Diffusion for a Markov, Divergence-form Generator
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1 Diffusion for a Markov, Divergence-form Generator Clark Department of Mathematics Michigan State University Arizona School of Analysis and Mathematical Physics March 16, 2012
2 Abstract We consider the long-time evolution of solutions to a Schrödinger-type wave equation on a lattice with a Markov random generator. We show that solutions to this problem possess a diffusive scaling limit and compute higher moments. Based on joint work with Jeffrey Schenker.
3 Statement of the Theorem Theorem If ψ t l 2 (Z d ) satisfies { i t ψ t (x) = θ ω(t) ψ t (x) ψ 0 (x) = δ 0 (x), then lim e i η 0 + x Z d ηk x E ( ψ t/η (x) 2) = e 4t e 1,e 2 (k e 1 )(k e 2 )D e1,e 2.
4 What do we mean by diffusion? Consider the standard heat equation { t u(x, t) = u(x, t) (x, t) R d R + u(x, 0) = δ 0 (x) x R d with solution u(x, t) = (2πt) d/2 e x 2 /4t. ( x c t u(x, t) is a p.d.f. on R d with c t = u(x, t) dt R d the normalizing constant. ) 1
5 What do we mean by diffusion? Consider the standard heat equation { t u(x, t) = u(x, t) (x, t) R d R + u(x, 0) = δ 0 (x) x R d with solution u(x, t) = (2πt) d/2 e x 2 /4t. ( x c t u(x, t) is a p.d.f. on R d with c t = u(x, t) dt R d the normalizing constant. ) 1
6 What do we mean by diffusion? The p th moment of position is given by x p c t u(x, t) dx = c tω d R d (2πt) d/2 r p+d 1 e r2 4t dr, 0 where ω d = B(0, 1) is the surface area of the unit ball in R d. The integrand is maximized when r t which leads us to define...
7 What do we mean by diffusion? The p th moment of position is given by x p c t u(x, t) dx = c tω d R d (2πt) d/2 r p+d 1 e r2 4t dr, 0 where ω d = B(0, 1) is the surface area of the unit ball in R d. The integrand is maximized when r t which leads us to define...
8 Diffusive Scaling Definition: Diffusive Scaling { t 1 η t x 1 as η 0 + η x Question: The problem under consideration is defined on the lattice Z d. How do we scale a discrete space? Answer: Mollify.
9 Diffusive Scaling Definition: Diffusive Scaling { t 1 η t x 1 as η 0 + η x Question: The problem under consideration is defined on the lattice Z d. How do we scale a discrete space? Answer: Mollify.
10 Diffusive Scaling Definition: Diffusive Scaling { t 1 η t x 1 as η 0 + η x Question: The problem under consideration is defined on the lattice Z d. How do we scale a discrete space? Answer: Mollify.
11 Characterization of Diffusion for a Discrete Problem h C c (R d ), h dx = 1, h 0 Under diffusive scaling, if the convolution h ψ t 2 converges (weakly) to a solution of the heat equation, then we say that the model exhibits diffusion. A Fourier transform removes the mollifier from our diffusion criterion. Diffusion Criterion: ηk x ψ t/η (x) 2 e Dt k 2, x Z d e i k T d
12 Characterization of Diffusion for a Discrete Problem h C c (R d ), h dx = 1, h 0 Under diffusive scaling, if the convolution h ψ t 2 converges (weakly) to a solution of the heat equation, then we say that the model exhibits diffusion. A Fourier transform removes the mollifier from our diffusion criterion. Diffusion Criterion: ηk x ψ t/η (x) 2 e Dt k 2, x Z d e i k T d
13 Characterization of Diffusion for a Discrete Problem h C c (R d ), h dx = 1, h 0 Under diffusive scaling, if the convolution h ψ t 2 converges (weakly) to a solution of the heat equation, then we say that the model exhibits diffusion. A Fourier transform removes the mollifier from our diffusion criterion. Diffusion Criterion: ηk x ψ t/η (x) 2 e Dt k 2, x Z d e i k T d
14 Characterization of Diffusion for a Discrete Problem h C c (R d ), h dx = 1, h 0 Under diffusive scaling, if the convolution h ψ t 2 converges (weakly) to a solution of the heat equation, then we say that the model exhibits diffusion. A Fourier transform removes the mollifier from our diffusion criterion. Diffusion Criterion: ηk x ψ t/η (x) 2 e Dt k 2, x Z d e i k T d
15 Resolvent Analysis Key step: ηk x E( ψ t/η (x) 2 ) x Z d e i = 1 2πi Γ e tz δ 0 1, η iˆl ηk + B ηz δ 0 1 dz Notes: The LHS is (almost) the diffusion criterion. The expectation allows us to use a Feynman-Kac-Pillet formula. FKP allows us to express the expectation as a matrix element of the semigroup e t(iˆl ηk +B), which can be understood by the holomorphic functional calculus: e t(iˆl ηk +B) = 1 2πi Γ e tz 1 iˆl ηk + B z dz
16 Resolvent Analysis Key step: ηk x E( ψ t/η (x) 2 ) x Z d e i = 1 2πi Γ e tz δ 0 1, η iˆl ηk + B ηz δ 0 1 dz Notes: The LHS is (almost) the diffusion criterion. The expectation allows us to use a Feynman-Kac-Pillet formula. FKP allows us to express the expectation as a matrix element of the semigroup e t(iˆl ηk +B), which can be understood by the holomorphic functional calculus: e t(iˆl ηk +B) = 1 2πi Γ e tz 1 iˆl ηk + B z dz
17 Resolvent Analysis Key step: ηk x E( ψ t/η (x) 2 ) x Z d e i = 1 2πi Γ e tz δ 0 1, η iˆl ηk + B ηz δ 0 1 dz Notes: The LHS is (almost) the diffusion criterion. The expectation allows us to use a Feynman-Kac-Pillet formula. FKP allows us to express the expectation as a matrix element of the semigroup e t(iˆl ηk +B), which can be understood by the holomorphic functional calculus: e t(iˆl ηk +B) = 1 2πi Γ e tz 1 iˆl ηk + B z dz
18 Resolvent Analysis Key step: ηk x E( ψ t/η (x) 2 ) x Z d e i = 1 2πi Γ e tz δ 0 1, η iˆl ηk + B ηz δ 0 1 dz Notes: The LHS is (almost) the diffusion criterion. The expectation allows us to use a Feynman-Kac-Pillet formula. FKP allows us to express the expectation as a matrix element of the semigroup e t(iˆl ηk +B), which can be understood by the holomorphic functional calculus: e t(iˆl ηk +B) = 1 2πi Γ e tz 1 iˆl ηk + B z dz
19 Resolvent Analysis Key step: ηk x E( ψ t/η (x) 2 ) x Z d e i = 1 2πi Γ e tz δ 0 1, η iˆl ηk + B ηz δ 0 1 dz Notes: The LHS is (almost) the diffusion criterion. The expectation allows us to use a Feynman-Kac-Pillet formula. FKP allows us to express the expectation as a matrix element of the semigroup e t(iˆl ηk +B), which can be understood by the holomorphic functional calculus: e t(iˆl ηk +B) = 1 2πi Γ e tz 1 iˆl ηk + B z dz
20 Resolvent Analysis Key step: ηk x E( ψ t/η (x) 2 ) x Z d e i = 1 2πi Γ e tz δ 0 1, η iˆl ηk + B ηz δ 0 1 dz Notes: The LHS is (almost) the diffusion criterion. The expectation allows us to use a Feynman-Kac-Pillet formula. FKP allows us to express the expectation as a matrix element of the semigroup e t(iˆl ηk +B), which can be understood by the holomorphic functional calculus: e t(iˆl ηk +B) = 1 2πi Γ e tz 1 iˆl ηk + B z dz
21 Resolvent Analysis We have reduced the problem to understanding: η lim δ 0 1, η 0 + iˆl ηk + B ηz δ 0 1. From here, use projections and the Schur complement formula. construct a symmetric operator D k, which is a lower bound for the matrix element in question. Use this to show the limit exists and is of the desired form. Higher Moments? lim e i η 0 + x Z d ηk x E ( ψ t/η (x) 2) = e 4t e 1,e 2 (k e 1 )(k e 2 )D e1,e 2, holomorphic for k B(0, ɛ) C d?
22 Resolvent Analysis We have reduced the problem to understanding: η lim δ 0 1, η 0 + iˆl ηk + B ηz δ 0 1. From here, use projections and the Schur complement formula. construct a symmetric operator D k, which is a lower bound for the matrix element in question. Use this to show the limit exists and is of the desired form. Higher Moments? lim e i η 0 + x Z d ηk x E ( ψ t/η (x) 2) = e 4t e 1,e 2 (k e 1 )(k e 2 )D e1,e 2, holomorphic for k B(0, ɛ) C d?
23 Resolvent Analysis We have reduced the problem to understanding: η lim δ 0 1, η 0 + iˆl ηk + B ηz δ 0 1. From here, use projections and the Schur complement formula. construct a symmetric operator D k, which is a lower bound for the matrix element in question. Use this to show the limit exists and is of the desired form. Higher Moments? lim e i η 0 + x Z d ηk x E ( ψ t/η (x) 2) = e 4t e 1,e 2 (k e 1 )(k e 2 )D e1,e 2, holomorphic for k B(0, ɛ) C d?
24 Resolvent Analysis We have reduced the problem to understanding: η lim δ 0 1, η 0 + iˆl ηk + B ηz δ 0 1. From here, use projections and the Schur complement formula. construct a symmetric operator D k, which is a lower bound for the matrix element in question. Use this to show the limit exists and is of the desired form. Higher Moments? lim e i η 0 + x Z d ηk x E ( ψ t/η (x) 2) = e 4t e 1,e 2 (k e 1 )(k e 2 )D e1,e 2, holomorphic for k B(0, ɛ) C d?
25 Resolvent Analysis We have reduced the problem to understanding: η lim δ 0 1, η 0 + iˆl ηk + B ηz δ 0 1. From here, use projections and the Schur complement formula. construct a symmetric operator D k, which is a lower bound for the matrix element in question. Use this to show the limit exists and is of the desired form. Higher Moments? lim e i η 0 + x Z d ηk x E ( ψ t/η (x) 2) = e 4t e 1,e 2 (k e 1 )(k e 2 )D e1,e 2, holomorphic for k B(0, ɛ) C d?
26 Thank You & References Thank you! Klaus-Jochen Engel and Rainer Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, Yang Kang and Jeffrey Schenker. Diffusion of wave packets in a markov random potential. Journal of Statistical Physics, 134: , Claude-Alain Pillet. Some results on the quantum dynamics of a particle in a markovian potential. Communications in Mathematical Physics, 102(2): , 1985.
27 Thank You & References Thank you! Klaus-Jochen Engel and Rainer Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, Yang Kang and Jeffrey Schenker. Diffusion of wave packets in a markov random potential. Journal of Statistical Physics, 134: , Claude-Alain Pillet. Some results on the quantum dynamics of a particle in a markovian potential. Communications in Mathematical Physics, 102(2): , 1985.
28 Thank You & References Thank you! Klaus-Jochen Engel and Rainer Nagel. One-parameter semigroups for linear evolution equations, volume 194 of Graduate Texts in Mathematics. Springer-Verlag, Yang Kang and Jeffrey Schenker. Diffusion of wave packets in a markov random potential. Journal of Statistical Physics, 134: , Claude-Alain Pillet. Some results on the quantum dynamics of a particle in a markovian potential. Communications in Mathematical Physics, 102(2): , 1985.
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