any m we may find z m ∈ D ( p, /m ) ⊆ C n such that g δ m ( z m ) is aregular value of g δ m with | g δ m ( z m ) | ≤ /m . Define, h m ( z ) = g δ m ( z + z m − p ) − g δ m ( z m ) ( z ∈ C n , m ≥ . Then h m ( p ) is a regular value of the holomorphic function h m : C n → C k . Moreover, h m −→ f as m → ∞ uniformly on compacts in C n . By the remark preceding the proof,the set Z = f − (0) contains the upper Hausdorff limit of ( Z m ) m ≥ , where Z m = h − m (0) .Moreover, p ∈ Z m and hence for any m ≥ , d (0 , Z m ) ≤ | p | . (52)Let us now apply Proposition 4.2 for the non-empty closed set Z m , the open set U = C n and the holomorphic function h m : U → C k for which is a regular value. By (52) and theconclusion of the proposition, for any m ≥ and R > , γ n ( Z m + R ) ≥ γ k ( D ( v, R ))) = γ n ( E + R ) , (53) here v ∈ C k is any point with | v | = | p | . Now (2) follows from (53) and Lemma 4.3,completing the proof. Proof of Corollary 1.3.
Set r K = inf {| z | ; z ∈ ∂K } . The infimum is attained by compact-ness, hence there exists z ∈ ∂K such that | z | = r K . Clearly, r K B n ⊆ K. (54)Since the origin belongs to the interior of K , necessarily z = 0 . Setting H = z ⊥ = { z ∈ C n ; z ∗ z = 0 } , we claim that K ⊆ H + r K B n . (55)Indeed, the Euclidean sphere of radius r K centered at the origin is tangent to ∂K at the point z . By convexity, K lies to one side of the (real) tangent hyperplane to that sphere at thepoint z . Equivalently, sup z ∈ K Re( z ∗ z ) = z ∗ z = | z | = r K . Since K is circled, then | z ∗ z | ≤ r K for all z ∈ K and (55) follows. In particular, K + H ⊆ H + r K B n . Let f : C n → C be a holomorphic function, set Z = f − (0) and assume that Z = ∅ . Let t ∈ R be defined via d (0 , tz + H ) = d (0 , Z ) . We now apply the inclusion (54) and Theorem 1.1 in order to obtain that for any r > , γ n ( Z + rK ) ≥ γ n ( Z + rr K B n ) ≥ γ n ( tz + H + rr K B n ) ≥ γ n ( tz + H + rK ) , where we also used (55). The desired conclusion follows with H = tz + H .We end this paper with a real analog of Corollary 1.3. With a slight abuse of notation,in the next corollary we write γ n for the standard Gaussian measure on R n whose density is x (2 π ) − n/ exp( −| x | / . Corollary 4.4.
Let K ⊆ R n be a convex body with K = − K . Then there exists an ( n − -dimensional subspace H ⊆ R n with the following property: For any continuous function f : R n → R there exists t ∈ R such that L = f − ( t ) satisfies γ n ( L + rK ) ≥ γ n ( H + rK ) for all r > . Proof.
Set r K = inf {| x | ; x ∈ ∂K } , and let x ∈ ∂K be such that | x | = r K . Then K ⊇ r K B n while for H = x ⊥ ⊆ R n we have K ⊆ H + r K B n . According to Gromov’sGaussian waist inequality ([11], see also [13]), for any continuous function f : R n → R there exists t ∈ R such that L = f − ( t ) satisfies γ n ( L + rK ) ≥ γ n ( L + r · r K B n ) ≥ γ n ( H + r · r K B n ) ≥ γ n ( H + rK ) . Remark 4.5.
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