ESSENTIAL SETS FOR RANDOM OPERATORS CONSTRUCTED FROM AN ARRATIA FLOW

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1 Communications on Stochastic Analysis Vol., No. 3 (07) 30-3 Serials Publications ESSENTIAL SETS FO ANDOM OPEATOS CONSTUCTED FOM AN AATIA FLOW A. A. DOOGOVTSEV AND IA. A. KOENOVSKA Abstract. In this paper we consider a strong random operator T t which describes a shift of functions from L () along an Arratia flow. We find a compact set in L () that doesn t disappear under T t, and estimate its Kolmogorov widths.. Introduction: Arratia Flow and andom Operators In this paper we consider random operators in L () which describe shifts of functions along an Arratia flow []. Let us recall the definition. Definition. ([]). A family of random processes x(u, s), u, s 0 is called an Arratia flow if ) for each u x(u, ) is a Wiener process with respect to the joint filtration such that x(u, 0) = u; ) for any u u and t 0 3) the joint characteristics are x(u, t) x(u, t) a.s. d < x(u, ), x(u, ) > (t) = I x(u,t)=x(u,t)dt. In the informal language, Arratia flow is a family of Wiener processes started from each point of, which move independently up to the meeting, coalesce, and move together. It was proved in [4, 8] that for any a, b and t > 0 the set x([a; b], t) is finite a.s. Since Arratia flow has a right-continuous modification [3], x(, t) : is a step function for any time t > 0. Hence, for any a, b and t > 0 with probability one there exists a random point y for which λu [a; b] : x(u, t) = y > 0, (.) where λ is Lebesgue measure on. Since x(, t) is a right-continuous step function, for a fixed countable set A Px(, t) A = Px(Q, t) A u Q Px(u, t) A = 0. (.) eceived ; Communicated by the editors. 00 Mathematics Subject Classification. Primary 60H5. Key words and phrases. Arratia flow, Kolmogorov widths, random operator. 30

2 30 A. A. DOOGOVTSEV AND IA. A. KOENOVSKA Since for any a < b the difference x(b, ) x(a, ) collision happens, and x(b,0) x(a,0) is a Wiener processes until the = b a, one can find the distribution of the time of coalescence τ a,b = infs 0 x(a, s) = x(b, s) of the processes x(a, ), x(b, ), i.e. for any t 0 Pτ a,b t = Px(a, t) = x(b, t) = π e v b a t dv. (.3) Let us notice that for a fixed time t > 0 and an Arratia flow X = x(u, s), u, s [0; t] there exists an Arratia flow Y = y(u, r), u, r [0; t] such that trajectories of X and Ỹ = y(u, t r), u, r [0; t] don t cross [, 7]. Y is called a conjugated (or dual) Arratia flow. It was proved in [0] the following change of variable formula for an Arratia flow. Theorem. ([0]). For any time t > 0 and nonnegative measurable function h : such that h(u)du < h(x(u, t))du = h(u)dy(u, t) a.s., (.4) where the last integral is in sense of Lebesgue-Stieltjes. In this paper we consider random operators T t, t > 0, in L () which are defined as follows (T t f) (u) = f(x(u, t)), where f L () and u. It was proved in [5] that T t is a strong random operator [] in L (), but, as it was shown in [0], is not a bounded one. eally, for the point y from (.) one can introduce a sequence of the intervals A i = [r i ; p i ] such that y A i for any i and p i r i 0, i. Thus for any i T t I Ai L () λu [a; b] : x(u, t) = y > 0, which can t be true if T t was a bounded random operator. Hence, the image of a compact set under T t may not be a random compact set. Moreover, as it was mentioned in [9], the image of a compact set under strong random operator may not exist. However, in [0] it was presented a family of compact sets in L () whose images under T t exist and are random compact sets. In this paper we consider a compact set of this type, and investigate the change of its Kolmogorov widths [] under T t.. T t -essential Functions If the support of the function f L () is bounded, suppf [a; b], then T t f equals to 0 with positive probability. eally, by (.4), one can check that

3 ESSENTIAL SETS FO ANDOM OPEATOS 303 P f (x(u, t))du = 0 P x(, t) [a; b] = = P I [a;b] (x(u, t))du = 0 = P I [a;b] (u)dy(u, t) = 0, where y(u, s), u, s [0; t] is a conjugated Arratia flow. Since, by (.3), P I [a;b] (u)dy(u, t) = 0 = P y(b, t) = y(a, t) > 0, then P T t f L() = 0 > 0. This leads to the following definition. Definition.. For a fixed t > 0 a function f L () is said to be a T t -essential if P T t f L () > 0 =. Example.. Let f L () be an analytic function which doesn t equal totally to zero. Denote the set of its zeroes Z f = u f(u) = 0. Then, by (.), P x(, t) Z f = =, so f is a T t -essential for any t > 0. Let us notice that if t t then T t -essential function may not be a T t - essential. To introduce a T -essential that is not T -essential function let us consider an increasing sequence such that u 0 = 0, u = and for any n N u n+ u n = n, u n = u n + n(ln ). Theorem.3. The function f = I [u n ;u n+ ] is a T -essential, and is not a T -essential. Proof. To prove that f is not a T essential we show that P T f L () > 0 <. Since [ ; + ] [u j ; u j+ ] = for any k j then, by (.4), P T f L > 0 = P () = P ( I [un;u n+](x(u, ))) du > 0 I [un;u n+](x(u, ))du > 0

4 304 A. A. DOOGOVTSEV AND IA. A. KOENOVSKA Thus by (.3), = P (y(u n+, ) y(u n, )) > 0 = P n 0 : y(u n+, ) y(u n, ) P y(u n+, ) y(u n, ). P y(u n+, ) y(u n, ) = 4π n+ e n+ v 4 dv π <. Consequently, the function f = I [u n ;u n+ ] is not a T -essential. To prove that f = I [u n ;u n+ ] is a T -essential one can show the following estimation. Lemma.4. Let w(u n, ) be a family of independent Wiener processes on [0; ] such that w(u n, 0) = u n. Then for any n N P max max w(u j, s) min w(u n, s) < s [0;] s [0;] j=0,n j=0 n π ln. Proof. Let w, w be an independent Wiener processes on [0; ] started from point 0, i.e. w (0) = w (0) = 0. It can be noticed that P max max w(u j, s) min w(u n, s) s [0;] j=0,n s [0;] = P j = 0, n : max w(u j, s) min w(u n, s) 0 s [0;] s [0;] n P max w(u j, s) min w(u n, s) 0 s [0;] s [0;] n j=0 P max w (s) min w (s) u n u j. s [0;] s [0;] From the fact that u n is an increasing sequence we can estimate the last expression and complete the proof n P max w (s) min w (s) u n u j s [0;] s [0;] j=0 n e (u n u j ) 4 π u n u j j=0 n (u n u n ) π(un u n ) e 4 n π ln.

5 ESSENTIAL SETS FO ANDOM OPEATOS 305 Let us prove that the function f = I [u n ;u n+ ] is a T -essential. Using the reasoning from the first part of the proof it can be checked that for the considered function f the following equality holds P T f L () > 0 = P (y(u n+, ) y(u n, )) > 0. Let us prove that P lim sup (y(u n+, ) y(u n, )) n =. (.) Build a new processes ỹ(u n, ) such that ỹ(u n, ) and y(u n, ) have the same distributions in C ([0; ]) in the following way [4]. Let w(u n, ) be a given family of Wiener processes on [0; ], w(u n, 0) = u n. Let us denote collision time of f, g C ([0; ]) by τ[f, g] := inft f(t) = g(t). Put ỹ(u 0, ) := w(u 0, ). Then for any n N, s [0; ] one can define ỹ(u n, s) = w(u n, s)i s < τ[w(u n, ), ỹ(u n, )] + ỹ(u n, s)is τ[w(u n, ), ỹ(u n, )]. According to constructions of stochastic processes ỹ(u n, ) Thus P N N : n N ỹ(u n, t) = w(u n, t), ỹ(u n+, t) = w(u n+, t)i t < τ[w(u n, ), w(u n+, )] +w(u n, t)i t τ[w(u n, ), w(u n+, )] =. (.) P N N : n N ỹ(u n+, t) ỹ(u n, t) = w(u n+, t) w(u n, t) =. For the considered sequence u n and any n N the following inequality holds P w(u n+, t) w(u n, t) = Therefore, by the Borel-Cantelli lemma and (.), ) e (v k 4 dv 4π 4π P lim sup(ỹ(u n+, t) ỹ(u n, t)) =. n Using the observation from Example. one can introduce a family of T t - essential functions for all t > 0. For any ε > 0 let us consider an integral operator K ε in L () with the kernel k ε (v, v ) = p ε (u v )p ε (u v )dy(u, t), (.3) e v 4 dv.

6 306 A. A. DOOGOVTSEV AND IA. A. KOENOVSKA where v, v, and p ε (u) = πε e u ε. By the change of variables formula for an Arratia flow [0], (K ε f, f) = (f p ε ) (x(u, t))du. (.4) Lemma.5. For any ε > 0 and nonzero function f L () P (K ε f, f) 0 =. Proof. According to (.) it is sufficient to note that f p ε is an analytic function. Consequently, for any t > 0 the following relations are true P (K f, f) > 0 = P T t (f p ) L() > 0 = P x(, t) Z f p = =. According to the last theorem and (.4), for any ε > 0 and nonzero f L () the function f p ε is a T t -essential for each t > Change of Compact Sets under a Strong andom Operator Generated by an Arratia Flow As it was noticed in the introduction any function with bounded support isn t a T t -essential. Consequently, if K L () is a compact set of functions with uniformly bounded supports such that T t (K) is well-defined, then the image T t (K) equals to 0 with positive probability. It was shown in [0] that T t may also change the geometry of K even in the case of a compact set K for which T t (K) 0 a.s. For example, the image T t (K) of a convergent sequence and its limiting point may not have limiting points. In this section we build a compact set K for which T t (K) 0 a.s. and investigate the change of its Kolmogorov-widths in L () under random operator T t. Definition 3. ([]). The Kolmogorov n-width of a set C H in a Hilbert space H is given by d n (C) = where L is a subspace of H. inf sup inf f g H, dim L n f C g L We consider the following compact set in L () K = f W () f (u)( + u ) 3 du + (f (u)) ( + u ) 7 du. (3.) Estimations on its Kolmogorov-widths in L () are presented in the next lemma. Lemma 3.. There exist positive constants C, C such that for any n N C n d n (K) C. n 3 0

7 ESSENTIAL SETS FO ANDOM OPEATOS 307 Proof. Let n N be fixed. To estimate d n (K) from above one can consider the partition n of [ n 5 ; n 5 ] into n segments [ ; + ], k = 0, n with equal lengths. Let us show that for the n-dimensional subspace L n = LSI [uk ;+ ], k = 0, n u >c sup f K inf f g L () C. g L n n 3 0 If f K then f (u)( + u ) 3 du. Thus for any C > 0 f (u)du ( + C) 3 f (u)( + u ) 3 du C 3. So, for the function g f = n f g f L () u >c f( )I [uk ;+ ] L n the following estimation is true + (f(u) g f (u)) du. u n 5 By the Cauchy inequality, for f K and u [ ; + ] u u f (v)dv dv ( + v ) 7 u. Consequently, u n 5 (f(u) g f (u)) du = n + u f (v)dv du n (+ ) =, and the upper estimation for d n (K) holds with the constant C = 3. To get a lower estimation for d n (K) we use the theorem about n-width of (n+)- dimensional ball []. Let (n+) be a partition of [0; ] into (n + ) segments [ ; + ], k = 0, n + with equal lengths. Consider (n+)-dimensional space L n+ = LSf k, k = 0, n, where the functions f k, k = 0, n, are defined as follows 0, u / [ ; + ],, u [ + 6(n+) f k = ; + 6(n+) ], 6(n + )(u ), u [ ; + 6(n+) ], (3.) 6(n + )(u + ), u [ + 6(n+) ; +]. We show that if c = 3 ( ) 5 then the ball B n+ = f L n+ f L () cn is a subset of K. Since f k L () = 5 8(n+), k = 0, n, then for any f B n+ such that f = n c k f k the following relation holds n c k 36 5cn. Thus according

8 308 A. A. DOOGOVTSEV AND IA. A. KOENOVSKA to (3.), f (u)( + u ) 3 du + 3 f L () + 7 n c k (f (u)) ( + u ) 7 du + 6(n+) (6(n + )) du + 3 cn + 0 3n 36 5cn c 3 ( ) =. 5 u k+ + 6(n+) (6(n + )) du Consequently, B n+ K and d n (K) d n (B n+ ). Due to the theorem about n-width of (n+)-dimensional ball, d n (B n+ ) = cn []. So the lower estimation for d n (K) holds with C := c. To show that estimations from above for the Kolmogorov-widths of the considered compact set K don t change under T t one may use the same idea as in Lemma. Theorem 3.3. There exists Ω of probability one such that for any ω Ω and n N where the constant C(ω) > 0 doesn t depend on n. d n (Tt ω (K)) C(ω), (3.3) n 3 0 Proof. For a fixed n N let us consider a partition n of [ n 5 ; n 5 ] into n segments with equal lengths. To prove (3.3) it s sufficient to show the following inequality for the linear space L ω n = LS T ω t I [uk ;+ ], k = 0, n with dimension at most n sup inf h Tt ω(k) h L ω n h h L () C(ω). n 3 0 According to the change of variable formula for an Arratia flow, one can check the equality for any f K T t ω f Tt ω ( n f( )I [uk ;+ ]) + u n 5 L () = u >n 5 f (u)dy(u, t, ω) ( n f(u) f( )I [uk ;+ ](u)) dy(u, t, ω).

9 ESSENTIAL SETS FO ANDOM OPEATOS 309 To estimate from above the last integral let us notice that u n 5 ( n f(u) f( )I [uk ;+ ](u)) dy(u, t, ω) n uk+ Due to (3.), for any f K and u [ ; + ] Thus n uk+ ( u ) f (v) dv dy(u, t, ω). ( u ) u f dv (v) dv ( + v ) 7 +. ( u ) n f (v) dv dy(u, t, ω) (+ ) uk+ dy(u, t, ω) = (y(n n 4 5, t, ω) y( n 5, t, ω)). 5 For an Arratia flow y(u, s), u, s [0; t] the following relation is true [] Consequently, for any ω Ω = ω Ω y(u, t) lim = a.s. u u y(u,t,ω lim ) u u = the estimation holds ( ) n f(u) f( )I [uk ;+ ](u) dy(u, t, ω) 4c(ω) u n 5 (3.4) with the constant c(ω) = sup u y(u, t, ω). (3.5) u Let us prove that for any ω Ω there exists a constant c(ω) such that u >n 5 f (u)dy(u, t, ω) c(ω). It can be noticed that u >n 5 f (u)dy(u, t) u >n 5 f (u)( + u ) 3 dy(u, t). Denote by θ j j= a sequence of jump points of the function y(, t) on +. Thus

10 30 A. A. DOOGOVTSEV AND IA. A. KOENOVSKA one may show f (u)( + u) 3 dy(u, t) = u>n 5 = θ i n 5 f (θ i )( + θ i ) 3 y(θ i, t) k= i: θ i [ k;k+) ( + k) 3 k= f (θ i )( + θ i ) 3 y(θ i, t) i: θ i [ k;k+) f (θ i ) y(θ i, t). According to the Cauchy inequality and (3.), for any u + the following relations hold f (u) (f (v)) ( + v) 7 dv dv u u ( + v) 7 6u 6. Consequently, due to (3.5), the inequalities are true ( + k) 3 f (θ i ) y(θ i, t) k= k= i: θ i [ k;k+) ( + k) 3 (y(k +, t) y(k, t)) 6k6 k= 6c 3 k. Hence, for any ω Ω there exists the constant C (ω) = 6c(ω) 3 such that f (u)dy(u, t, ω) C (ω). u>n 5 Similarly, it can be proved that u< n 5 f (u)dy(u, t, ω) C(ω). According to this and (3.4), for any ω Ω an upper estimation for d n (T ω t (K)) is true. The functions from Lemma that were used to build the (n + )-dimensional subspace are not T t -essential for any t > 0. Thus the image of this subspace under the random operator T t may be equal to 0 with positive probability. So, one can ask about the existence of a finite-dimensional subspace such that for any t > 0 its image under T t is a linear subspace with the same dimension. 4. A Subspace Preserving the Dimension under a andom Operator Generated by an Arratia Flow In this section for any t > 0 and n N we present a family g k, k = 0, n of linearly independent T t -essential functions such that their images under T t are linearly independent. Such a family generates a subspace which preserves the

11 ESSENTIAL SETS FO ANDOM OPEATOS 3 dimension under a random operator generated by an Arratia flow. It can be used to get a lower estimation of d n (T t (K)). Let us fix any n N, and build a family of (n+) linearly independent functions in the following way. Let (n+) be a partition of [0; n ] into (n+) segments with equal lengths. For any k = 0, n define f k by 0, u / [ ; + ],, u [ + n 6(n+) f k = ; + n 6(n+) ], 6(n+) n (u u k ), u [ ; + n 6(n+) ], (4.) 6(n+) n (u u k+ ), u [ + n 6(n+) ; +]. Lemma 4.. There exists ε 0 > 0 such that for any 0 < ε ε 0 the functions f k p ε, k = 0, n are linearly independent. Proof. Since the considered functions f k, k = 0, n are linearly independent, its Gram determinant doesn t equal to 0, i.e. G(f 0,..., f n ) 0. For each k = 0, n f k p ε f k, ε 0. Hence, due to the continuity of the Gram determinant, one may notice that there exists ε 0 > 0 such that for any 0 < ε ε 0 G(f 0 p ε,..., f n p ε ) 0, and the desired result is proved. Theorem 4.. There exists a set Ω 0 of probability one such that for any ω Ω 0 the functions T ω t (f 0 p ε ),..., T ω t (f n p ε ) are linearly independent. Proof. Denote by K ε the integral operator in L () with the kernel k ε. To prove the statement of the theorem it s enough to show that on some Ω 0 of probability one the following inequality holds (K ε f, f) > 0, for any nonzero f LSf 0,..., f n. Due to (.4) (K ε f, f) = θ (f p ε ) (θ) y(θ, t), (4.) where θ is a point of jump of the function y(, t). It was proved in [6] that there exists Ω 0 of probability one such that for any ω Ω 0 a linear span of the functions p ε ( θ(ω)) [0;] θ(ω) is dense in L ([0; ]). Thus on the set Ω 0 for any f LSf 0,..., f n L ([0; ]) one can find a random point θ f such that (f( ), p ε ( θ f )) 0. Since y(, t) : is nondecreasing, y(θ, t) > 0 for any jump-point θ. Consequently, on the set Ω 0 (f p ε ) (θ) y(θ, t) = (f( ), p ε ( θ)) y(θ, t) θ θ (f( ), p ε ( θ f )) y(θ f, t) > 0, which proves the theorem.

12 3 A. A. DOOGOVTSEV AND IA. A. KOENOVSKA Acknowledgment. The first author acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) within the project Stochastic Calculus and Geometry of Stochastic Flows with Singular Interaction for initiation of international collaboration between the Institute of Mathematics of the Friedrich- Schiller University Jena (Germany) and the Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev. The second author is grateful to the Institute of Mathematics of the Friedrich- Schiller University Jena (Germany) for hospitality and financial support within Erasmus+ programme 05/6-07/8 between Jena Friedrich-Schiller University (Germany) and the Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev. eferences. Arratia,.: Coalescing Brownian motions on the line, PhD Thesis, University of Wisconsin, Madison, Chernega, P. P.: Local time at zero for Arratia flow, Ukr. Mat. Zh. 64 (04), no. 4, Dorogovtsev, A. A.: Some remarks about Brownian flow with coalescence, Ukr. Mat. Zh. 57 (005), no. 0, Dorogovtsev, A. A.: Measure-valued processes and stochastic flows (in ussian), Institute of Mathematics of the NAS of Ukraine, Kiev, Dorogovtsev, A. A.: Krylov-Veretennikov expansion for coalescing stochastic flows, Commun. Stoch. Anal. 6 (0), no. 3, Dorogovtsev, A. A. and Korenovska, Ia. A.: Some random integral operators related to a point process, in: Theory of Stochastic Processes. (38) (07), no.. 7. Fontes, L.. G., Isopi, M., Newman, C. M., and avishankar, K.: The Brownian web, Proc. Nat. Acad. Sciences. 99 (00), no. 5, Harris, T. E.: Coalescing and noncoalescing stochastic flows in, Stochastic Processes and their Applications 7 (984), no., Korenovska, I. A.: andom maps and Kolmogorov widths, Theory of Stochastic Processes 0(36) (05), no., Korenovska, Ia. A.: Properties of strong random operators generated by an Arratia flow (in ussian), Ukr. Mat. Zh. 69 (07), no., Skorokhod, A. V.: andom Linear Operators, D.eidel Publishing Company, Dordrecht, Holland, Tikhomirov, V. M.: Some questions in approximation theory (in ussian), Izdat. Moskov. Univ., Moscow, 976. A. A. Dorogovtsev: Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine, and National Technical University of Ukraine Igor Sikorsky Kyiv Polytechnic Institute, Institute of Physics and Technology, Kiev, Ukraine. address: andrey.dorogovtsev@gmail.com Ia. A. Korenovska: Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine address: iaroslava.korenovska@gmail.com