Finite Dimensional Distributions. What can you do with them?

Transcription

1 Finite Dimensional Distributions. What can you do with them? Rodrigo Bañuelos 1 Purdue University Department of Mathematics banuelos@math.purdue.edu banuelos June 25, 26, Supported in part by NSF Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

2 Outline 1 Lecture 1 Question: Is the lowest eigenvalue for stable processes in an interval knowable (as in the case of Brownian motion)? Best known bounds irrelevant K.L. Chung s (and Taylor s) LIL also irrelevant Quick review of Lévy Processes (probably not necessary nor sufficient) Heat semigroup for Brownian motion and stable processes and their eigenvalues and eigenfunctions Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

3 Outline 1 Lecture 1 Question: Is the lowest eigenvalue for stable processes in an interval knowable (as in the case of Brownian motion)? Best known bounds irrelevant K.L. Chung s (and Taylor s) LIL also irrelevant Quick review of Lévy Processes (probably not necessary nor sufficient) Heat semigroup for Brownian motion and stable processes and their eigenvalues and eigenfunctions 2 Lecture 2 Eigenvalues and eigenfunctions as survival time (lifetimes) probabilities My long and twisted road to Finite Dimensional Distributions (F.D.D.) Isoperimetric and Isoperimetric type inequalities: Fixed volume, inradius, diameter, problems, questions, etc... Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

4 My Wants: To study properties of the functions: Φ m (x, D) = P x {B t1 D, B t2 D,..., B tm D} B t = Brownian motion (twice the speed) in R d, D R d open connected (referred to as domains ), x D, 0 < t 1 < t 2 < t m Same as studying Multiple Integrals: Φ m (x, D) = Π m j=1p (2) t j t j 1 (x j x j 1 )dx 1... dx m, D D x 0 = x and p (2) 1 t (y) = 2 /4t e y (4πt) d/2 More general, study for any times: Φ m (x, D) = Π m j=1p (2) t j (x j x j 1 )dx 1... dx m, 0 < t j < D D Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

5 But Why? Not Terribly Exciting, you may say. I agree! Question What is the smallest Dirichelt eigenvalue λ 1,α for the rotationally invariant stable processes of order 0 < α < 2 for the interval ( 1, 1)? Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

6 But Why? Not Terribly Exciting, you may say. I agree! Question What is the smallest Dirichelt eigenvalue λ 1,α for the rotationally invariant stable processes of order 0 < α < 2 for the interval ( 1, 1)? Note: I learned this from Davar Khoshnevisan about 8 years ago. Has been investigated by Investigated by: M.Kac-H. Pollar (1950). H. Widom (1961), J. Taylor (1967), B. Fristedt (1974), J. Bertoin (1996), Khosnevisan Z. Shi (1998). I don t know the answer and, to be perfectly honest, don t care. In the process of investigating this simple question we discovered that little is known about the fine spectral theoretic properties of stables. More Exciting: The techniques give new Theorem for the Laplacian (BM). Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

7 Best known bounds for λ 1,α R.B.and R. Latala and P. Méndez (2001) and R.B.and T. Kulczycki (2004) B(0, 1) = unit ball in R d. C α,d = Γ( d 2 ) 2 α Γ(1 + d d+α 2 )Γ( 2 ) 1 λ 1,α (B(0, 1)) 1 B(d/2, α/2 + 1) C α,d C α,d B(α/2, α + 1) For α = 1 (Cauchy processes), B(0, 1) = ( 1, 1) (as in Davar s question) 1 λ 1,1 3π Note: 3π 8 < π π 2 = 2 4 That is, eigenvalue for Cauchy is not the square root of the one for Brownian motion! Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

8 Variational Formula: For any D R d The Dirichlet form, (E, F), for stables processes, 0 < α < 2, in R d is: (f (x) f (y))(g(x) g(y)) E(f, g) = A α,d x y α+d dx dy R d R d and with F = { Rd f L 2 (R d f (x) f (y) 2 } ) : R x y α+d dx dy < d A α,d = αγ( d+α 2 ) 2 1 α π d/2 Γ(1 α 2 ) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

9 From this we have for any region D R d : u(x) u(y) λ 1,α (D) = inf {A 2 } α,d D D x y α+d dx dy + 2A α,d u(x) 2 k D (x)dx D where inf is over all u C 0 with u(y) 2 dy = 1. D K D (x) = D c dy x y α+d Reference: Potential Theory of subbordinate Killed Brownian motion R.Song-Z.Vondracek, Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

10 Eigenvalues and eigenfunctions enter into path properties of BM Theorem (Chungs s LIL. Set B t = sup 0 s t B s ) lim inf t ( log log t t ) 1/2 Bt = π, a.s. (1) 2 But, is π 2 really just our good-old-friend π 2 or is it something else? (1) comes from Borel Cantelli arguments and the small balls probability estimate: (See Bertoin, Lévy Processes page 224, Bañuelos Moore, Probabilistic behavior of harmonic functions, Chapter 4.) P 0 {B 1 < ε} e π2 4ε 2, ε 0 { } 1 { } P 0 {B1 < ε} = P 0 ε B t < 1 = P 0 B 1 < 1 = P 0 {τ ( 1,1) > 1 } ε 2 ɛ 2 τ ( 1,1) = inf{t > 0 : B t ( 1, 1)} = first exit time from the interval Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

11 As we shall see, P 0 { τ( 1,1) > t } e λ1t ϕ 1 (0) 1 1 ϕ 1 (y) dy, t, where λ 1 is the smallest eigenvalue for one half of the Laplacian in the interval ( 1, 1) with Dirichlet boundary conditions and ϕ 1 is the corresponding eigenfunction. That is, π 2 /4 and the sin function. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

12 As we shall see, P 0 { τ( 1,1) > t } e λ1t ϕ 1 (0) 1 1 ϕ 1 (y) dy, t, where λ 1 is the smallest eigenvalue for one half of the Laplacian in the interval ( 1, 1) with Dirichlet boundary conditions and ϕ 1 is the corresponding eigenfunction. That is, π 2 /4 and the sin function. For any 0 < α < 2, let X α t be the rotationally invariant stable process of order α. A similar statement holds for the small ball probabilities and there is Theorem (J. Taylor 1967) ( ) 1/α log log t lim inf Xt = (λ 1,α ) 1/α, t t a.s. (2) For several other occurrences of the eigenvalue in sample path behavior, see Erkan Nane: Higher order PDE s and iterated Processes and Iterated Brownian motion in bounded domains in R n Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

13 Lévy Processes Constructed by Paul Lévy in the 30 s (shortly after Wiener constructed Brownian motion). Other names: de Finetti, Kolmogorov, Khintchine, Itô. Rich stochastic processes, generalizing several basic processes in probability: Brownian motion, Poisson processes, stable processes, subordinators,... Regular enough for interesting analysis and applications. Their paths consist of continuous pieces intermingled with jump discontinuities at random times. Probabilistic and analytic properties studied by many. Many Developments in Recent Years: Applied: Queueing Theory, Math Finance, Control Theory, Porous Media... Pure: Investigations on the fine potential and spectral theoretic properties for subclasses of Lévy processes Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

14 Some References General Theory 1. J. Bertoin, Lévy Processes, Cambridge University Press, D. Applebaum. Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004 [3. K. Sato. Lévy Processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, 1999 Applications 1. B. Øksendal and A. Sulem. Applied Stochastic Control of Jump Diffusions, Springer, R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, 2004 Recent analytical, probabilistic, geometric, developments: Work of R. Bass, K. Burdzy, K. Bogdan, Z.Q. Chen, T. Kulczycki, R. Song, J-M. Wu, Z. Vondracek,... banuelos (for spectral theoretic stuff) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

15 Definition A Lévy Process is a stochastic process X = (X t ), t 0 with X has independent and stationary increments X 0 = 0 (with probability 1) X is stochastically continuous: For all ε > 0, lim P{ X t X s > ε} = 0 t s Note: Not the same as a.s. continuous paths. However, it gives cadlag paths: Right continuous with left limits. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

16 Stationary increments: 0 < s < t <, A R d Borel P{X t X s A} = P{X t s A} Independent increments: For any given sequence of ordered times 0 < t 1 < t 2 < < t m <, the random variables are independent. X t1 X 0, X t2 X t1,..., X tm X tm 1 The characteristic function of X t is ϕ t (ξ) = E ( e ) iξ Xt = e iξ x p t (dx) = (2π) d/2 p t (ξ) R d where p t is the distribution of X t. Notation (same with measures) 1 f (ξ) = e ix ξ 1 f (x)dx, f (x = e ix ξ f (ξ)dξ (2π) d/2 R (2π) d d/2 R d Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

17 The Lévy Khintchine Formula The characteristic function has the form ϕ t (ξ) = e tρ(ξ), where ρ(ξ) = ib ξ 1 ( ) 2 ξ Aξ + e iξ x 1 iξ x1 { x <1} (x) ν(dx) R d for some b R d, a non negative definite symmetric n n matrix A and a Borel measure ν on R d with ν{0} = 0 and ( ) min x 2, 1 ν(dx) < R d ρ(ξ) is called the symbol of the process or the characteristic exponent. The triple (b, A, ν) is called the characteristics of the process. Converse also true. Given such a triple we can construct a Lévy process. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

18 Examples 1. Standard Brownian motion: With (0, I, 0), I the identity matrix, X t = B t, Standard Brownian motion 2. Gaussian Processes, General Brownian motion : (0, A, 0), X t is generalized Brownian motion, mean zero, covariance E(X j s X i t ) = a ij min(s, t) X t has the normal distribution (assume here that det(a) > 0) ( 1 (2πt) d/2 det(a) exp 1 ) 2t x A 1 x 3. Brownian motion plus drift: With (b, A, 0) get Brownian motion with a drift: X t = bt + B t Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

19 4. Poisson Process: The Poisson Process X t = N λ (t) of intensity λ > 0 is a Lévy process with (0, 0, λδ 1 ) where δ 1 is the Dirac delta at 1. P{N λ (t) = m} = e λt (λt) m, m = 1, 2,... m! N λ (t) has continuous paths except for jumps of size 1 at the random times τ m = inf{t > 0 : N λ (t) = m} 5. Compound Poisson Process obtained by summing iid random variables up to a Poisson Process. 6. Relativistic Brownian motion According to quantum mechanics, a particle of rest mass m moving with momentum p has kinetic energy E(p) = m 2 c 4 + c 2 p 2 mc 2 where c is speed of light. Then ρ(p) = E(p) is the symbol of a Lévy process, called relativistic Brownian motion. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

20 7. The zeta process: Consider the Riemann zeta function ζ(z) = 1 n z = 1, z = x + iy C 1 p z n=1 p P Khintchine: For every fix x > 1, ( ) ζ(x + iy) ρ x (y) = log ζ(y) is the symbol of a Lévy process in fact, limits of Poissons. (See Applebaum page 39.) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

21 7. The zeta process: Consider the Riemann zeta function ζ(z) = 1 n z = 1, z = x + iy C 1 p z n=1 p P Khintchine: For every fix x > 1, ( ) ζ(x + iy) ρ x (y) = log ζ(y) is the symbol of a Lévy process in fact, limits of Poissons. (See Applebaum page 39.) Biane Pitman-Yor: Probability laws related to the Jacobi theta and Riemann Zeta functions and Brownian excursions, Bull. Amer math. Soc., M Yor: A note about Selberg s integrals with relation with the beta gamma algebra, Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

22 8. The rotationally invariant stable processes: These are self similar processes, denoted by X α t, in R d with symbol ρ(ξ) = ξ α, 0 < α 2. That is, ϕ t (ξ) = E ( e iξ X α t ) = e t ξ α α = 2 is Brownian motion. α = 1 is the Cauchy processes. α = 3/2 is called the Haltsmark distribution used to model gravitational fields of stars. (See V.M. Zolotarev (1986) One dimensional Stable Distributions.) Transition probabilities: P x {Xt α A} = A p α t (x y)dy, any Borel A R d pt α (x) = 1 (2π) d e iξ x e t ξ α dξ R d Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

23 p 2 t (x) = p 1 t (x) = 1 x 2 e 4t, α = 2, Brownian motion (4πt) d/2 C d t ( x 2 + t 2 ) d+1 2, α = 1, Cauchy Process For any a > 0, the two processes {η (at) ; t 0} and {a 1/α η t ; t 0}, have the same finite dimensional distributions (self-similarity). In the same way, the transition probabilities scale similarly to those for BM: p α t (x) = t d/α p α 1 (t 1/α x) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

24 Rodrigo Bañuelos (Purdue University) α/2 F.D.D 2/α June 25, 26, / Subordinators A subordinator is a one-dimensional Lévy process {T t } such that (i) T t 0 a.s. for each t > 0 (ii) T t1 T t2 a.s. whenever t 1 t 2 Theorem (Bertoin, p.73: Laplace transforms) E(e λtt ) = e tψ(λ), λ > 0, ψ(λ) = bλ + 0 ( 1 e λs ) ν(ds) b 0 and the Lévy measure satisfies ν(, 0) = 0 and 0 min(s, 1)ν(ds) <. ψ is called the Laplace exponent of the subordinator. Example (α/2 Stable subordinator): ψ(λ) = λ α/2, 0 < α < 2 gives the with b = 0 and α/2 ν(ds) = Γ(1 α/2) s 1 α/2 ds Example 2 (Relativistic stable subordinator): 0 < α < 2 and m > 0, Ψ(λ) = (λ + m 2/α ) α/2 m.

25 α/2 ν(ds) = 2/αs Γ(1 α/2) e m s 1 α/2 ds Example 3 (Gamma subordinator): Ψ(λ) = log(1 + λ). ν(ds) = e s s Many others: Geometric stable subordinators, iterated geometric stable subordinators, Bessel subordinators,... Theorem (Applebaum, p. 53) If X is an arbitrary Lévy process and T is a subordinator independent of X, then Z t = X Tt is a Lévy process. For any Borel A R d, p Zt (A) = 0 ds p Xs (A)p Tt (ds) When X t = B t Brownian motion, Z t is called subordinate Brownian motion. When T t is the α/2 stable subordinator and X is BM, Z is the α rotationally invariant stable process of Example 8. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

26 Lévy semigroup For the Lévy process {X (t); t 0}, define T t f (x) = E[f (X (t)) X 0 = x] = E 0 [f (X (t) + x)], f S(R d ). This is a Feller semigroup (takes C 0 (R d ) into itself). Setting p t (A) = P 0 {X t A} (the distribution of X t ) we see that (by Fourier inversion formula) T t f (x) = f (x + y)p t (dy) = p t f (x) = 1 R (2π) d e ix ξ e tρ(ξ) f (ξ)dξ d R d with generator Af (x) = T tf (x) 1 ( ) = lim E x [f (X (t)] f (x) t t=0 t 0 t 1 = (2π) d e ix ξ ρ(ξ)ˆf (ξ)dξ = a pseudo diff operator, in general R d Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

27 From the Lévy Khintchine formula (and properties of the Foruier transform), Af (x) = b i i f (x) + 1 a i,j i j f (x) 2 i=1 i,j + [ f (x + y) f (x) y f (x)χ { y <1}] ν(dy) Examples: Standard Brownian motion: Af (x) = 1 f (x) 2 Poisson Process of intensity λ: [ ] Af (x) = λ f (x + 1) f (x) Rotationally Invariant Stable Processes of order 0 < α < 2, Fractional Diffusions: Af (x) = ( ) α/2 f (x) f (y) f (x) = A α,d dy x y d+α Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

28 Many questions on the fine potential theoretic properties of solutions for ( ) α/2 have been studied by many authors in recent years. Examples: Regularity of heat kernels, general solutions of heat equation, Sobolev, log-sobolev inequalities, intrinsic ultracontractivity,... Boundary regularity of solutions, including boundary Harnack principle, gauge theorems, Fatou theorems, Martin boundary,... Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

29 Many questions on the fine potential theoretic properties of solutions for ( ) α/2 have been studied by many authors in recent years. Examples: Regularity of heat kernels, general solutions of heat equation, Sobolev, log-sobolev inequalities, intrinsic ultracontractivity,... Boundary regularity of solutions, including boundary Harnack principle, gauge theorems, Fatou theorems, Martin boundary,... (1) Potential theory of suborndinate BM, (2007), Song-Vondracek (2) Potential Theory for Lévy Processes, (2002) Bogdan Stozs Sztonyk. (3) Multidimensional symmetric stable processes, (1999), Z.Q. Chen (4) Also, have a look at the references given in the above papers and search for other recent papers of the authors already mentioned. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

30 Many questions on the fine potential theoretic properties of solutions for ( ) α/2 have been studied by many authors in recent years. Examples: Regularity of heat kernels, general solutions of heat equation, Sobolev, log-sobolev inequalities, intrinsic ultracontractivity,... Boundary regularity of solutions, including boundary Harnack principle, gauge theorems, Fatou theorems, Martin boundary,... (1) Potential theory of suborndinate BM, (2007), Song-Vondracek (2) Potential Theory for Lévy Processes, (2002) Bogdan Stozs Sztonyk. (3) Multidimensional symmetric stable processes, (1999), Z.Q. Chen (4) Also, have a look at the references given in the above papers and search for other recent papers of the authors already mentioned. I am interested in the fine spectral theoretic properties of these processes Estimates on eigenvalues, including the ground state λ 1,α and the spectral gap λ 2,α λ 1,α, Number of nodal domains (Courant Hilbert Nodal domain Theorem), geometric properties of eigenfunctions, including a Brascamp Lieb log concavity type theorem for ϕ 1,α,... Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

31 The semigroup for regions D R d From now on X t = X α t is rotationally invariant stable with symbol ρ(ξ) = ξ α, 0 < α 2. Let D be a bounded connected subset of R d. The first exit time of X α t from D is τ D = inf{t > 0 : X α t / D} Heat Semigroup in D is the self-adjoint operator ] Tt D f (x) = E x [f (Xt α ); τ D > t, f L 2 (D) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

32 The semigroup for regions D R d From now on X t = X α t is rotationally invariant stable with symbol ρ(ξ) = ξ α, 0 < α 2. Let D be a bounded connected subset of R d. The first exit time of X α t from D is τ D = inf{t > 0 : X α t / D} Heat Semigroup in D is the self-adjoint operator ] Tt D f (x) = E x [f (Xt α ); τ D > t, f L 2 (D) = D pt D,α (x, y)f (y)dy, Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

33 The semigroup for regions D R d From now on X t = X α t is rotationally invariant stable with symbol ρ(ξ) = ξ α, 0 < α 2. Let D be a bounded connected subset of R d. The first exit time of X α t from D is τ D = inf{t > 0 : X α t / D} Heat Semigroup in D is the self-adjoint operator ] Tt D f (x) = E x [f (Xt α ); τ D > t, f L 2 (D) = D pt D,α (x, y)f (y)dy, pt D,α (x, y) = pt α (x y) E x (τ D < t; pt τ α D (Xτ α D, y)). (K.L. Chung and Z. Zhao, From Brownian motion to Schrödinger equations Springer) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

34 p D,α t (x, y) is called the Heat Kernel for the stable process in D. ( ) pt D,α 1 (x, y) pt α (x y) p1 α (0)t d/α α = (2π) d dξ e ξ R d = t d/α ω d (2π) d α = t d/α ω dγ(d/α) (2π) d α 0 t d/α e s s ( n α 1) ds The general theory of heat semigroups (R. Bass Probabilistic Techniques in Analysis, B. Davies, Heat Semigroups ) gives an orthonormal basis of eigenfunctions {ϕ m,α } m=1 on L 2 (D) with eigenvalues {λ m,α } satisfying That is, 0 < λ 1,α < λ 2,α λ 3,α T D t ϕ m,α (x) = e λm,αt ϕ m,α (x), x D. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

35 p D,α t (x, y) = e λm,αt ϕ m,α (x)ϕ m,α (y) m=1 = e λ1,αt ϕ 1,α (x)ϕ 1,α (y) + e λm,αt ϕ m,α (x)ϕ m,α (y) m=2 Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

36 p D,α t (x, y) = e λm,αt ϕ m,α (x)ϕ m,α (y) m=1 = e λ1,αt ϕ 1,α (x)ϕ 1,α (y) + Theorem (From Intrinsic Ultracontractivity ) e (λ2,α λ1,α)t sup x,y D e λm,αt ϕ m,α (x)ϕ m,α (y) m=2 e λ1,αt pt D,α (x, y) ϕ 1,α (x)ϕ 1,α (y) 1 C(D, α)e (λ2,α λ1,α)t, t 1. For α = 2 this is valid for many domains but not all. For 0 < α < 2, valid for any bounded domain. See Smits, Spectral Gaps and Rates to Equilibrium Mich. Math. J. p. Vol 43 (1997), page 154. Studied in the 1990 s for Brownian motion by: Brian Davies, B. Simon, Bass-Burdzy, R.B. Burgess Davis, and countless others, especially in connections to conditioned Brownian motion and Schrödinger operators. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

37 p D,α t (x, y) = e λm,αt ϕ m,α (x)ϕ m,α (y) m=1 = e λ1,αt ϕ 1,α (x)ϕ 1,α (y) + Theorem (From Intrinsic Ultracontractivity ) e (λ2,α λ1,α)t sup x,y D e λm,αt ϕ m,α (x)ϕ m,α (y) m=2 e λ1,αt pt D,α (x, y) ϕ 1,α (x)ϕ 1,α (y) 1 C(D, α)e (λ2,α λ1,α)t, t 1. For α = 2 this is valid for many domains but not all. For 0 < α < 2, valid for any bounded domain. See Smits, Spectral Gaps and Rates to Equilibrium Mich. Math. J. p. Vol 43 (1997), page 154. Studied in the 1990 s for Brownian motion by: Brian Davies, B. Simon, Bass-Burdzy, R.B. Burgess Davis, and countless others, especially in connections to conditioned Brownian motion and Schrödinger operators. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

38 Apply the semigroup to the function f (x) = 1, x D Tt D f (x) = E x [1 D (Xt α ); τ D > t] = pt D,α (x, y)dy So that P x {τ D > t} = e tλm,α ϕ m,α (x) ϕ m,α (y)dy m=1 D = e tλ1,α ϕ 1,α (x) ϕ 1,α (y)dy + e tλm,α ϕ m,α (x) D m=2 D D ϕ m,α (y)dy Theorem (Implied by the Intrinsic Ultracontractivity result) lim t etλ1,α P x {τ D > t} = ϕ 1,α (x) ϕ 1,α (y)dy (3) and uniformly for x D. D 1 lim t t log P x{τ D > t} = λ 1,α, (4) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

39 The Long and Twisted Conclusion If I want to study the eigenfunction ϕ 1,α and λ 1,α and how these are affected by the geometry of the domain D, I should (better, must,... ) study the distribution of the exit time τ D of the process. That is, study as a function of D, x D, t > 0. P x {τ D > t} But: P x {τ D > t} = P z {X α s D; s, 0 < s t} = lim P z{x α m jt/m D, j = 1, 2,..., m} = lim m D pt/m α (x x 1) pt/m α (x m x m 1 )dx 1... dx m D p α t (x) = 1 (2π) d R d e iξ x e t ξ Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56 α dξ

40 In the same way (integrating against a delta function at y) pt D,α (x, y) = lim p α m t/m (x x 1) pt/m α (y x m 1)dx 1... dx m 1, D D Alternatively, for Brownian motion, if 0 = t 0 < t 1 <... < t m < t, then the conditional finite-dimensional distribution is given by Reference: P z0 {B t1 dx 1,..., B tm dx m B t = y}, p 2 t t m (z n, y) p 2 t (z 0, y) m pt 2 i t i 1 (z i, z i 1 ), i=1 I. Karatzas and E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlang, New York, 1991 page 395. Similar formula holds for stable bridges, Bertoin, page 226. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

41 Do we really need the stables? With X α t = B Tt with we have E(e λtt ) = e tλα/2, λ > 0, where p α t (x) = 0 p s (2) 1 (x) g α/2 (t, s)ds = 2 e x /4s g 0 (4πs) d/2 α/2 (t, s) ds g α/2 (t, s) = density of T t Note: α = 1/2, T t = inf{s > 0 : B s = t 2 } and g 1/2 (t, s) = ( ) t 2 s 3/2 e t2 /4s, π Bochner subordination Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

42 P x {Xt α 1 D,..., Xt α m D} m = pt α i t i 1 (x i x i 1 ) dx 1... dx n = = D 0... D i=1 0 ( D D ) m ps 2 i (x i x i 1 ) dx 1... dx n i=1 n g α/2 (t i t i 1, s i ) ds 1... ds m i= P x {B 2s1 D, B 2(s1+s 2) D,..., B 2(s1+s 2+ +s n) D} m g α/2 (t i t i 1, s i ) ds 1... ds m. i=1 Must study the function Φ m (x, D) = No order on t i. D D m pt 2 i (x i x i 1 ) dx 1... dx m, i=1 x 0 = x Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

43 For A R d, A =ball centered at the origin and same volume as A. χ A = χ A f : R d R, f (x) = (Compare this with) f (x) = Properties: 0 0 χ { f >t} (x)dt χ { f >t} (x)dt f (x) = f (y), x = y, f (x) f (y), x y {x : f (x) > t} = {x : f (x) > t} (same level sets) m{x : f (x) > λ} = m{x : f (x) > λ} Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

44 Theorem (Luttinger 1973) Let f 1,..., f m be nonnegative functions in R d and let f 1,..., f m be their symmetric decreasing rearrangement. Then for any x 0 D we have D m j=1 m f j (x j x j 1 )dx 1 dx m {D } m f 1 (x 1 ) D =ball center at zero and and same volume as D Theorem (Brascamp Lieb Lutinger (1975), (1977)) m j=2 f j (x j x j 1 )dx 1 dx m. Q j : R d [0, ) and 1 j r. a jk, 1 j r, 1 k m real numbers. (R d ) m j=1 r ( m ) Q j a jk z k dz1 dz m k=1 r Qj (R d ) m j=1 ( m ) a jk z k dz1 dz m Roots lie in inequalities of Hardy Littlewood Pólya Riesz F 1 (x 1 )H(x 2 x 1 )F 2 (x 2 )dx 1 dx 2 R d R d Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56 k=1

45 Theorem (Luttinger 1973) Let f 1,..., f m be nonnegative functions in R d and let f 1,..., f m be their symmetric decreasing rearrangement. Then for any x 0 D we have D m j=1 m f j (x j x j 1 )dx 1 dx m {D } m f 1 (x 1 ) D =ball center at zero and and same volume as D Theorem (Brascamp Lieb Lutinger (1975), (1977)) m j=2 f j (x j x j 1 )dx 1 dx m. Q j : R d [0, ) and 1 j r. a jk, 1 j r, 1 k m real numbers. (R d ) m j=1 r ( m ) Q j a jk z k dz1 dz m k=1 r Qj (R d ) m j=1 ( m ) a jk z k dz1 dz m Roots lie in inequalities of Hardy Littlewood Pólya Riesz F 1 (x 1 )H(x 2 x 1 )F 2 (x 2 )dx 1 dx 2 R d R d Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56 k=1

46 Theorem (R. B. Latala, Méndez, 2001 (d = 2), Méndez 2003, d 3) D R d convex of finite inradius r D and S infinite strip of inradius r D Let f 1,..., f m be nonnegative radially symmetric decreasing on R d. For any z 0 R d, Theorem S D D... S m f j (z j z j 1 ) dz 1 dz m j=1 f 1 (z 1 ) m f j (z j z j 1 ) dz 1 dz m. j=2 f j : R d [0, 1], h j : R d [0, ), 1 j m, radial symmetric decreasing. Then [1 [1 m m (1 f j (z j ))] h j (z j z j 1 )dz 0... dz m j=1 j=1 j=1 m m (1 fj (z j ))] h j (z j z j 1 )dz 0... dz m j=1 Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

47 Corollary (Isoperimetric Inequality for stable processes and eigenvalues) Φ m (x, D) Φ m (0, D ) P x {τ α D > t} P 0 {τ α D > t} sup E x (τd α ) E 0 (τd α ) x D Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

48 Corollary (Isoperimetric Inequality for stable processes and eigenvalues) λ 1,α (D ) λ 1,α (D) Φ m (x, D) Φ m (0, D ) P x {τ α D > t} P 0 {τ α D > t} sup E x (τd α ) E 0 (τd α ) x D The Faber-Krahn Theorem Cap α (A) Cap α (A ), (α-capacity version of a theorem of Polya Szego. Proved by Watanabe 1984, conjectured by Mattila 1990, Proved by Betsakos 2003, P. Méndez 2006) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

49 Corollary (Isoperimetric Inequality for stable processes and eigenvalues) λ 1,α (D ) λ 1,α (D) Φ m (x, D) Φ m (0, D ) P x {τ α D > t} P 0 {τ α D > t} sup E x (τd α ) E 0 (τd α ) x D The Faber-Krahn Theorem Cap α (A) Cap α (A ), (α-capacity version of a theorem of Polya Szego. Proved by Watanabe 1984, conjectured by Mattila 1990, Proved by Betsakos 2003, P. Méndez 2006) Corollary (Isoperimetric Inequality for the partition function ) Z α t (D) = e tλm,α(d) = m=1 D pt α,d (x, x)dx e tλm,α(d ) = Zt α (D ) p α,d t (x, x)dx D m=1 Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

50 Classical Isoperimetric Inequality Amongst all regions of equal volume the ball minimizes surface area. It follows from trace inequality and Theorem (M. Kac, Can one hear the shape of a drum? ) With α = 2, D =surface area of boundary of D, Z 2 t (D) C d t d/2 vol(d) C dt (d 1)/2 D + o(t (d 1)/2, t 0 The first term is trivial from P 2,D t (x, y) = 1 (4πt) x y 2 e d/2 4t P x {τ D > t B t = y} = free motion times Brownian bridge in D Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

51 A detour into Weyl s asymptotics lim t 0 tγ 0 e tλ dµ(λ) = A lim a a γ µ[0, a) = A Γ(γ + 1) Theorem (Weyl s Formula, α = 2. N D (λ) = #{j 1 : λ j λ}) N D (λ) C d vol(d)λ d/2, More difficult (and no probabilistic treatment exists): λ N D (λ) C d vol(d)λ d/2 C d D λ (d 1)/2 + o(λ (d 1)/2 ) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

52 A detour into Weyl s asymptotics lim t 0 tγ 0 e tλ dµ(λ) = A lim a a γ µ[0, a) = A Γ(γ + 1) Theorem (Weyl s Formula, α = 2. N D (λ) = #{j 1 : λ j λ}) N D (λ) C d vol(d)λ d/2, More difficult (and no probabilistic treatment exists): λ N D (λ) C d vol(d)λ d/2 C d D λ (d 1)/2 + o(λ (d 1)/2 ) Theorem (R.B. and T. Kulczycki (2007). 0 < α 2) pt α,d (x, x)dx C d,α t d/α vol(d) C dt (d 1)/α D + o(t (d 1)/α D as t 0. This gives Weyl s version for all 0 < α 2. The $$ Question: Is there an α version of the more general Weyl? Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

53 Back on main road: Fixing other parameters besides volume Question Amongst all convex domains D R d of inradius 1, which one has the largest exit time for Brownian motion? Also, lowest eigenvalue? Answer:The infinite strip: S = R d 1 ( 1, 1) Theorem (For D convex with inradius 1.) Φ m (x, D) Φ m (0, S), x D R.B. Méndez-Latala (2001), d = 2 and (2003), d 3. (Convexity is essential here!) Corollary (For D convex with inradius 1 and 0 < α 2.) P x {τ D > t} P 0 {τ S > t} = P 0 {τ ( 1,1) > t} (5) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

54 Back on main road: Fixing other parameters besides volume Question Amongst all convex domains D R d of inradius 1, which one has the largest exit time for Brownian motion? Also, lowest eigenvalue? Answer:The infinite strip: S = R d 1 ( 1, 1) Theorem (For D convex with inradius 1.) Φ m (x, D) Φ m (0, S), x D R.B. Méndez-Latala (2001), d = 2 and (2003), d 3. (Convexity is essential here!) Corollary (For D convex with inradius 1 and 0 < α 2.) P x {τ D > t} P 0 {τ S > t} = P 0 {τ ( 1,1) > t} (5) λ 1,α ( 1, 1) λ 1,α (D) (6) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

55 The Brascamp Lieb log concavity result Definition: F : R d R is said to be log-concave if or Examples: and log F (βx + (1 β)y) β log F (x) + (1 β) log F (y), x, y R d F (βx + (1 β)y) F (x) β F (y) 1 β F (x) = D R d is convex, are log concave. Theorem (Prékopa (1971)) 1 2 /4 e x (4π) d/2 F (x) = χ D (x), Convolutions of log-concave functions are log-concave. Corollary (D R d convex) For Brownian motion, the function Φ m (x, D) is log-concave. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

56 Corollary (Brascamp-Lieb (1979)) For any bounded convex domain D R d and for Brownian motion, the function P x {τ D > t} is log-concave and therefore so is the ground state eigenfunction ϕ 1,2 (x). In fact, this holds for the ground state eigenfunction for the Scrödinger operator + V where V : D [0, ) is convex. Note: Unfortunately we cannot conclude the same for 0 < α < 2. Why? Question (D R d convex, 0 < α < 2) Are the functions P x {τ D > t} and/or ϕ 1,α (x) log-convex? Known only for α = 1, D = ( 1, 1). In fact, in this case the functions are concave, just like for α = 2. Several other partial results are known for special doubly symmetric domains in the plane. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

57 Definition D R d be a convex symmetric relative to each coordinate axes. J any line segment in D parallel to the x 1 -axis intersecting D only at the two points. F : D R, is mid concave on J if it is concave on mid half of J. F mid concave along the x 1 axis if it is mid concave on every such segment contained in D parallel to the x 1 axis. Same for mid concavity along the x 2 -axis,. F mid concave on D if it is mid concave along each coordinate axes. Theorem (R.B.-Méndez-Kulczycki, 2006) Q R d a rectangle. Φ m (x, Q) = P x {X α t 1 Q,..., X α t m Q} is mid concave in Q for any 0 < α 2. In addition, if x = (x 1,..., x n ) Q, then But (recall, x 0 = x) Φ m (x) = 1 1 x i F (x) 0, if x i < 0, and 1 1 m i=1 x i F (x) 0, if x i > 0. p (2) t i (x i 1 x i ) dx 1... dx m, not concave on ( 1, 1) for all t i. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

58 Corollary Let Q = ( a 1, a 1 ) ( a 2, a 2 ) ( a d, a d ), 0 < a i < for all i = 1, 2,..., d, be a rectangle in R d. ϕ 1,α 0 < α < 2 is mid concave on Q. In addition, if x = (x 1,..., x n ) Q, then Theorem For n 1, set D(n) = x i ϕ α 1 (x) 0, if x i < 0, and { (x 1, x 2 ) R 2 : x 1 ( n, n), x 2 For n large enough, ϕ 1,2 is not mid concave on D(n). x i ϕ α 1 (x) 0, if x i > 0. ( 1 + x 1 n, 1 x ) } 1. n Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

59 Why is log-concavity important? Probabilistic reasons: Many people (C. Borell, S. Bobkov, C. Houdré, M. Ledoux) have studied isoperimetric inequalties Cheeger s inequalities, etc., for log concave measures. Here is a beautiful result: For any probability measure µ on R d define its spectral gap by Specg(µ) = inf f Lip(1) { R d f 2 dµ R d f 2 dµ : R d f dµ = 0 } Theorem (Bobkov 1996) Suppose dµ = f (x)dx, f log-concave and diam(µ) = diam(supp(f )) <. Then Specg(µ) c diam 2 (µ) Theorem (R. Smits Under same assumptions as Bobkov) Specg(µ) π 2 diam 2 (µ) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

60 Conjecture of Michiel van den Berg 1983 H = + V with Dirichlet conditions in the bounded convex domain D R d of finite diameter d D, V 0 is bounded and convex in D. We have 0 < λ 1 < λ 2 λ Conjecture: (Problem #44 in Yau s 1990 open problems in geometry ) gap(d, V ) = λ 2 λ 1 > 3π2 d 2 D with the lower bound approached when V = 0 and the domain becomes a thin rectangular box. False for nonconvex domains even with V = 0. ( b > a ) R a ( 4π 2 λ 2 (R) λ 1 (R) = b 2 + π2 a 2 b ) ( π 2 b 2 + π2 a 2 ) = 3π2 b 2, Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

61 Theorem ( I.M. Singer, B. Wang, S.T. Yau and S.S.T. Yau (1985)) gap(d, V ) π2 4d 2 D Several improvements by many, many, others. Full conjecture only known for D = ( 1, 1) (R. Lavin (1994) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

62 Theorem ( I.M. Singer, B. Wang, S.T. Yau and S.S.T. Yau (1985)) gap(d, V ) π2 4d 2 D Several improvements by many, many, others. Full conjecture only known for D = ( 1, 1) (R. Lavin (1994) If dµ = ϕ 2 1 (x) dx, then Specg(µ) = λ 2 λ 1. Since ϕ 2 1 is log concave, Smits gives: Corollary (R. Smits, 1997 best general bound up-to-date) gap(d, V ) π2 d 2 D Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

63 Take V = 0 for now. (A. Melas, 1998 UCLA Ph.D Thesis. The nodal domains for ϕ 2 look like in the picture) Domain D _ D D + φ 2 < 0 φ2 > 0 φ (D)= φ (D+ 2 1 ) λ (D)= λ (D 2 1 +) More General Conjecture: D convex of diameter d D. Let I = ( d D 2, d D 2 ), I + = (0, d D 2 ) P D D+ + t (z, z)dz D PD t (z, z)dz I + P I + t (z, z)dz I PI t (z, z)dz Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

64 Conjecture Equivalent to (In terms of Partition Function): Zt 2 (D + ) Zt 2 (D) = j=1 e tλ j (D + ) j=1 e tλ j (I + ) j=1 e tλ j (D) = Z t 2 (I + ) j=1 e tλ j (I ) Zt 2 (I ) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

65 Conjecture Equivalent to (In terms of Partition Function): Zt 2 (D + ) Zt 2 (D) = j=1 e tλ j (D + ) j=1 e tλ j (I + ) j=1 e tλ j (D) = Z t 2 (I + ) j=1 e tλ j (I ) Zt 2 (I ) L. Payne (here at Cornell) (1973) Nodal line either: a) or b) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

66 Symmetric in y, Convex in x a ẉ*. w D={(x, y): -a <y < a, -f(y) <x < f(y)} D+ R+ -b (x, 0).. b -a Theorem (R.B. Médez-Hernández, 2002) Suppose D, and D + and b are as in the picture. P (x,0) {τ D + > t} P (x,0) {τ D > t} P x{τ (0,b) > t} P x {τ ( b,b) > t} Same as: Φ m (D +, (x, 0)) Φ m (D, (x, 0)) Φ m((0, b), x) Φ m (( b, b), x) (F.D.D., again!) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

67 Theorem about Brownian motion in cylinders Two domains in 3-dimensions: Right Half and Bottom Half Diameter= Height Which Half has the smallest Dirichlet eigenvalue? D D- symmetric relative to y Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

68 Hot spots conjecture of Jeff Rauch (University of Michigan) 1973 The maximum (and the minimum) of the first non-constant Neumann eigenfunction for bounded convex domains are attained on the boundary and only on the boundary of the domain. Many partial results: R.B.-K.Burdzy (1999), D.Jerison-N.Darishavilli (2000), M. Pascu (2001), R. Bass K. Burdzy (2000), R.B.-M. Pang (2003), R.B. M.Pang-Pascu (2004), R.Atar K.Burdzy (2005) Counterexample: K. Burdzy-W. Werner (2000), K. Burdzy (2005) Believed to be true for any simply connected domain, conjectured to be true for any convex domain. Unknown even for an arbitrary triangle in the plane! Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

69 Hot spots Conjecture for conditioned Brownian motion Conjecture: The maximum and minimum for the first nonconstant eigenfunction for the semigroup of Brownian motion conditioned to remain forever in a convex domain are attained on the boundary and only on the boundary of the domain. That is, the function ϕ 2 /ϕ 1 attains its maximum and minimum on the boundary and only on the boundary of D. Theorem (R.B. Médez-Hernández, 2006) The conditional Hot Spots conjecture is true for symmetric domains in the plane as those shown above. The maximum (and minimum) of the function Ψ(z) = ϕ 2(z) ϕ 1 (z) is attained on the boundary and only on the boundary of D. Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

70 Hot spots Conjecture for conditioned Brownian motion Conjecture: The maximum and minimum for the first nonconstant eigenfunction for the semigroup of Brownian motion conditioned to remain forever in a convex domain are attained on the boundary and only on the boundary of the domain. That is, the function ϕ 2 /ϕ 1 attains its maximum and minimum on the boundary and only on the boundary of D. Theorem (R.B. Médez-Hernández, 2006) The conditional Hot Spots conjecture is true for symmetric domains in the plane as those shown above. The maximum (and minimum) of the function Ψ(z) = ϕ 2(z) ϕ 1 (z) is attained on the boundary and only on the boundary of D. Proof: Via finite dimensional distributions! Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

71 Theorem Let D be a bounded domain in R 2 which is symmetric and convex with respect to both axes. (i) If z 1 = (x, y 1 ) D +, z 2 = (x, y 2 ) D + and y 1 < y 2, then P z1 {τ D + > t} P z1 {τ D > t} < P z 2 {τ D+ > t} P z2 {τ D > t}, for any t > 0. In particular, the function Ψ(z, t) = P z{τ D + > t} P z {τ D > t}, for each t > 0 arbitrarily fixed, cannot have a maximum at an interior point of D +. (ii) If z 1 = (x 1, y) D + and z 2 = (x 2, y) D + with x 2 x 1, then for any t > 0. P z1 {τ D + > t} P z1 {τ D > t} P z 2 {τ D+ > t} P z2 {τ D > t}, Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

72 Corollary D R 2 as in Theorem ϕ 2 be such that its nodal line is the intersection of the x-axis with the domain. Without LOG, ϕ 2 > 0 in D + and ϕ 2 < 0 in D. Set Ψ = ϕ 2 /ϕ 1. (i) If z 1 = (x, y 1 ) D + and z 2 = (x, y 2 ) D + with y 1 < y 2, then Ψ(z 1 ) < Ψ(z 2 ). (ii) If z 1 = (x, y 1 ) D and z 2 = (x, y 2 ) D with y 2 < y 1, then Ψ(z 1 ) < Ψ(z 2 ). In particular, Ψ cannot attain a maximum nor a minimum in the interior of D. (iii) If z 1 = (x 1, y) D + and z 2 = (x 2, y) D + with x 2 < x 1, then Ψ(z 1 ) Ψ(z 2 ). (7) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

73 Corollary (Exact analogue of D. Jerison and N. Nadirashvili (2000) for classical hot spots ) Suppose D R 2 is a bounded domain with piecewise smooth boundary which is symmetric and convex with respect to both coordinate axes and that ϕ 2 is as in Theorem 1.2. Then strict inequality holds in (7) unless D is a rectangle. The maximum and minimum of Ψ on D are achieved at the points where the y-axis meets D and, except for the rectangle, at no other points. Nodal line either: a) or b) Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

74 To a hammer everything is a nail Can finite dimensional distributions make my morning coffee? Mow my lawn? don t know that,... Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56

75 To a hammer everything is a nail Can finite dimensional distributions make my morning coffee? Mow my lawn? don t know that,... yet! Rodrigo Bañuelos (Purdue University) F.D.D June 25, 26, / 56