Free Multiplicative Brownian Motion, and Brown Measure Extended Probabilistic Operator Algebras Seminar UC Berkeley Todd Kemp UC San Diego November 10, 2017 1 / 27
Giving Credit where Credit is Due Based partly on joint work with Bruce Driver and Brian Hall, and highlighting the work of Philippe Biane. • Biane, P .: Free Brownian motion, free stochastic calculus and random matrices . Fields Inst. Commun. vol. 12, Amer. Math. Soc., PRovidence, RI, 1-19 (1997) • Biane, P .: Segal-Bargmann transform, functional calculus on matrix spaces and the theory of semi-circular and circular systems . J. Funct. Anal. 144, 1, 232-286 (1997) Driver; Hall; K: The large- N limit of the Segal–Bargmann transform on U N . • J. Funct. Anal. 265, 2585-2644 (2013) K: The Large- N Limits of Brownian Motions on GL N . Int. Math. Res. Not. • IMRN, no. 13, 4012-4057 (2016) • K: Heat kernel empirical laws on U N and GL N . J. Theoret. Probab. 30, no. 2, 397-451 (2017) 2 / 27
• Citations Brownian Motion • BM on Lie Groups • U & GL • Free + BM • Free × BM • Free Unitary BM • Transforms • Free Mult. BM Brownian Motion on U( N ) , • GL Spectrum GL( N, C ) , and the Large- N Limit Brown Measure Segal–Bargmann 3 / 27
Brownian Motion on Lie Groups On any Riemannian manifold M , there’s a Laplace operator ∆ M . • Citations And where there’s a Laplacian, there’s a Brownian motion: the Brownian Motion Markov process ( B x t ) t ≥ 0 on M with generator 1 2 ∆ M , started at • BM on Lie Groups • U & GL B x 0 = x ∈ M . • Free + BM • Free × BM • Free Unitary BM • Transforms • Free Mult. BM • GL Spectrum Brown Measure Segal–Bargmann 4 / 27
Brownian Motion on Lie Groups On any Riemannian manifold M , there’s a Laplace operator ∆ M . • Citations And where there’s a Laplacian, there’s a Brownian motion: the Brownian Motion Markov process ( B x t ) t ≥ 0 on M with generator 1 2 ∆ M , started at • BM on Lie Groups • U & GL B x 0 = x ∈ M . • Free + BM • Free × BM Let Γ be a (matrix) Lie group. Any inner product on Lie(Γ) = T I Γ • Free Unitary BM • Transforms gives rise to a unique left-invariant Riemannian metric, and • Free Mult. BM corresponding Laplacian ∆ Γ . On Γ we canonically start the • GL Spectrum Brownian motion ( B t ) t ≥ 0 at I ∈ Γ . Brown Measure Segal–Bargmann 4 / 27
Brownian Motion on Lie Groups On any Riemannian manifold M , there’s a Laplace operator ∆ M . • Citations And where there’s a Laplacian, there’s a Brownian motion: the Brownian Motion Markov process ( B x t ) t ≥ 0 on M with generator 1 2 ∆ M , started at • BM on Lie Groups • U & GL B x 0 = x ∈ M . • Free + BM • Free × BM Let Γ be a (matrix) Lie group. Any inner product on Lie(Γ) = T I Γ • Free Unitary BM • Transforms gives rise to a unique left-invariant Riemannian metric, and • Free Mult. BM corresponding Laplacian ∆ Γ . On Γ we canonically start the • GL Spectrum Brownian motion ( B t ) t ≥ 0 at I ∈ Γ . Brown Measure Segal–Bargmann There is a beautiful relationship between the Brownian motion W t on the Lie algebra Lie(Γ) and the Brownian motion B t : the rolling map � t dB t = B t ◦ dW t , B t = I + B t ◦ dW t . i.e. 0 Here ◦ denotes the Stratonovich stochastic integral. This can always be converted into an Itˆ o integral; but the answer depends on the structure of the group Γ (and the chosen inner product). 4 / 27
Brownian Motion on U( N ) and GL( N, C ) Fix the reverse normalized Hilbert–Schmidt inner product on • Citations M N ( C ) for all matrix Lie algebras: Brownian Motion • BM on Lie Groups � A, B � = N Tr( B ∗ A ) . • U & GL • Free + BM • Free × BM Let X t = X N and Y t = Y N • Free Unitary BM be independent Hermitian Brownian t t • Transforms motions of variance t/N . • Free Mult. BM • GL Spectrum Brown Measure Segal–Bargmann 5 / 27
Brownian Motion on U( N ) and GL( N, C ) Fix the reverse normalized Hilbert–Schmidt inner product on • Citations M N ( C ) for all matrix Lie algebras: Brownian Motion • BM on Lie Groups � A, B � = N Tr( B ∗ A ) . • U & GL • Free + BM • Free × BM Let X t = X N and Y t = Y N • Free Unitary BM be independent Hermitian Brownian t t • Transforms motions of variance t/N . • Free Mult. BM • GL Spectrum The Brownian motion on Lie(U( N )) is iX t ; the Brownian motion Brown Measure U t on U( N ) satisfies Segal–Bargmann dU t = iU t dX t − 1 2 U t dt. 5 / 27
Brownian Motion on U( N ) and GL( N, C ) Fix the reverse normalized Hilbert–Schmidt inner product on • Citations M N ( C ) for all matrix Lie algebras: Brownian Motion • BM on Lie Groups � A, B � = N Tr( B ∗ A ) . • U & GL • Free + BM • Free × BM Let X t = X N and Y t = Y N • Free Unitary BM be independent Hermitian Brownian t t • Transforms motions of variance t/N . • Free Mult. BM • GL Spectrum The Brownian motion on Lie(U( N )) is iX t ; the Brownian motion Brown Measure U t on U( N ) satisfies Segal–Bargmann dU t = iU t dX t − 1 2 U t dt. The Brownian motion on Lie(GL( N, C )) = M N ( C ) is Z t = 2 − 1 / 2 i ( X t + iY t ) ; the Brownian motion G t on GL( N, C ) satisfies dG t = G t dZ t . 5 / 27
1 1 Free Additive Brownian Motion If X t = X N • Citations is a Hermitian Brownian motion process, then at each t Brownian Motion time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law • BM on Lie Groups then shows that the empirical spectral distribution of X N • U & GL converges t • Free + BM 1 � (4 t − x 2 ) + dx . to the semicircle law ς t = • Free × BM 2 πt • Free Unitary BM • Transforms • Free Mult. BM • GL Spectrum Brown Measure Segal–Bargmann 6 / 27
1 1 Free Additive Brownian Motion If X t = X N • Citations is a Hermitian Brownian motion process, then at each t Brownian Motion time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law • BM on Lie Groups then shows that the empirical spectral distribution of X N • U & GL converges t • Free + BM 1 � (4 t − x 2 ) + dx . In fact, it converges to the semicircle law ς t = • Free × BM 2 πt • Free Unitary BM as a process . • Transforms • Free Mult. BM • GL Spectrum Brown Measure Segal–Bargmann 6 / 27
1 1 Free Additive Brownian Motion If X t = X N • Citations is a Hermitian Brownian motion process, then at each t Brownian Motion time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law • BM on Lie Groups then shows that the empirical spectral distribution of X N • U & GL converges t • Free + BM 1 � (4 t − x 2 ) + dx . In fact, it converges to the semicircle law ς t = • Free × BM 2 πt • Free Unitary BM as a process . • Transforms • Free Mult. BM A process ( x t ) t ≥ 0 (in a W ∗ -probability space with trace τ ) is a free • GL Spectrum additive Brownian motion if its increments are freely independent Brown Measure — x t − x s is free from { x r : r ≤ s } — and x t − x s has the Segal–Bargmann semicircular distribution ς t − s , for all t > s . 6 / 27
Free Additive Brownian Motion If X t = X N • Citations is a Hermitian Brownian motion process, then at each t Brownian Motion time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law • BM on Lie Groups then shows that the empirical spectral distribution of X N • U & GL converges t • Free + BM 1 � (4 t − x 2 ) + dx . In fact, it converges to the semicircle law ς t = • Free × BM 2 πt • Free Unitary BM as a process . • Transforms • Free Mult. BM A process ( x t ) t ≥ 0 (in a W ∗ -probability space with trace τ ) is a free • GL Spectrum additive Brownian motion if its increments are freely independent Brown Measure — x t − x s is free from { x r : r ≤ s } — and x t − x s has the Segal–Bargmann semicircular distribution ς t − s , for all t > s . It can be constructed on the free Fock space over L 2 ( R + ) : x t = l ( 1 [0 ,t ] ) + l ∗ ( 1 [0 ,t ] ) . 6 / 27
Free Additive Brownian Motion If X t = X N • Citations is a Hermitian Brownian motion process, then at each t Brownian Motion time t > 0 it is a GUE N with entries of variance t/N . Wigner’s law • BM on Lie Groups then shows that the empirical spectral distribution of X N • U & GL converges t • Free + BM 1 � (4 t − x 2 ) + dx . In fact, it converges to the semicircle law ς t = • Free × BM 2 πt • Free Unitary BM as a process . • Transforms • Free Mult. BM A process ( x t ) t ≥ 0 (in a W ∗ -probability space with trace τ ) is a free • GL Spectrum additive Brownian motion if its increments are freely independent Brown Measure — x t − x s is free from { x r : r ≤ s } — and x t − x s has the Segal–Bargmann semicircular distribution ς t − s , for all t > s . It can be constructed on the free Fock space over L 2 ( R + ) : x t = l ( 1 [0 ,t ] ) + l ∗ ( 1 [0 ,t ] ) . In 1991, Voiculescu showed that the Hermitian Brownian motion ( X N t ) t ≥ 0 converges to ( x t ) t ≥ 0 in finite-dimensional non-commutative distributions: 1 N Tr( P ( X t 1 , . . . , X t n )) → τ ( P ( x t 1 , . . . , x t n )) ∀ P. 6 / 27
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