A local time scaling exponent for compact metric spaces

Transcription

1 A local time scaling exponent for compact metric spaces John Dever School of Mathematics Georgia Institute of Technology Fractals Cornell, June 15, 2017 Dever (GaTech) Exit time exponent Fractals 6 1 / 19

2 Local Hausdorff Dimension Let (X, d) a metric space. If A B X then dim H (A) dim H (B) 1. So define the local Hausdorff dimension α by... Definition α(x) := lim r 0 + dim H (B r (x)) = inf r>0 dim H (B r (x)). 1 dim H means the Hausdorff dimension Dever (GaTech) Exit time exponent Fractals 6 2 / 19

3 Carathéodory construction of a metric outer measure For δ > 0 and A X let C δ (A) be the collection of all covers of A by an at most countable number of subsets of X of diameter at most δ. Then let µ (A) = sup δ>0 inf{ U U diam(u)dim H(U) U C δ (A)}. If M := {A X E X [ µ (E) = µ (E A) + µ (E A c ) ] }, then M is a σ-algebra containing the Borel measurable sets and µ := µ M is a complete measure. We call µ the local Hausdorff measure of variable dimension α( ). Dever (GaTech) Exit time exponent Fractals 6 3 / 19

4 Variable Ahlfors Regularity By f g we mean there exists a constant C > 0 (independent of the arguments of f and g) such that 1 C f g Cg. Definition If Q : X (0, ), a borel measure ν is called variable Ahlfors Q( ) regular if ν(b r (x)) r Q(x). Theorem a Let X be a compact metric space. Then if ν is an Ahlfors Q-regular Borel measure then Q = α and ν µ. a J. Dever, Local Hausdorff Measure, ArXiv e-prints (2016). Dever (GaTech) Exit time exponent Fractals 6 4 / 19

5 A Koch curve of continuously varying local dimension. 2 Constructed by recursively varying the angles of the generator. Any constant dimensional Hausdorff measure gives measure 0 or. But the µ measure is positive and finite. 2 Laurent Nottale Fractal space time and microphysics: towards a theory of scale relativity, Petteri Harjulehto, Peter Hästö, and Visa Latvala, Sobolev embeddings in metric measure spaces with variable dimension, Mathematische Zeitschrift 254 (2006), no. 3, [8] Dever (GaTech) Exit time exponent Fractals 6 5 / 19

6 Time scaling exponent: discrete approach From now on: X connected, compact, Ahlfors α( ) regular. An ɛ net on X is a set N X such that x N B ɛ (x) = X and if x, y X with x y then d(x, y) ɛ. We way define an induced graph G N with vertex set N in many ways. Some choices for edge relations are as follows: Given c > 0 set x y if B cɛ (x) B cɛ (y). For x N let T (x) = {z X d(x, z) d(y, z) for all y N} be the Voronoi tile of x. Set x y it T (x) T (y). For x N let deg(x) = #{y N y x}. Let D N be the degree matrix and A N the adjacency matrix of G N...i.e. D N x,y = deg(x)δ x,y and A N x,y = 1 if x y and 0 otherwise. Define a random walk (X k ) k=1 by the transition matrix P N = D 1 N A N, P N x,y = p N (x, y) = 1 deg(x) A N x,y.[2] Dever (GaTech) Exit time exponent Fractals 6 6 / 19

7 Time scaling exponent: discrete approach Let B = B R (x 0 ) be a ball. Let τ B,N = inf{k X k / B}. Let E B,N (x) = E x τ B,N. Then we have the exit time equation E B,N (x) = 1 + y x p N(x, y)e B,N (y) for x B and E(x) = 0 for x / B, i.e. (I P B,N )E B = 1 for x B, E(x) = 0 for x / B. Example Dever (GaTech) Exit time exponent Fractals 6 7 / 19

8 Time scaling exponent: discrete approach Set E + B,N = max x N E B,N (x). Then let ω β (B) = inf δ>0 sup{e + B,N ɛβ N an ɛ net 3 with ɛ < δ} Note if β < β then E + N,B ɛβ δ β β E + N,B ɛβ It follows that if ω β (B) > 0 then ω β (B) = and if ω β (B) < then ω β (B) = 0. Hence we may define β(b) := sup{β 0 ω β (B) = } = inf{β 0 ω β (B) = 0}. Then if B B, β(b ) β(b). We define... Definition β(x) := inf R>0 β(b R (x)) = lim R 0 + β(b R (x)). 3 Another (weaker) possibility is to take lim sup only along a chosen sequence of nested ɛ nets or dyadic cube approximations with the length scale going to 0. Dever (GaTech) Exit time exponent Fractals 6 8 / 19

9 Example Here β = 2α. Hence β can vary continuously as well! Example On R n with standard euclidean lattice approximations, with the (weaker) definition of β we have β(x) = 2. Example On the Sierpinski gasket the exit time from a ball of graph distance n is of order 5 n. Since length scale is 1 2 n, (with weaker def.) β(x) = log(5) log(2). Dever (GaTech) Exit time exponent Fractals 6 9 / 19

10 Continuous space approach Since we have an Ahlfors α( ) regular measure µ we may define a random walk at stage r by the transition kernel p r (x, y) = 1 µ(b r (x)) χ B r (x)(y), x, y X. Define P r on L 2 (µ) by P r f (x) = p r (x, y)dµ(y). This defines a (continuous space, discrete time) random walk (Y k ) k=0. If B = B R (x 0 ) is a ball let τ B = inf{k Y k / B}. Let E r,b (x) = E x τ B,r (x). Again we have the exit time equation E r,b (x) = 1 + X p r (x, y)e r,b (y)dµ(y) for x B and E r,b (x) = 0 for x / B. So (I P r )E r,b = 1 on B and E r,b (x) = 0 for x / B. Dever (GaTech) Exit time exponent Fractals 6 10 / 19

11 Time scale exponent: continuous space approach Let E + r,b = sup x B E r,b. Then let ω β (B) = lim sup r 0 + E + r,b r β. As before there is a critical exponent β(b) = sup{β > 0 ω β (B) = } = inf{β > 0 ω β (B) = 0}. Definition β(x) := inf R>0 β(b R (x)) = lim R 0 + β(b R (x)). Example Let B = B R (0) a ball about the origin in R n with euclidean norm. Let r small enough so that B r (x) B. Then we have Hence β(x) = 2 on R n. ( ) n + 2 R 2 x 2 E r (x) = n r 2. Dever (GaTech) Exit time exponent Fractals 6 11 / 19

12 Continuous time re-normalization Let τ r (x) = r β(x) for x X. Let Ω = (R + X) Z+. For x = x 0, r > 0 define a measure on cylinder sets by P x 0 ( r ({ω (R + X) Z+ ω(k) I k A k for k = 1,..., n}) = n ) e k=1 Ik Ak t k /τr (x k 1 ) τ r (x k 1 ) p r (x k 1, x k ) dt n dµ(x n )...dt 1 dµ(x 1 ). 4 This is Kolmogorov consistent. By the Kolmogorov Extension Theorem we may extend to a prob. measure P x r on Ω with product σ-algebra. For basepoint x = x 0 set ω(0) = (0, x 0 ). Then for t 0, ω Ω, let ˆt(ω) = inf{k k j=0 ω(k) 1 t}. Then define (X (r) t ) t 0 by X (r) t (ω) = ω(ˆt(ω)) 2. 4 A measure with local exponential waiting times in the case of a graph was considered by Bellissard in [3]. A formula for the generator was also given there. Dever (GaTech) Exit time exponent Fractals 6 12 / 19

13 Generator Then (X (r) t ) t 0 has generator L r on L 2 (µ) defined by E x 1 f (X t ) = L r f (x) := (f (x) f (y))dµ(y). [3] t=0 d dt r β(x) µ(b r (x)) B r (x) There is an equilibrium probability measure ν r with density dν r (x)/dµ(x) := r β(x) µ(b r (x)) Z r, where Z r is the normalization factor Z r = X r β(z) µ(b r (z))dµ(z). Then L r is self-adjoint on L 2 (ν r ). Let τ B,r := inf{t X (r) t / B}. For x X let φ r,b (x) := E x τ B,r. Let φ + r,b := sup y B φ r,b (y). Dever (GaTech) Exit time exponent Fractals 6 13 / 19

14 Variable Time Regularity Condition For β > 0 let T(B) := lim sup r 0 + φ + r,b. Definition (Time regularity, general case) For β(x) the local time exponent, we say X satisfies E(β) if β is bounded and for all x X, 0 < r < diam(x) 2, we have T(B r (x)) r β(x). a a [7] considers a similar condition under assumption that a heat kernel exists. Example Let B the open ball of radius R about the origin in R n under the euclidean norm. If B r (x) B we have φ r (x) = ( ) n+2 n (R 2 x 2 ). Hence T(B r (x)) = ( n+2 n )r 2. Dever (GaTech) Exit time exponent Fractals 6 14 / 19

15 Dirichlet spectrum lower bound Let B = B R (x 0 ) and assume µ((b c ) ) > 0. Then define L B r on L 2 (B, ν r ) by L B r f (x) = χ B (x)l r (χ B f )(x) where we define χ B f outside B to be 0. We have the exit time equation L B r φ B,r (x) = 1. 5 Then one can show L B r has a ν r -symmetric Green s function g B (x, y) such that g B (x, y)dν r (y) = φ B,r (x). Theorem If λ B r is the bottom of the spectrum of L B r on L 2 (B, ν r ) then λ B r c R β(x 0) for some constant c > 0 independent of r, R, and x 0. 5 Compare to torsion function and torsional rigidity on a Riemannian manifold. Dever (GaTech) Exit time exponent Fractals 6 15 / 19

16 Γ Convergence 6 If X is a separable second countable topological space with neighborhood basis at x B(x) then (Γ- lim inf n f n )(x) := sup U B(x) lim inf n inf y U f n (y) and (Γ- lim sup n f n )(x) := sup U B(x) lim sup n inf y U f n (y). We say f n Γ converges to f if the two are equal. Theorem a Every sequence of non-negative Markovian forms forms on a separable Hilbert space L 2 (X, µ) has a subsequence Γ converging to a closed, non-negative, Markovian form (i.e. a Dirichlet form). a proven in Mosco Composite media and asymptotic Dirichlet forms[4]. This Theorem was applied in the Kumagai-Sturm construction in [1]. In that paper Γ limits of approximate Dirichlet forms E r (u) = X X u(x) u(y) 2 k r (x, y)dµ(x)dµ(y) are considered. 1 Particular attention is paid to the choice k r (x, y) = h(r)µ(b r (x)) χ B r (x). 6 For more information see book by Dal Maso [5] Dever (GaTech) Exit time exponent Fractals 6 16 / 19

17 In our case the process (X (r) t ) t 0 leads us to consider approximate forms given by E r (f ) = f, L r f L 2 (X,ν r ) = 1 Z r X B r (x) f (y) f (x) 2 dµ(y)dµ(x), where Z r = v r r β µ = X µ(b r (x))r β(x) dµ(x), dν(x) = µ(b r (x))r β(x) dµ(x). Theorem A sequence (r n ) n=1 of positive numbers decreasing to 0 can be chosen such that E = Γ- lim n E rn exists, and D(E) contains an algebra of bounded measurable functions separating points a. a this algebra is generated by pointwise limits of exit time functions from balls in a neighborhood basis. Dever (GaTech) Exit time exponent Fractals 6 17 / 19

18 [1]Takashi Kumagai, Karl-Theodor Sturm, Construction of diffusion processes on fractals, d-sets, and general metric measure spaces(2005). [2]András Telcs, The art of random walks, [3]Jean Bellissard, Diffusion on compact metric spaces, Unpublished private notes. [4]Umberto Mosco, Composite media and asymptotic Dirichlet forms, [5]Gianni Dal Maso, An introduction to Γ-convergence, [6]J. Dever, Local Hausdorff measure, ArXiv (2016). [7]Alexander Grigor yan, Heat kernels on manifolds, graphs and fractals, [8]Harjulehto, Hästö, and Latvala, Sobolev embeddings in metric measure spaces with variable dimension(2006). Dever (GaTech) Exit time exponent Fractals 6 18 / 19

19 Thank you. Dever (GaTech) Exit time exponent Fractals 6 19 / 19