On the backbone exponent
|
|
- Sharlene Snow
- 6 years ago
- Views:
Transcription
1 On the backbone exponent Christophe Garban Université Lyon 1 joint work with Jean-Christophe Mourrat (ENS Lyon) Cargèse, September 2016 C. Garban (univ. Lyon 1) On the backbone exponent 1 / 30
2 Critical percolation C. Garban (univ. Lyon 1) On the backbone exponent 2 / 30
3 SLE κ processes (Schramm) H γ t+s γ t g t g s Domain Markov Property g t+s (law) = g s g t Definition (Schramm, 1999) { t = t t k g t (law) = g (k) t k... g (2) t 2 g (1) t 1 Let κ R +. SLE κ is the Loewner chain driven by a Brownian motion κb t, i.e. 2 t g t (z) = g t (z) z H, t < T (z) κb t C. Garban (univ. Lyon 1) On the backbone exponent 3 / 30
4 Different phases of SLE κ C. Garban (univ. Lyon 1) On the backbone exponent 4 / 30
5 Zoology SLE 3 SLE 6
6 Critical exponents R Theorem (Lawler, Schramm & Werner 2002) α 1 (R) = R 5/48+o(1) 0 Corollary As η 0, large clusters converge to random compact sets in the plane of dimension d H = C. Garban (univ. Lyon 1) On the backbone exponent 6 / 30
7 Critical exponents One-arm event Polychromatic two-arm event Monochromatic two-arm event R R R 0 Expect R α1 α 1 = 5 48 LSW 2002 Expect R α2 α 2 = 1/4 LSW and Smirnov-Werner Expect R α 2 α 2 =?? (Backbone exponent) C. Garban (univ. Lyon 1) On the backbone exponent 7 / 30
8 Picturing the backbone In a finite box R R, this looks as follows: C. Garban (univ. Lyon 1) On the backbone exponent 8 / 30
9 Picturing the backbone In a finite box R R, this looks as follows: C. Garban (univ. Lyon 1) On the backbone exponent 8 / 30
10 How does one compute a critical exponent? Two main steps: I. Find an exponent for the scaling limit using SLE 6 P D ɛ ɛ α II. Connect this with the actual discrete model (using quasi-multiplicativity). P [ ε ω D ] ε α P [ 0 B(0, R) ] = R α+o(1) C. Garban (univ. Lyon 1) On the backbone exponent 9 / 30
11 Requires a radial version of SLE κ θ 0 g t θ t g t is the conformal map D \ γ([0, t]) D such that g t(0) = e t Radial version of Loewner s Theorem { t g t (z) = g t (z) gt(z)+eiλ t g 0 (z) = z g t(z) e iλ t z D, t < T (z) Consequence Radial SLE 6 corresponds to λ t 6B t. It follows that t θ t satisfies the SDE: dθ t = 6dB t + cot( θ t 2 )dt
12 How did LSW find α 1 = 5/48? e iθ Introduce h(θ, t) := P [ r(θ) e t]
13 How did LSW find α 1 = 5/48? e iθ Introduce h(θ, t) := P [ r(θ) e t] Q(θ)
14 How did LSW find α 1 = 5/48? e iθ Introduce h(θ, t) := P [ r(θ) e t] Q(θ) Claim The function h(θ, t) solves the following PDE: t h = 3 θh 2 + cot( θ 2 ) θh h(θ, t = 0) := 1 where: Dirichlet b.c. on {θ = 0, t > 0} Neumann b.c. on {θ = 2π, t > 0}
15 Short sketch of proof e iθ e iθ g u e iθ u Q(θ) h(θ, t) := P [ r(θ) e t]
16 Short sketch of proof e iθ e iθ g u e iθ u Q(θ) h(θ, t) := P [ r(θ) e t] = E [ P [ r(θ) e t] Fu ]
17 Short sketch of proof e iθ e iθ g u e iθ u Q(θ) h(θ, t) := P [ r(θ) e t] = E [ P [ r(θ) e t] ] Fu E [ P [ r(θ u ) e u t]] = E [ h(θ u, t u) ]
18 Short sketch of proof e iθ e iθ g u e iθ u Q(θ) h(θ, t) := P [ r(θ) e t] = E [ P [ r(θ) e t] Fu ] E [ P [ r(θ u ) e u t]] = E [ h(θ u, t u) ] 1 From the above argument, u h(θ u, t u) is a martingale. 2 u θ u evolves according to the SDE dθ u = 6dB u + cot( θu 2 )du. 3 From Itô s formula, get t h = 3 2 θ h + cot( θ 2 ) θh
19 Short sketch of proof e iθ e iθ g u e iθ u Q(θ) h(θ, t) := P [ r(θ) e t] = E [ P [ r(θ) e t] Fu ] E [ P [ r(θ u ) e u t]] = E [ h(θ u, t u) ] 1 From the above argument, u h(θ u, t u) is a martingale. 2 u θ u evolves according to the SDE dθ u = 6dB u + cot( θu 2 )du. 3 From Itô s formula, get t h = 3 2 θ h + cot( θ 2 ) θh Finally, some more work (which uses RSW on the discrete level) is still needed to justify the Neumann boundary condition.
20 Now that h(θ, t) := P [ r(θ) e t] is known to solve the PDE t h = 3 2 θ h + cot( θ 2 ) θh, it remains to prove: Theorem (Lawler, Schramm, Werner 2002) The one-arm exponent α 1 is the leading eigenvalue of L where L is the following differential operator on [0, 2π]: L = 3 2 θ + cot( θ 2 ) θ with Neumann b.c. at 0 and Dirichlet b.c. at 2π. This gives α 1 = 5 48, i.e. P [ 0 B R ] = R 5/48+o(1)
21 Now that h(θ, t) := P [ r(θ) e t] is known to solve the PDE t h = 3 2 θ h + cot( θ 2 ) θh, it remains to prove: Theorem (Lawler, Schramm, Werner 2002) The one-arm exponent α 1 is the leading eigenvalue of L where L is the following differential operator on [0, 2π]: L = 3 2 θ + cot( θ 2 ) θ with Neumann b.c. at 0 and Dirichlet b.c. at 2π. This gives α 1 = 5 48, i.e. P [ 0 B R ] = R 5/48+o(1) By a quite elaborate use of the maximum principle, they prove that 1 C e α1t (sin(θ/4)) 1/3 h(θ, t) Ce α1t (sin(θ/4)) 1/3
22 Now that h(θ, t) := P [ r(θ) e t] is known to solve the PDE t h = 3 2 θ h + cot( θ 2 ) θh, it remains to prove: Theorem (Lawler, Schramm, Werner 2002) The one-arm exponent α 1 is the leading eigenvalue of L where L is the following differential operator on [0, 2π]: L = 3 2 θ + cot( θ 2 ) θ with Neumann b.c. at 0 and Dirichlet b.c. at 2π. This gives α 1 = 5 48, i.e. P [ 0 B R ] = R 5/48+o(1) By a quite elaborate use of the maximum principle, they prove that 1 C e α1t (sin(θ/4)) 1/3 h(θ, t) Ce α1t (sin(θ/4)) 1/3 This approach by LSW is in some sense of purely PDE nature. There is also a tangential approach (in the end very similar) which relies directly on an exploration procedure (SLE 6 ).
23 The one-arm exponent from a direct exploration procedure e iθ θ t 2π θ t In the bulk, dθ t = 6dB t + cot( θt 2 )dt From this exploration path perspective, want to consider: where T := inf{u > 0, θ u = 0}. h(θ, t) := P θ [ T > t ], If one carefully characterizes the reflexion on 2π as being Neumann, then indeed h satisfies the same PDE as h, with same initialisation h(t = 0) = h(t = 0) 1 so that h = h
24 Polychromatic two-arm exponent α 2 = 1/4 Theorem (LSW and Smirnov, Werner 2002) The (polychromatic) two-arm exponent α 2 is the leading eigenvalue of the same! differential operator on [0, 2π]: L = 3 2 θ + cot( θ 2 ) θ with different b.c: Dirichlet at 0 and 2π. This gives α 2 = 1 4. e iθ θ t 2π θ t Here, the situation is somewhat easier as the diffusion does not touch the Bdy.
25 Backbone exponent α 2 Warm-up remark: it is a priori not clear at all that SLE 6 will help here! C. Garban (univ. Lyon 1) On the backbone exponent 16 / 30
26 Backbone exponent α 2 Warm-up remark: it is a priori not clear at all that SLE 6 will help here! Yet, the following very nice approach is sketched in the appendix of LSW02: Keep track of TWO angles: α, β (and call γ := 2π α β). eiβ e iγ e iα C. Garban (univ. Lyon 1) On the backbone exponent 16 / 30
27 Backbone exponent α 2 Warm-up remark: it is a priori not clear at all that SLE 6 will help here! Yet, the following very nice approach is sketched in the appendix of LSW02: Keep track of TWO angles: α, β (and call γ := 2π α β). eiβ e iγ - Let Q(α, β) be the set of points which have two dispoint open paths to the γ-grey arc. N.B. these path may use the β-black arc! e iα C. Garban (univ. Lyon 1) On the backbone exponent 16 / 30
28 Backbone exponent α 2 Warm-up remark: it is a priori not clear at all that SLE 6 will help here! Yet, the following very nice approach is sketched in the appendix of LSW02: Keep track of TWO angles: α, β (and call γ := 2π α β). eiβ e iγ - Let Q(α, β) be the set of points which have two dispoint open paths to the γ-grey arc. N.B. these path may use the β-black arc! e iα - Introduce the quantity G(α, β, t) := P [ r(α, β) e t] C. Garban (univ. Lyon 1) On the backbone exponent 16 / 30
29 eiβ e iγ G(α, β, t) := P [ r(α, β) e t] Claim: the function G(α, β, t) solves the following parabolic PDE on {α + β 2π}. 2π e iα t G = ( 3 α β 2 + cot( α 2 ) α + cot( β 2 ) β) G := L[G] With boundary conditions: 1 Neumann on bottom 2 Constant on left, i.e. G t (0, β) C t β 3 Dirichlet on {(0, 2π)} {(2π, 0)} diagonal. 0 Neumann α 2π C. Garban (univ. Lyon 1) On the backbone exponent 17 / 30
30 In the appendix of LSW02, it is stated that the Monochromatic two-arms exponent α 2 is the leading eigenvalue of the differential operator on the triangle: L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β with the above b.c.s (Neumann+Dirichlet+"Constant").
31 In the appendix of LSW02, it is stated that the Monochromatic two-arms exponent α 2 is the leading eigenvalue of the differential operator on the triangle: L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β with the above b.c.s (Neumann+Dirichlet+"Constant"). A meta-statement All examples of known critical exponents in percolation are obtained as leading eigenvalues of some differential operator. Examples 1 α 1, α 2, α 2. 2 Plane exponents α j, j 3 (through derivative exponents which also require a leading eigenvalue analysis). 3 Brownian intersection exponents such as ξ(3, 3) etc...
32 In the appendix of LSW02, it is stated that the Monochromatic two-arms exponent α 2 is the leading eigenvalue of the differential operator on the triangle: L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β with the above b.c.s (Neumann+Dirichlet+"Constant"). A meta-statement All examples of known critical exponents in percolation are obtained as leading eigenvalues of some differential operator. Examples 1 α 1, α 2, α 2. 2 Plane exponents α j, j 3 (through derivative exponents which also require a leading eigenvalue analysis). 3 Brownian intersection exponents such as ξ(3, 3) etc... counter-examples 1 α 5, α2 H, αh 3 (integer valued) 2 α H j 3 α(2, λ) obtained via analytic continuation
33 Stochastic process associated to the backbone As in the one-arm case, it is natural to consider (if well-defined) the stochastic process X t associated to the generator 2π L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β β With previous boundary conditions: 1 Neumann on bottom 2 Constant on left. 3 Dirichlet on {(0, 2π)} {(2π, 0)} diagonal. 0 Neumann α 2π C. Garban (univ. Lyon 1) On the backbone exponent 19 / 30
34 Stochastic process associated to the backbone As in the one-arm case, it is natural to consider (if well-defined) the stochastic process X t associated to the generator 2π L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β β With previous boundary conditions: 1 Neumann on bottom 2 Constant on left. 3 Dirichlet on {(0, 2π)} {(2π, 0)} diagonal. 0 Neumann α 2π C. Garban (univ. Lyon 1) On the backbone exponent 19 / 30
35 Stochastic process associated to the backbone As in the one-arm case, it is natural to consider (if well-defined) the stochastic process X t associated to the generator 2π L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β β With previous boundary conditions: 1 Neumann on bottom 2 Constant on left. 3 Dirichlet on {(0, 2π)} {(2π, 0)} diagonal. 0 Neumann α 2π This stochastic process is reminiscent of a striking algorithm to detect a monochromatic two-arm event! A closer inspection of this stochastic process leads to a few mathematical issues on the initial PDE approach. C. Garban (univ. Lyon 1) On the backbone exponent 19 / 30
36 Mathematical issues raised L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β 1 A.s., the process X t never reaches the diagonal {γ = 2π} (except both corners). What does it mean to require Dirichlet boundary conditions there? C. Garban (univ. Lyon 1) On the backbone exponent 20 / 30
37 Mathematical issues raised L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β 1 A.s., the process X t never reaches the diagonal {γ = 2π} (except both corners). What does it mean to require Dirichlet boundary conditions there? 2 The process is highly non-reversible. There are no measure µ and no domain for which the differential operator L is self-adjoint. How can one justify the existence of a (leading) positive eigenfunction? C. Garban (univ. Lyon 1) On the backbone exponent 20 / 30
38 Mathematical issues raised L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β 1 A.s., the process X t never reaches the diagonal {γ = 2π} (except both corners). What does it mean to require Dirichlet boundary conditions there? 2 The process is highly non-reversible. There are no measure µ and no domain for which the differential operator L is self-adjoint. How can one justify the existence of a (leading) positive eigenfunction? 3 (facultative) is the process X t well-defined? (Rather unusual b.c.s). C. Garban (univ. Lyon 1) On the backbone exponent 20 / 30
39 Mathematical issues raised L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β 1 A.s., the process X t never reaches the diagonal {γ = 2π} (except both corners). What does it mean to require Dirichlet boundary conditions there? 2 The process is highly non-reversible. There are no measure µ and no domain for which the differential operator L is self-adjoint. How can one justify the existence of a (leading) positive eigenfunction? 3 (facultative) is the process X t well-defined? (Rather unusual b.c.s). 4 Finally, assuming the process X t is well-defined, let Ĝ(α, β, t) := P α,β [ Tsurvival > t ] then Ĝ satisfies the same PDE also starting from Ĝ(t = 0) 1 (so that Ĝ G). t Ĝ = ( 3 2 α β + cot( α 2 ) α + cot( β 2 ) β)ĝ = L Ĝ C. Garban (univ. Lyon 1) On the backbone exponent 20 / 30
40 Mathematical issues raised L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β 1 A.s., the process X t never reaches the diagonal {γ = 2π} (except both corners). What does it mean to require Dirichlet boundary conditions there? 2 The process is highly non-reversible. There are no measure µ and no domain for which the differential operator L is self-adjoint. How can one justify the existence of a (leading) positive eigenfunction? 3 (facultative) is the process X t well-defined? (Rather unusual b.c.s). 4 Finally, assuming the process X t is well-defined, let Ĝ(α, β, t) := P α,β [ Tsurvival > t ] then Ĝ satisfies the same PDE also starting from Ĝ(t = 0) 1 (so that Ĝ G). t Ĝ = ( 3 2 α β + cot( α 2 ) α + cot( β 2 ) β)ĝ = L Ĝ C. Garban (univ. Lyon 1) On the backbone exponent 20 / 30
41 [ Ĝ(α, β, t) := P α,β Tsurvival > t ] = G(α, β, t) Two possible strategies: 2π 2π β OR β 0 α 2π 0 α 2π
42 [ Ĝ(α, β, t) := P α,β Tsurvival > t ] = G(α, β, t) Two possible strategies: 2π 2π β OR β 0 α 2π 0 α 2π The second strategy corresponds to the polychromatic two-arm event. As such, we would have e α 2 t G( π 2, π, t) e α2t 2
43 [ Ĝ(α, β, t) := P α,β Tsurvival > t ] = G(α, β, t) Two possible strategies: 2π 2π β OR β 0 α 2π 0 α 2π The second strategy corresponds to the polychromatic two-arm event. As such, we would have e α 2 t G( π 2, π, t) e α2t 2 This is in conflict with Theorem (Beffara, Nolin 2009) α 2 > α 2
44 Where does issue 4 come from? eiβ e iα e iγ - Recall G(α, β, t) := P [ r(α, β) e t] - Let T be the stopping time of touching Triangle. - It is tempting to write as in the one-arm case G(α, β, t) = E α,β [ G(XT, t T ) ] It thus appears that the previous heuristics is not true here: stopping time T DOES have a non-trivial effect! C. Garban (univ. Lyon 1) On the backbone exponent 22 / 30
45 Where does issue 4 come from? eiβ e iα e iγ - Recall G(α, β, t) := P [ r(α, β) e t] - Let T be the stopping time of touching Triangle. - It is tempting to write as in the one-arm case G(α, β, t) = E α,β [ G(XT, t T ) ] It thus appears that the previous heuristics is not true here: stopping time T DOES have a non-trivial effect! t G [ 3 2 α β + cot( α 2 ) α + cot( β 2 ) β] G C. Garban (univ. Lyon 1) On the backbone exponent 22 / 30
46 How to proceed to deal with these difficulties? A) Ignore them! and rely on Camia/Newman or Schramm/Smirnov + Beffara/Nolin. C. Garban (univ. Lyon 1) On the backbone exponent 23 / 30
47 How to proceed to deal with these difficulties? A) Ignore them! and rely on Camia/Newman or Schramm/Smirnov + Beffara/Nolin. Still, having a direct SLE 6 proof is desirable C. Garban (univ. Lyon 1) On the backbone exponent 23 / 30
48 How to proceed to deal with these difficulties? A) Ignore them! and rely on Camia/Newman or Schramm/Smirnov + Beffara/Nolin. Still, having a direct SLE 6 proof is desirable B) Look for a direct PDE proof C) Look for a proof based on the suggested exploration procedure (process X t ) C. Garban (univ. Lyon 1) On the backbone exponent 23 / 30
49 Let s investigate the PDE approach Issues: The operator L is not self-adjoint Existence of a positive eigenfunction? No Hilbertian structure... Which PDE? t G LG. Boundary conditions are rather singular Which domain for this operator? C. Garban (univ. Lyon 1) On the backbone exponent 24 / 30
50 Let s investigate the PDE approach Issues: The operator L is not self-adjoint Existence of a positive eigenfunction? No Hilbertian structure... Which PDE? t G LG. Boundary conditions are rather singular Which domain for this operator? Viscosity solutions? Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators. Berestycki, Capuzzo Dolcetta, Porretta, Rossi, Due to the possible loss of regularity, as well as of boundary conditions, which is caused by degeneracy of ellipticity, the appropriate framework to deal with this problem is, even in the linear case, that of viscosity solutions. C. Garban (univ. Lyon 1) On the backbone exponent 24 / 30
51 Instead, look for a proof based on the suggested exploration procedure This goes into three steps: 1 Prove the convergence of the exploration procedure towards a well-defined stochastic process X t 2 Obtain a candidate (λ) for the exponent from a sub-additivity property. 3 Identify λ α 2 C. Garban (univ. Lyon 1) On the backbone exponent 25 / 30
52 Step 1: convergence of the exploration procedure 2π L = 3 2 α β + cot( α 2 ) α + cot( β 2 ) β β 0 α 2π dα t dβ t = κdb t + cot( αt 2 )dt = κdb t + cot( βt 2 )dt Plus Neumann on β t Plus Constant bdy condition C. Garban (univ. Lyon 1) On the backbone exponent 26 / 30
53 Step 2: a sub-additivity property. For f in the domain of L, we expect P t+s f (x) P t f (x)p s f (x) Note that if needed, one may rely here on Hörmander s theorem C. Garban (univ. Lyon 1) On the backbone exponent 27 / 30
54 Step 2: a sub-additivity property. For f in the domain of L, we expect P t+s f (x) P t f (x)p s f (x) Note that if needed, one may rely here on Hörmander s theorem There are more directs ways to extract an exponent. For example, introduce p δ t := P (0,0) [ u [t, t + δ], so that Xu = (0, 0) ] Easy observation: p 2δ t+s p δ t p δ s C. Garban (univ. Lyon 1) On the backbone exponent 27 / 30
55 Step 2: a sub-additivity property. For f in the domain of L, we expect P t+s f (x) P t f (x)p s f (x) Note that if needed, one may rely here on Hörmander s theorem There are more directs ways to extract an exponent. For example, introduce p δ t := P (0,0) [ u [t, t + δ], so that Xu = (0, 0) ] Easy observation: p 2δ t+s p δ t p δ s lemma There is a constant C > 0 so that for any u 0, p 2δ u Cp δ u+δ This gives for any t, s > 0, C p δ t+s+δ pδ t p δ s which implies the existence of λ := lim t log pδ t t C. Garban (univ. Lyon 1) On the backbone exponent 27 / 30
56 Step 3: identifying λ α 2 Let A t be the backbone event from e t to D. Our goal is to show that P [ A t ] = e λt+o(t) The lower bound is clear: P [ A t ] e λ(t 10)+o(t)
57 Step 3: identifying λ α 2 Let A t be the backbone event from e t to D. Our goal is to show that P [ A t ] = e λt+o(t) The lower bound is clear: P [ A t ] e λ(t 10)+o(t) For the upper bound: tempting to decompose as follows: P [ ] A t = P [ A t, Grey angle of X t [2 k, 2 k+1 ] ] k=1
58 Step 3: identifying λ α 2 Let A t be the backbone event from e t to D. Our goal is to show that P [ A t ] = e λt+o(t) The lower bound is clear: P [ A t ] e λ(t 10)+o(t) For the upper bound: tempting to decompose as follows: P [ ] A t = P [ A t, Grey angle of X t [2 k, 2 k+1 ] ] k=1 But, this is not at all suitable to analysis!
59 Step 3: identifying λ α 2 Let A t be the backbone event from e t to D. Our goal is to show that P [ A t ] = e λt+o(t) The lower bound is clear: P [ A t ] e λ(t 10)+o(t) For the upper bound: tempting to decompose as follows: P [ ] A t = P [ A t, Grey angle of X t [2 k, 2 k+1 ] ] k=1 But, this is not at all suitable to analysis! Instead, decompose as follows: P [ A t ] = P [ A t, T [t (k + 1), t k] ] k=1 k k p δ=1 t k P (0,0) [ { T > k} At ] e λ(t k)+o(t k)
60 Step 3: identifying λ α 2 Let A t be the backbone event from e t to D. Our goal is to show that P [ A t ] = e λt+o(t) The lower bound is clear: P [ A t ] e λ(t 10)+o(t) For the upper bound: tempting to decompose as follows: P [ ] A t = P [ A t, Grey angle of X t [2 k, 2 k+1 ] ] k=1 But, this is not at all suitable to analysis! Instead, decompose as follows: P [ A t ] = P [ A t, T [t (k + 1), t k] ] k=1 k k p δ=1 t k P (0,0) [ { T > k} At ] e λ(t k)+o(t k) e α3 k
61 Step 3: identifying λ α 2 Let A t be the backbone event from e t to D. Our goal is to show that P [ A t ] = e λt+o(t) The lower bound is clear: P [ A t ] e λ(t 10)+o(t) For the upper bound: tempting to decompose as follows: P [ ] A t = P [ A t, Grey angle of X t [2 k, 2 k+1 ] ] k=1 But, this is not at all suitable to analysis! Instead, decompose as follows: P [ A t ] = P [ A t, T [t (k + 1), t k] ] k=1 k p δ=1 t k P (0,0) [ { T > k} At ] Case study: 1 What if λ > α 3? 2 λ = α 3? k e λ(t k)+o(t k) e α3 k
62 Step 3: identifying λ α 2 Let A t be the backbone event from e t to D. Our goal is to show that P [ A t ] = e λt+o(t) The lower bound is clear: P [ A t ] e λ(t 10)+o(t) For the upper bound: tempting to decompose as follows: P [ ] A t = P [ A t, Grey angle of X t [2 k, 2 k+1 ] ] k=1 But, this is not at all suitable to analysis! Instead, decompose as follows: P [ A t ] = P [ A t, T [t (k + 1), t k] ] k=1 k k p δ=1 t k P (0,0) [ { T > k} At ] e λ(t k)+o(t k) e α3 k Case study: 1 What if λ > α 3? 2 λ = α 3? 3 Therefore λ < α 3 and we get λ = α 2
63 Backbone exponent for κ 6? { dα t dβ t = κdb t + κ 4 2 cot( αt = κdb t + 1 cot( βt 2 2 )dt 6 κ )dt + 2 cot( αt 2 )dt Which gives the following PDE: t G = L κ [G] with L κ := κ 2 2 α β + κ 4 2 One can check L 6 = L. cot( α ( 2 ) α + cot( β 2 ) + 6 κ cot( α ) 2 2 ) β The vector field is very different when κ 6! C. Garban (univ. Lyon 1) On the backbone exponent 29 / 30
64 Conclusion SLE have provided a way to derive critical exponents rigorously (as opposed to Conformal Field Theory CFT.) The earlier meta-statement that most exponents are obtained as leading eigenvalues... may still hold if one wants to include the backbone exponent, but then what we mean by leading eigenvalue needs to be generalized accordingly (Viscosity solutions?... ) NUMERICS: viscosity solutions? Use this strengthened understanding of α 2 to compute a very good approximate value of α 2. This sheds some light on why the backbone exponent α 2 from the others and why it looks harder to compute singles out C. Garban (univ. Lyon 1) On the backbone exponent 30 / 30
65 Conclusion SLE have provided a way to derive critical exponents rigorously (as opposed to Conformal Field Theory CFT.) The earlier meta-statement that most exponents are obtained as leading eigenvalues... may still hold if one wants to include the backbone exponent, but then what we mean by leading eigenvalue needs to be generalized accordingly (Viscosity solutions?... ) NUMERICS: viscosity solutions? Use this strengthened understanding of α 2 to compute a very good approximate value of α 2. This sheds some light on why the backbone exponent α 2 from the others and why it looks harder to compute singles out Conjecture (Beffara, Nolin) α 2 = α 1 + α 2 C. Garban (univ. Lyon 1) On the backbone exponent 30 / 30
GEOMETRIC AND FRACTAL PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE)
GEOMETRIC AND FRACTAL PROPERTIES OF SCHRAMM-LOEWNER EVOLUTION (SLE) Triennial Ahlfors-Bers Colloquium Gregory F. Lawler Department of Mathematics Department of Statistics University of Chicago 5734 S.
More informationPlan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM) and percolation (c) Loop-erased random walk (LERW) (d
Percolation, Brownian Motion and SLE Oded Schramm The Weizmann Institute of Science and Microsoft Research Plan 1. This talk: (a) Percolation and criticality (b) Conformal invariance Brownian motion (BM)
More informationGradient Percolation and related questions
and related questions Pierre Nolin (École Normale Supérieure & Université Paris-Sud) PhD Thesis supervised by W. Werner July 16th 2007 Introduction is a model of inhomogeneous percolation introduced by
More informationTowards conformal invariance of 2-dim lattice models
Towards conformal invariance of 2-dim lattice models Stanislav Smirnov Université de Genève September 4, 2006 2-dim lattice models of natural phenomena: Ising, percolation, self-avoiding polymers,... Realistic
More informationAn Introduction to Percolation
An Introduction to Percolation Michael J. Kozdron University of Regina http://stat.math.uregina.ca/ kozdron/ Colloquium Department of Mathematics & Statistics September 28, 2007 Abstract Percolation was
More informationImaginary Geometry and the Gaussian Free Field
Imaginary Geometry and the Gaussian Free Field Jason Miller and Scott Sheffield Massachusetts Institute of Technology May 23, 2013 Jason Miller and Scott Sheffield (MIT) Imaginary Geometry and the Gaussian
More informationThe near-critical planar Ising Random Cluster model
The near-critical planar Ising Random Cluster model Gábor Pete http://www.math.bme.hu/ gabor Joint work with and Hugo Duminil-Copin (Université de Genève) Christophe Garban (ENS Lyon, CNRS) arxiv:1111.0144
More informationPlan 1. Brownian motion 2. Loop-erased random walk 3. SLE 4. Percolation 5. Uniform spanning trees (UST) 6. UST Peano curve 7. Self-avoiding walk 1
Conformally invariant scaling limits: Brownian motion, percolation, and loop-erased random walk Oded Schramm Microsoft Research Weizmann Institute of Science (on leave) Plan 1. Brownian motion 2. Loop-erased
More informationAn Introduction to the Schramm-Loewner Evolution Tom Alberts C o u(r)a n (t) Institute. March 14, 2008
An Introduction to the Schramm-Loewner Evolution Tom Alberts C o u(r)a n (t) Institute March 14, 2008 Outline Lattice models whose time evolution is not Markovian. Conformal invariance of their scaling
More informationStochastic Loewner Evolution: another way of thinking about Conformal Field Theory
Stochastic Loewner Evolution: another way of thinking about Conformal Field Theory John Cardy University of Oxford October 2005 Centre for Mathematical Physics, Hamburg Outline recall some facts about
More informationCRITICAL PERCOLATION AND CONFORMAL INVARIANCE
CRITICAL PERCOLATION AND CONFORMAL INVARIANCE STANISLAV SMIRNOV Royal Institute of Technology, Department of Mathematics, Stockholm, S10044, Sweden E-mail : stas@math.kth.se Many 2D critical lattice models
More informationConvergence of loop erased random walks on a planar graph to a chordal SLE(2) curve
Convergence of loop erased random walks on a planar graph to a chordal SLE(2) curve Hiroyuki Suzuki Chuo University International Workshop on Conformal Dynamics and Loewner Theory 2014/11/23 1 / 27 Introduction(1)
More information2D Critical Systems, Fractals and SLE
2D Critical Systems, Fractals and SLE Meik Hellmund Leipzig University, Institute of Mathematics Statistical models, clusters, loops Fractal dimensions Stochastic/Schramm Loewner evolution (SLE) Outlook
More informationCritical percolation under conservative dynamics
Critical percolation under conservative dynamics Christophe Garban ENS Lyon and CNRS Joint work with and Erik Broman (Uppsala University) Jeffrey E. Steif (Chalmers University, Göteborg) PASI conference,
More informationRandom planar curves Schramm-Loewner Evolution and Conformal Field Theory
Random planar curves Schramm-Loewner Evolution and Conformal Field Theory John Cardy University of Oxford WIMCS Annual Meeting December 2009 Introduction - lattice models in two dimensions and random planar
More informationErdős-Renyi random graphs basics
Erdős-Renyi random graphs basics Nathanaël Berestycki U.B.C. - class on percolation We take n vertices and a number p = p(n) with < p < 1. Let G(n, p(n)) be the graph such that there is an edge between
More informationFractal Properties of the Schramm-Loewner Evolution (SLE)
Fractal Properties of the Schramm-Loewner Evolution (SLE) Gregory F. Lawler Department of Mathematics University of Chicago 5734 S. University Ave. Chicago, IL 60637 lawler@math.uchicago.edu December 12,
More informationSLE and nodal lines. Eugene Bogomolny, Charles Schmit & Rémy Dubertrand. LPTMS, Orsay, France. SLE and nodal lines p.
SLE and nodal lines p. SLE and nodal lines Eugene Bogomolny, Charles Schmit & Rémy Dubertrand LPTMS, Orsay, France SLE and nodal lines p. Outline Motivations Loewner Equation Stochastic Loewner Equation
More informationUNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9947(XX)0000-0 UNIQUENESS OF POSITIVE SOLUTION TO SOME COUPLED COOPERATIVE VARIATIONAL ELLIPTIC SYSTEMS YULIAN
More informationLOGARITHMIC COEFFICIENTS AND GENERALIZED MULTIFRACTALITY OF WHOLE-PLANE SLE
LOGARITHMIC COEFFICIENTS AND GENERALIZED MULTIFRACTALITY OF WHOLE-PLANE SLE BERTRAND DUPLANTIER 1), XUAN HIEU HO ), THANH BINH LE ), AND MICHEL ZINSMEISTER ) Abstract. It has been shown that for f an instance
More informationarxiv:math/ v1 [math.pr] 21 Dec 2001
Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk arxiv:math/0112246v1 [math.pr] 21 Dec 2001 Tom Kennedy Departments of Mathematics and Physics University of Arizona, Tucson, AZ, 85721
More informationIntroduction to Random Diffusions
Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales
More informationAdvanced Topics in Probability
Advanced Topics in Probability Conformal Methods in 2D Statistical Mechanics Pierre Nolin Different lattices discrete models on lattices, in two dimensions: square lattice Z 2 : simplest one triangular
More informationNumerical simulation of random curves - lecture 1
Numerical simulation of random curves - lecture 1 Department of Mathematics, University of Arizona Supported by NSF grant DMS-0501168 http://www.math.arizona.edu/ e tgk 2008 Enrage Topical School ON GROWTH
More informationMathematical Research Letters 8, (2001) THE DIMENSION OF THE PLANAR BROWNIAN FRONTIER IS 4/3
Mathematical Research Letters 8, 401 411 (2001) THE DIMENSION OF THE PLANAR BROWNIAN FRONTIER IS 4/3 Gregory F. Lawler 1, Oded Schramm 2, and Wendelin Werner 3 1. Introduction The purpose of this note
More informationTHE WORK OF WENDELIN WERNER
THE WORK OF WENDELIN WERNER International Congress of Mathematicians Madrid August 22, 2006 C. M. Newman Courant Institute of Mathematical Sciences New York University It is my pleasure to report on some
More informationarxiv:math-ph/ v2 6 Jun 2005
On Conformal Field Theory of SLE(κ, ρ) arxiv:math-ph/0504057v 6 Jun 005 Kalle Kytölä kalle.kytola@helsinki.fi Department of Mathematics, P.O. Box 68 FIN-00014 University of Helsinki, Finland. Abstract
More informationShock Reflection-Diffraction, Nonlinear Conservation Laws of Mixed Type, and von Neumann s Conjectures 1
Contents Preface xi I Shock Reflection-Diffraction, Nonlinear Conservation Laws of Mixed Type, and von Neumann s Conjectures 1 1 Shock Reflection-Diffraction, Nonlinear Partial Differential Equations of
More informationNonlinear Analysis 71 (2009) Contents lists available at ScienceDirect. Nonlinear Analysis. journal homepage:
Nonlinear Analysis 71 2009 2744 2752 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A nonlinear inequality and applications N.S. Hoang A.G. Ramm
More informationGeneralized Gaussian Bridges of Prediction-Invertible Processes
Generalized Gaussian Bridges of Prediction-Invertible Processes Tommi Sottinen 1 and Adil Yazigi University of Vaasa, Finland Modern Stochastics: Theory and Applications III September 1, 212, Kyiv, Ukraine
More informationStochastic evolutions in superspace and superconformal field theory
Stochastic evolutions in superspace and superconformal field theory arxiv:math-ph/031010v1 1 Dec 003 Jørgen Rasmussen Centre de recherches mathématiques, Université de Montréal Case postale 618, succursale
More informationTowards conformal invariance of 2D lattice models
Towards conformal invariance of 2D lattice models Stanislav Smirnov Abstract. Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising
More informationRichard F. Bass Krzysztof Burdzy University of Washington
ON DOMAIN MONOTONICITY OF THE NEUMANN HEAT KERNEL Richard F. Bass Krzysztof Burdzy University of Washington Abstract. Some examples are given of convex domains for which domain monotonicity of the Neumann
More informationLocality property and a related continuity problem for SLE and SKLE I
Locality property and a related continuity problem for SLE and SKLE I Masatoshi Fukushima (Osaka) joint work with Zhen Qing Chen (Seattle) October 20, 2015 Osaka University, Σ-hall 1 Locality property
More informationGaussian Fields and Percolation
Gaussian Fields and Percolation Dmitry Beliaev Mathematical Institute University of Oxford RANDOM WAVES IN OXFORD 18 June 2018 Berry s conjecture In 1977 M. Berry conjectured that high energy eigenfunctions
More informationOn conformally invariant CLE explorations
On conformally invariant CLE explorations Wendelin Werner Hao Wu arxiv:1112.1211v2 [math.pr] 17 Dec 2012 Abstract We study some conformally invariant dynamic ways to construct the Conformal Loop Ensembles
More informationAsymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends
Asymptotics of generalized eigenfunctions on manifold with Euclidean and/or hyperbolic ends Kenichi ITO (University of Tokyo) joint work with Erik SKIBSTED (Aarhus University) 3 July 2018 Example: Free
More informationSome new estimates on the Liouville heat kernel
Some new estimates on the Liouville heat kernel Vincent Vargas 1 2 ENS Paris 1 first part in collaboration with: Maillard, Rhodes, Zeitouni 2 second part in collaboration with: David, Kupiainen, Rhodes
More informationProblem set 3: Solutions Math 207B, Winter Suppose that u(x) is a non-zero solution of the eigenvalue problem. (u ) 2 dx, u 2 dx.
Problem set 3: Solutions Math 27B, Winter 216 1. Suppose that u(x) is a non-zero solution of the eigenvalue problem u = λu < x < 1, u() =, u(1) =. Show that λ = (u ) 2 dx u2 dx. Deduce that every eigenvalue
More informationFlat 2D Tori with Sparse Spectra. Michael Taylor
Flat 2D Tori with Sparse Spectra Michael Taylor Abstract. We identify a class of 2D flat tori T ω, quotients of the plane by certain lattices, on which the Laplace operator has spectrum contained in the
More informationCauchy s Theorem (rigorous) In this lecture, we will study a rigorous proof of Cauchy s Theorem. We start by considering the case of a triangle.
Cauchy s Theorem (rigorous) In this lecture, we will study a rigorous proof of Cauchy s Theorem. We start by considering the case of a triangle. Given a certain complex-valued analytic function f(z), for
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationReversibility of Some Chordal SLE(κ; ρ) Traces
Reversibility of Some Chordal SLE(κ; ρ Traces Dapeng Zhan May 14, 010 Abstract We prove that, for κ (0, 4 and ρ (κ 4/, the chordal SLE(κ; ρ trace started from (0; 0 + or (0; 0 satisfies the reversibility
More informationSelf-avoiding walk ensembles that should converge to SLE
Tom Kennedy UC Davis, May 9, 2012 p. 1/4 Self-avoiding walk ensembles that should converge to SLE Tom Kennedy University of Arizona, MSRI Tom Kennedy UC Davis, May 9, 2012 p. 2/4 Outline Variety of ensembles
More informationNumerical Approximation of Phase Field Models
Numerical Approximation of Phase Field Models Lecture 2: Allen Cahn and Cahn Hilliard Equations with Smooth Potentials Robert Nürnberg Department of Mathematics Imperial College London TUM Summer School
More informationProf. Erhan Bayraktar (University of Michigan)
September 17, 2012 KAP 414 2:15 PM- 3:15 PM Prof. (University of Michigan) Abstract: We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled
More informationReflected Brownian motion in generic triangles and wedges
Stochastic Processes and their Applications 117 (2007) 539 549 www.elsevier.com/locate/spa Reflected Brownian motion in generic triangles and wedges Wouter Kager Instituut voor Theoretische Fysica, Universiteit
More informationExit times of diffusions with incompressible drifts
Exit times of diffusions with incompressible drifts Andrej Zlatoš University of Chicago Joint work with: Gautam Iyer (Carnegie Mellon University) Alexei Novikov (Pennylvania State University) Lenya Ryzhik
More informationThe dimer model: universality and conformal invariance. Nathanaël Berestycki University of Cambridge. Colloque des sciences mathématiques du Québec
The dimer model: universality and conformal invariance Nathanaël Berestycki University of Cambridge Colloque des sciences mathématiques du Québec The dimer model Definition G = bipartite finite graph,
More informationSolution formula and time periodicity for the motion of relativistic strings in the Minkowski space R 1+n
Solution formula and time periodicity for the motion of relativistic strings in the Minkowski space R 1+n De-Xing Kong and Qiang Zhang Abstract In this paper we study the motion of relativistic strings
More informationKrzysztof Burdzy Wendelin Werner
A COUNTEREXAMPLE TO THE HOT SPOTS CONJECTURE Krzysztof Burdzy Wendelin Werner Abstract. We construct a counterexample to the hot spots conjecture; there exists a bounded connected planar domain (with two
More informationClass Meeting # 2: The Diffusion (aka Heat) Equation
MATH 8.52 COURSE NOTES - CLASS MEETING # 2 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 2: The Diffusion (aka Heat) Equation The heat equation for a function u(, x (.0.). Introduction
More informationGaussian Processes. 1. Basic Notions
Gaussian Processes 1. Basic Notions Let T be a set, and X : {X } T a stochastic process, defined on a suitable probability space (Ω P), that is indexed by T. Definition 1.1. We say that X is a Gaussian
More informationViscosity Solutions for Dummies (including Economists)
Viscosity Solutions for Dummies (including Economists) Online Appendix to Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach written by Benjamin Moll August 13, 2017 1 Viscosity
More informationFRACTAL AND MULTIFRACTAL PROPERTIES OF SLE
Clay Mathematics Proceedings FRACTAL AND MULTIFRACTAL PROPERTIES OF SLE Gregory F. Lawler Introduction This is a slightly expanded version of my lectures at the 2010 Clay Mathematics Institute summer (winter)
More informationConvergence of 2D critical percolation to SLE 6
Convergence of 2D critical percolation to SLE 6 Michael J. Kozdron University of Regina http://stat.math.uregina.ca/ kozdron/ Lecture given at the Mathematisches Forschungsinstitut Oberwolfach (MFO) during
More informationContinuous LERW started from interior points
Stochastic Processes and their Applications 120 (2010) 1267 1316 www.elsevier.com/locate/spa Continuous LERW started from interior points Dapeng Zhan Department of Mathematics, Michigan State University,
More informationALTERNATING ARM EXPONENTS FOR THE CRITICAL PLANAR ISING MODEL. By Hao Wu Yau Mathematical Sciences Center, Tsinghua University, China
Submitted to the Annals of Probability arxiv: arxiv:1605.00985 ALTERNATING ARM EXPONENTS FOR THE CRITICAL PLANAR ISING MODEL By Hao Wu Yau Mathematical Sciences Center, Tsinghua University, China We derive
More informationConformal invariance and covariance of the 2d self-avoiding walk
Conformal invariance and covariance of the 2d self-avoiding walk Department of Mathematics, University of Arizona AMS Western Sectional Meeting, April 17, 2010 p.1/24 Outline Conformal invariance/covariance
More informationSurface x(u, v) and curve α(t) on it given by u(t) & v(t). Math 4140/5530: Differential Geometry
Surface x(u, v) and curve α(t) on it given by u(t) & v(t). α du dv (t) x u dt + x v dt Surface x(u, v) and curve α(t) on it given by u(t) & v(t). α du dv (t) x u dt + x v dt ( ds dt )2 Surface x(u, v)
More informationFirst Passage Time Calculations
First Passage Time Calculations Friday, April 24, 2015 2:01 PM Homework 4 will be posted over the weekend; due Wednesday, May 13 at 5 PM. We'll now develop some framework for calculating properties of
More informationInterfaces between Probability and Geometry
Interfaces between Probability and Geometry (Prospects in Mathematics Durham, 15 December 2007) Wilfrid Kendall w.s.kendall@warwick.ac.uk Department of Statistics, University of Warwick Introduction Brownian
More information4.10 Dirichlet problem in the circle and the Poisson kernel
220 CHAPTER 4. FOURIER SERIES AND PDES 4.10 Dirichlet problem in the circle and the Poisson kernel Note: 2 lectures,, 9.7 in [EP], 10.8 in [BD] 4.10.1 Laplace in polar coordinates Perhaps a more natural
More informationA new class of pseudodifferential operators with mixed homogenities
A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a
More informationSolutions to Homework 11
Solutions to Homework 11 Read the statement of Proposition 5.4 of Chapter 3, Section 5. Write a summary of the proof. Comment on the following details: Does the proof work if g is piecewise C 1? Or did
More informationS chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.
Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable
More informationStochastic nonlinear Schrödinger equations and modulation of solitary waves
Stochastic nonlinear Schrödinger equations and modulation of solitary waves A. de Bouard CMAP, Ecole Polytechnique, France joint work with R. Fukuizumi (Sendai, Japan) Deterministic and stochastic front
More informationarxiv:math-ph/ v3 4 May 2006
On Conformal Field Theory of SLE(κ, ρ arxiv:math-ph/0504057v3 4 May 006 Kalle Kytölä kalle.kytola@helsinki.fi Department of Mathematics and Statistics, P.O. Box 68 FIN-00014 University of Helsinki, Finland.
More informationPivotal, cluster and interface measures for critical planar percolation
Pivotal, cluster and interface measures for critical planar percolation arxiv:1008.1378v5 [math.pr 13 Feb 2014 Christophe Garban Gábor Pete Oded Schramm Abstract This work is the first in a series of papers
More informationHypothesis testing for Stochastic PDEs. Igor Cialenco
Hypothesis testing for Stochastic PDEs Igor Cialenco Department of Applied Mathematics Illinois Institute of Technology igor@math.iit.edu Joint work with Liaosha Xu Research partially funded by NSF grants
More information1 Lyapunov theory of stability
M.Kawski, APM 581 Diff Equns Intro to Lyapunov theory. November 15, 29 1 1 Lyapunov theory of stability Introduction. Lyapunov s second (or direct) method provides tools for studying (asymptotic) stability
More informationNatural parametrization of percolation interface and pivotal points
Natural parametrization of percolation interface and pivotal points Nina Holden Xinyi Li Xin Sun arxiv:1804.07286v2 [math.pr] 3 Apr 2019 Abstract We prove that the interface of critical site percolation
More informationCFT and SLE and 2D statistical physics. Stanislav Smirnov
CFT and SLE and 2D statistical physics Stanislav Smirnov Recently much of the progress in understanding 2-dimensional critical phenomena resulted from Conformal Field Theory (last 30 years) Schramm-Loewner
More informationSingular Perturbations of Stochastic Control Problems with Unbounded Fast Variables
Singular Perturbations of Stochastic Control Problems with Unbounded Fast Variables Joao Meireles joint work with Martino Bardi and Guy Barles University of Padua, Italy Workshop "New Perspectives in Optimal
More informationWEYL S LEMMA, ONE OF MANY. Daniel W. Stroock
WEYL S LEMMA, ONE OF MANY Daniel W Stroock Abstract This note is a brief, and somewhat biased, account of the evolution of what people working in PDE s call Weyl s Lemma about the regularity of solutions
More informationON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS
Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)
More informationMakarov s LIL for SLE
Makarov s LIL for SLE Nam-Gyu Kang Department of Mathematics, M.I.T. Workshop at IPAM, 2007 Nam-Gyu Kang (M.I.T.) Makarov s LIL for SLE Workshop at IPAM, 2007 1 / 24 Outline 1 Introduction and Preliminaries
More informationComplex Analysis Homework 3
Complex Analysis Homework 3 Steve Clanton David Holz March 3, 009 Problem 3 Solution. Let z = re iθ. Then, we see the mapping leaves the modulus unchanged while multiplying the argument by -3: Ω z = z
More informationTOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017
TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 Abstracts of the talks Spectral stability under removal of small capacity
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationREACTION-DIFFUSION EQUATIONS FOR POPULATION DYNAMICS WITH FORCED SPEED II - CYLINDRICAL-TYPE DOMAINS. Henri Berestycki and Luca Rossi
Manuscript submitted to Website: http://aimsciences.org AIMS Journals Volume 00, Number 0, Xxxx XXXX pp. 000 000 REACTION-DIFFUSION EQUATIONS FOR POPULATION DYNAMICS WITH FORCED SPEED II - CYLINDRICAL-TYPE
More informationEigenvalues of Robin Laplacians on infinite sectors and application to polygons
Eigenvalues of Robin Laplacians on infinite sectors and application to polygons Magda Khalile (joint work with Konstantin Pankrashkin) Université Paris Sud /25 Robin Laplacians on infinite sectors 1 /
More informationExplosive Solution of the Nonlinear Equation of a Parabolic Type
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 5, 233-239 Explosive Solution of the Nonlinear Equation of a Parabolic Type T. S. Hajiev Institute of Mathematics and Mechanics, Acad. of Sciences Baku,
More informationMATH 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, < 1, 1 +, 1 < < 0, ψ() = 1, 0 < < 1, 0, > 1, so that it verifies ψ 0, ψ() = 0 if 1 and ψ()d = 1. Consider (ψ j ) j 1 constructed as
More informationarxiv:math/ v1 [math.pr] 27 Mar 2003
1 arxiv:math/0303354v1 [math.pr] 27 Mar 2003 Random planar curves and Schramm-Loewner evolutions Lecture Notes from the 2002 Saint-Flour summer school (final version) Wendelin Werner Université Paris-Sud
More informationSymplectic critical surfaces in Kähler surfaces
Symplectic critical surfaces in Kähler surfaces Jiayu Li ( Joint work with X. Han) ICTP-UNESCO and AMSS-CAS November, 2008 Symplectic surfaces Let M be a compact Kähler surface, let ω be the Kähler form.
More informationGeometric projection of stochastic differential equations
Geometric projection of stochastic differential equations John Armstrong (King s College London) Damiano Brigo (Imperial) August 9, 2018 Idea: Projection Idea: Projection Projection gives a method of systematically
More informationParticles I, Tutorial notes Sessions I-III: Roots & Weights
Particles I, Tutorial notes Sessions I-III: Roots & Weights Kfir Blum June, 008 Comments/corrections regarding these notes will be appreciated. My Email address is: kf ir.blum@weizmann.ac.il Contents 1
More informationRégularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen
Régularité des équations de Hamilton-Jacobi du premier ordre et applications aux jeux à champ moyen Daniela Tonon en collaboration avec P. Cardaliaguet et A. Porretta CEREMADE, Université Paris-Dauphine,
More informationWeak solutions for the Cahn-Hilliard equation with degenerate mobility
Archive for Rational Mechanics and Analysis manuscript No. (will be inserted by the editor) Shibin Dai Qiang Du Weak solutions for the Cahn-Hilliard equation with degenerate mobility Abstract In this paper,
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More informationAncient solutions to Geometric Flows Lecture No 2
Ancient solutions to Geometric Flows Lecture No 2 Panagiota Daskalopoulos Columbia University Frontiers of Mathematics and Applications IV UIMP 2015 July 20-24, 2015 Topics to be discussed In this lecture
More informationConvergence at first and second order of some approximations of stochastic integrals
Convergence at first and second order of some approximations of stochastic integrals Bérard Bergery Blandine, Vallois Pierre IECN, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes B.P. 239 F-5456
More informationarxiv:math/ v3 [math.pr] 27 Aug 2008
The Annals of Probability 2008, Vol. 36, No. 4, 1421 1452 DOI: 10.1214/07-AOP364 c Institute of Mathematical Statistics, 2008 arxiv:math/0211322v3 [math.pr] 27 Aug 2008 THE DIMENSION OF THE SLE CURVES
More informationPERCOLATION IN SELFISH SOCIAL NETWORKS
PERCOLATION IN SELFISH SOCIAL NETWORKS Charles Bordenave UC Berkeley joint work with David Aldous (UC Berkeley) PARADIGM OF SOCIAL NETWORKS OBJECTIVE - Understand better the dynamics of social behaviors
More informationAN OPEN FOUR-MANIFOLD HAVING NO INSTANTON
N OPEN FOUR-MNIFOLD HVING NO INSNON MSKI SUKMOO bstract. aubes proved that all compact oriented four-manifolds admit non-flat instantons. We show that there exists a non-compact oriented four-manifold
More informationStochastic Homogenization for Reaction-Diffusion Equations
Stochastic Homogenization for Reaction-Diffusion Equations Jessica Lin McGill University Joint Work with Andrej Zlatoš June 18, 2018 Motivation: Forest Fires ç ç ç ç ç ç ç ç ç ç Motivation: Forest Fires
More informationSome SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen
Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text
More informationStochastic optimal control with rough paths
Stochastic optimal control with rough paths Paul Gassiat TU Berlin Stochastic processes and their statistics in Finance, Okinawa, October 28, 2013 Joint work with Joscha Diehl and Peter Friz Introduction
More informationOptimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience
Optimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience Ulrich Horst 1 Humboldt-Universität zu Berlin Department of Mathematics and School of Business and Economics Vienna, Nov.
More information