Free diffusions and Matrix models with strictly convex interaction

0.35 MB·English

by  A. Guionnet

7
Free diffusions and Matrix models with strictly convex interaction
0
0
2
n Alice Guionnet∗ D. Shlyakhtenko†
a
J February 2, 2008

6
2

]
A Abstract
O
1
We study solutions to the free stochastic differential equation dX = dS DV(X )dt,
t t 2 t
. −
h where V is a locally convex polynomial potential in m non-commuting variables. We show
t that for self-adjoint V, the law µ of a stationary solution is the limit law of a random
a V
m matrix model, in which an m-tuple of self-adjoint matrices are chosen according to the law
[ exp(−NTr(V(A1,...,Am)))dA1···dAm. We show that if V =Vβ depends on complex param-
eters β1,...,βk, then the law µV is analytic in β at least for those β for which Vβ is locally
1 convex. In particular, this gives information on the region of convergence of the generating
v
function for planar maps.
7
We show that the solutiondX has nice convergencepropertieswith respectto the operator
8 t
∗ ∗
7 norm. This allows us to derive several properties of C and W algebras generated by an m-
1 tuple with law µ . Among them is lack of projections, exactness, the Haagerup property, and
V
0 embeddability into the ultrapower of the hyperfinite II1 factor. We show that the microstates
7
free entropy χ(τ ) is finite.
0 V
A corollary of these results is the fact that the support of the law of any self-adjoint poly-
/
h nomial in X1,...,Xn under the law µV is connected, vastly generalizing the case of a single
t
a random matrix.
m
:
v 1 Introduction
i
X
r There has been a great deal of interest in studying matrix integrals in physics since the work
a
of ’t Hooft who made the connection between the problem of enumerating maps and estimating
integrals of the form
Z (V)= e NTr(V(X1,...,Xm))dX dX
N − 1 m
···
Z
where dX denotes the Lebesgue measure on N N Hermitian matrices and Tr the non-normalized
trace Tr(A) = N A . V is a polynomial ×in m-indeterminates. Let us recall that a map of
i=1 ii
genus g is a graph which is embedded into a surface of genus g in such a way that the edges
P
do not intersect and so that disecting the surface along the edges decomposes it into faces, each
homeomorphic to a disk. We shall consider maps with colored edges and enumerate them when
the degrees of the vertices, as well as the ditribution of color of the edges around each vertex, are
∗Ecole NormaleSup´erieuredeLyon,Unit´edeMath´ematiquespuresetappliqu´ees,FranceandMiller institutefor
Basic Research in Science, Universityof California Berkeley. E-mail: [e-mail].
†Department of Mathematics, UCLA, Los Angeles, CA 90095. E-mail: [e-mail]. Research sup-
ported byNSFgrants DMS-0355226 and DMS-0555680.
1
prescribed. The number of colors will be m, and the colors will be simply refered by the numbers
1,...,m . A vertex with colored half-edges, an orientation and a distinguished half-edge, can be
{ }
associated bijectivly with a non-commutative monomial q(X) = X X as follows; the first (or
i1··· ip
distinguished) half-edge has color i , second has color i , etc., the last half-edge having color i .
1 2 p
Such a vertex, equipped with its colored half-edges, distinguished half-edge and orientation, will be
called a star of type q. We will denote by M ((q ,k ) ) the number of maps with genus g and
g i i 1 i n
≤≤
with k stars of type q for 1 i n, the maps being constructed by gluing pairwise the half-edges
i i
≤ ≤
of the stars (the counting is done up to homomorphism of the surface and stars are labelled).
’t Hooft showed that if
m n
1
V(X ,...,X ) = W (X ,...,X ) = X2+ β q (X ,...,X )
1 m (βi,qi)1≤i≤n 1 m 2 i i i 1 m
i=1 i=1
X X
where q , i = 1,...,n are monomials in m-non commutative indeterminates, then
i
1 log ZN(W(βi,qi)1≤i≤n) = 1 (−βi)kiM ((q ,k ) )
N2 ZN(W(0,qi)1≤i≤n Xg≥0 N2g k1,..X.,kn∈Nn1≤Yi≤n ki! g i i 1≤i≤n
where the equality holds in the sense of formal series. Differentiating formally this equality, one
also finds that if we consider the Gibbs measure
dµNW(βi,qi)1≤i≤n(X1,...,Xm) := ZN(W(β1i,qi)1≤i≤n)e−NTr(W(βi,qi)1≤i≤n(X1,...,Xm))dX1···dXm (1)
then we have, for any monomial P, the formal expansion
1
µ¯N (P) := Tr(P(X ,...,X ))dµN (X ,...,X )
W(βi,qi)1≤i≤n N 1 m W(βi,qi)1≤i≤n 1 m
Z
1 ( βi)ki
= − M ((q ,k ) ,(P,1)).
N2g ki! g i i 1≤i≤n
Xg≥0 k1X,...,kn1≤Yi≤n
We refer the reader to the survey papers [13, 19] to see diverse uses of this fact in the physics
literature.
In [1, 12] for m = 1 and then in [20, 21, 22] for all m N, these formal equalities were shown
∈
to hold in the sense of large N expansion when V satisfies some convexity hypothesis (or one adds
a cutoff to make the integral finite) and the parameters (β ) are chosen to be small enough.
i 1 i n
In [20], one of the key steps towards this analysis is to notice ≤th≤at µ¯N converges towards
W
(βi,qi)1≤i≤n
a limit, denoted later τ , which satisfies the so-called Schwinger-Dyson equation
W
(βi,qi)1≤i≤n
τ τ(∂ P)= τ(D VP) (2)
i i
⊗
for all polynomials P and all i 1,...,m , and with V = W . Here, ∂ and D are
∈ { } (βi,qi)1≤i≤n i i
respectively the non-commutative derivative and the cyclic derivative with respect to the variable
X (see the next section for a definition). One then shows that for sufficiently small parameters β ,
i i
this equation has a unique solution, which is exactly the generating function for planar maps:
( βi)ki
τ (P) = (P) := − M ((q ,k ) ,(P,1)).
W(βi,qi)1≤i≤n M(βi,qi)1≤i≤n ki! 0 i i 1≤i≤n
k1X,...,kn1≤Yi≤n
2
It is natural to wonder how to use these expansions to study the numbers M ((q ,k ) ) and
g i i 1 i n
≤≤
in particular their asymptotics as k ,...,k go to infinity. The answer to this question is still open
1 n
in such a general context. It is, however, quite well understood in the case m = 1. Let us highlight
this point in the case of quadrangulations, corresponding to the potential V(x) = βx4, β R, even
∈
though the enumeration of quadrangulations was achieved by direct combinatorial arguments by
Tutte [23] long ago. In this case, the Schwinger-Dyson equation (2), taken at P(x) = (z x) 1
−
−
shows that the Cauchy transform
1 1
G(z) = dτ (x)= d (x)
z x Wβ,x4 z x Mβ,x4
Z − Z −
satisfies an algebraic equation of degree two
x3 z3
G(z)2 = 4βz3G(z)+zG(z)+P(z) with P(z) = 4β − dτ (x) 1.
x z Wβ,x4 −
Z −
We can solve this equation in terms of P(z) which is a polynomial of degree two with two unknown
coefficients; we then find that G is given by a polynomial plus the square root of a polynomial of
degree six, that we denote Q, with two unknown coefficients. Until this point, all the arguments
follow the induction relations already found by Tutte. However, the difference now is that we know
that = τ is a probability measureon R. Assumewe can argue that for sufficiently small
Mβ,x4 Wβ,x4
β, the support of τ is connected. Then, we see that, because this means that G is analytic
Wβ,x4
outside an interval, the polynomial Q must have two double roots. This actually determines Q,
and thus G, uniquely. Since G is the generating function for planar maps, we are done. Hence, we
see in this context that the a priori information that (P) is the Cauchy transform of
M(βi,qi)1≤i≤n
a measure on the real line (which is not clear from its definition as a generating function of maps),
and with a connected support, is enough to conclude.
The goal of this article is to push forward the analysis of the limiting tracial state τ
W
(βi,qi)1≤i≤n
in the multi-matrix context. We prove in particular that when W satisfies a certain local
(βi,qi)1≤i≤n
convexity property (see (3)), the support of the limiting spectral measure of the random matrices
with law µN is connected. In fact, the same is true for an arbitrary non-commutative
W
(βi,qi)1≤i≤n
polynomial in the random matrices. Note that this information is enough to solve the enumeration
problem when m = 1 as we have seen above for quadrangulations (though connectivity of support
can in this case be proved by other techniques, see e.g [10]).
The tracial states τ = τ under consideration will be solution to the Schwinger-Dyson equation
V
(2) for some general potential V. Non-commutative laws arising as limits of laws of random matrix
models given by (1) have also naturally appeared in free probability theory. There, the fact that
they satisfy the Schwinger-Dyson type equation is restated as the fact that the free conjugate
variables of the law are equal to the cyclic gradient of a polynomial potential, see also [8, 3].
In the multi-matrix setting, uniqueness of the solution to the Schwinger-Dyson equation is
unclear in general. It was proved in [20] that, when V = W , there exists a unique
(βi,qi)1≤i≤n
solution such that τ(X X ) Rk for all choices of i 1, ,m and all k = 1,2,...,
| i1··· ik | ≤ j ∈ { ··· }
provided the β ’s are sufficiently small. In this paper we define a notion of locally strictly convex
i
potential, which generalizes to non-commutative variables the standard notion of local convexity
for functions on the real line. One of the central result of this paper will be the uniqueness of the
solution to Schwinger-Dyson equation for locally strictly convex potentials V, when the domain of
strict convexity is large enough.
3
We also show that if V = V = β q with some monomials q and β = (β ) , β τ (P)
β i i i i 1≤i≤n → Vβ
is analytic in the whole region of local convexity of V , for any monomial P. Because
β
P
( βi)ki
(P) = τ = − M ((q ,k ) ,(P,1)),
M(βi,qi)1≤i≤n W(qi,βi)1≤i≤n ki! 0 i i 1≤i≤n
k1X,...,kn1≤Yi≤n
this result shows that there is no breaking of analyticity of (β ) (P) in the
i 1≤i≤n → M(βi,qi)1≤i≤n
domain where W stays locally strictly convex, and thus provides valuable information on
(qi,βi)1≤i≤n
the asymptotics of the numbers M ((q ,k ) ,(P,1)).
0 i i 1 i n
≤≤
Of particular interest are the C -algebras (A ,τ ) and the W -algebras (M ,τ ) generated
∗ V V ∗ V V
by operators having a law satisfying the Schwinger-Dyson equation with a fixed locally convex
potential V. We derive several properties of such algebras, showing that they are projectionless,
exact [28]andpossessthecompact approximation propertyof Haagerup [16], andcan beembedded
into the ultrapower of the hyperfinite II factor. We also show that the algebras M are factors
1 V
and that the generating operators have a law with finite free entropy. This has as consequences a
number of properties of the algebras M (including primeness and lack of Cartan subalgebras), see
V
[26] and [15]. All of these properties are similar to (and are often derived from) the corresponding
propertiesoffreegroupfactors(whicharevonNeumannalgebrasassociatedtoaquadraticpotential
V(X ,...,X )= 1 X2). This adds evidence towards a positive answer to Voiculescu’s question
1 n 2 i i
of whether M is isomorphic to a free group factor for a fairly arbitrary potential V.
V
P
While somewhat technical, it should be noted that studying properties of A and M is of
V V
substantial interest. Indeed,connectivity of supportof limit distributionsof randommatrix models
is directly related to the lack of projections in the C -algebra A (which in turn is derived from
∗ V
the famous result of Pimsner and Voiculescu [24], essentially dealing with the quadratic V).
Finally, in the remaining sections of the paper we show that the random matrices following the
law (1) give a very good approximation to the non-commutative law τ . If V is locally convex, we
V
show that the limsup in the definition of the microstates free entropy of τ [25] can be replaced by
V
a limit. In the case that V is (globally) convex, we show that operator norms of arbitrary polyno-
mials in such random matrices almost surely approximate the operator norms of such polynomials
computed in the C -algebra A . This extends the results of [14], which correspond to the case of
∗ V
quadratic V (our proof, though, relies on their result).
The main technical tools used in the present paper involve the study of free Langevin-type
diffusion and its convergence to a stationary measure which corresponds to the limit law τ of
V
random matrices following the measure (1). We extend some of the results of [5] to the setting
of locally convex potentials (see below). This way, we are able to show that operators having
a specific limit law can be approximated in the operator norm by continuous functions of free
Brownian motion. This enables us to carry over a number of properties of the algebra generated
by free Brownian motion to the algebras A and M .
V V
1.1 Definitions, notations and statment of the results.
Let us now state more precisely our setup and results.
We let C X ,...,X bethe set of polynomials in m non-commutative variables (X ,...,X ).
1 m 1 m
h i
We shall not assume in general that (X ,...,X ) are self-adjoint but let (X ,...,X ) be their
1 m 1∗ m∗
adjoints for some involution . We denote C X ,...,X ,X ,...,X the set of polynomials in
∗ h 1 m 1∗ m∗i
4
the non-commutative variables (X ,...,X ,X ,...,X ). This set is endowed with the linear
1 m 1∗ m∗
involution so that
(Xiε11···Xiεkk)∗ =Xi−kεkXi−k−εk1−1···Xi−1ε1
where we denoted in short X1 = X and X 1 = X and the (ε ,...,ε ) belong to 1,1 k. We
i i i− i∗ 1 k {− }
shall denote below for two sets of non-commutative variables (X ,...,X ) and (Y ,...,Y ) and
1 m 1 m
an involution
∗ m
1
X.Y = (X Y +Y X ).
2 i i∗ i i∗
i=1
X
will denote an operator norm such that the completion of (C X ,...,X ,X ,...,X , )
k·k∞ h 1 m 1∗ m∗i ∗
for this norm is a C -algebra. For a m-dimensional vector X = (X ,...,X ) we denote in short
∗ 1 m
X = max X .
1 i m i
k kW∞e let V ≤≤C Xk ,.k.∞.,X be a polynomial in m non-commutative variables. We will say
1 m
∈ h i
that V is (c,M) convex iff for any m-tuples of non-commutative variables X = (X ,...,X ) and
1 m
Y = (Y ,...,Y ) in some C -algebra ( , ) satisfying X , Y M, i = 1,...,m we
1 m ∗ i i
A k·k∞ k k∞ k k∞ ≤
have
[DV(X) DV(Y)].(X Y) c(X Y).(X Y) (3)
− − ≥ − −
where the inequality is understood in the sense of operators (X Y iff X Y is self adjoint and
≥ −
has non negative spectrum). D = (D ,...,D ) denotes the cyclic gradient which is linear and
1 m
given, for any monomial P, by
D P = P P .
i 2 1
P=XP1XiP2
Later, we shall also need the non-commutative gradient ∂ which is given, for any monomial P, any
i 1,...,m by
∈{ }
∂ P = P P .
i 1 2
⊗
P=XP1XiP2
We occasionally shall consider polynomials in C X ,...,X ,X ,...,X ; in that case we extend
h 1 m 1∗ m∗ i
∂ and D by setting ∂ X = 1 1 1 and ∂ X = 0 0 whereas we have also the derivative ∂
i i i j i=j ⊗ i j∗ ⊗ i,∗
and D with respect to X which satisfy ∂ X = 0 0 but ∂ X = 1 1, and extending by
i,∗ i∗ i,∗ j ⊗ i,∗ j∗ ⊗
linearity and Leibnitz rule.
InthecasethatX ,...,X areself-adjoint, weshallmaketheconvention that∂ X = 1 1 1,
1 n i j i=j
⊗
while ∂ X = 0 for all j (in other words, we shall continue to think of all quantities as functions
i, j
∗
of X ,...,X alone).
1 n
NotethatbytakingX = Y+ǫZ andlettingǫgoingtozerowefindthatforanyboundedoperator
Z and any operator Y with norm strictly smaller than M, the condition that V is (c,M)-convex
implies that
m m
∂ D V(Y)♯Z Z +Z (∂ D V(Y)♯Z ) 2cZ.Z.
i j i× j∗ j × i j i ∗ ≥
i=1 j=1
XX(cid:0) (cid:1)
We shall say that V C X ,...,X is self-adjoint iff for any self-adjoint variables X =
1 m
∈ h i
(X ,...,X ), [V(X ,...,X )] = V(X ,...,X ). V is self-adjoint (c,M)-convex if V is self-
1 m 1 m ∗ 1 m
adjoint and the above holds once restricted to self-adjoint variables, i.e for any m-tuples of self-
adjoint variables X = (X ,...,X ) and Y = (Y ,...,Y ) living in some C -algebra ( , )
1 m 1 m ∗
A k·k∞
5
which are bounded in norm by M, we have
[DV(X) DV(Y)].(X Y) c(X Y).(X Y)
− − ≥ − −
In this case X.Y = X,Y := 1 m (X Y +Y X ) is simply the anti-commutator of X and Y.
{ } 2 i=1 i i i i
If we specialize this assumption to matrices and consider to bethe algebra of N N matrices
with complex entries equipped wPith the usual involution (A )A = A¯ and the spectral×norm ,
∗ ij ji
k·k∞
we find that if V is self-adjoint (c,M)-convex, (X ) Tr[V(X)] is strictly convex on the set of
ij i j
≤ →
entries where X is Hermitian and with spectral radius bounded by M since
1
TrV(X) TrV(Y) = Tr(DV(αX +(1 α)Y).(X Y))dα
− − −
Z0
1
= Tr(DV(Y).(X Y))+ Tr([DV(αX +(1 α)Y) DV(X)].(X Y))dα
− − − −
Z0
c
Tr(DV(Y).(X Y))+ Tr((X Y)2).
≥ − 2 −
Taking Y = (X +Z)/2 and X to be X or Z and summing the resulting inequalities gives
X +Z
TrV(X)+TrV(Z) 2TrV cTr((X Z)2)
− 2 ≥ −
(cid:18) (cid:19)
and hence the Hessian of (X ) TrV(X) is bounded below by cI, at least on matrices X with
ij i j
≤ →
norm bounded by M.
This kind of hypothesis was shown to be very useful in [20]. The interest in relaxing the
hypothesis of convexity to hold in a bounded domain is related with matrix models where V =
1X.X+W with W = n β q for some monomials and complex parameters (β ) Cn. It is
2 i=1 i i i 1≤i≤n ∈
clear now that for all M finitewe can choose the β ’s sufficiently small so that V is (1/2,M)-convex
i
P
(whereas it would not work with no bounds). Indeed, in that case
(DV(X) DV(Y)).(X Y) = (X Y).(X Y)+(DW(X) DW(Y)).(X Y)
− − − − − −
But when the norms of X and Y are bounded by M,
(DW(X) DW(Y)).(X Y) C(M)max β (X Y).(X Y)
i
| − − | ≤ i | | − −
with C(M) a constant which only depends on M and the q . Hence, we can now choose t small
i
enough so that C(M)max t < 1/2 and so V is then (1/2,M)-convex. This is analogous to what
i i
| |
was done in [20] in case of non-convex interaction; it was shown that then if one adds a cut-off, the
large N expansion is still valid provided the parameters in W are small enough.
HereafterweassumethatV is(c,M)-convex. Welet( , ,φ)beanon-commutativeprobability
A ∗
spacegenerated byafreeBrownian motionS (wereferto[5,4]foranintroductiontofreeBrownian
motion and its related free Itˆo calculus). We shall denote by the operator norm in ( ,φ).
k·k∞ A
We prove (see Lemma 2.1 and Theorem 2.2):
Theorem 1.1. Let V be a (c,M)-convex polynomial in X ,...,X .
1 m
Then there exist M = M (c, DV(0).DV(0) ), B = B (c, DV(0).DV(0) ), and b =
0 0 0 0
k k∞ k k∞
b(c, DV(0).DV(0) ,M) B finite constants, so that whenever M M and Z is an m-tuple
0 0
k k∞ ≥ ≥
with Z < b,
k k
6
i. There exists a unique solution XZ to
t
1
dXZ = dS DV(XZ)dt, t [0,+ ), (4)
t t− 2 t ∈ ∞
with the initial data XZ = Z. Moreover, in this case,
0
XZ M, t [0,+ ),
k t k∞ ≤ ∀ ∈ ∞
limsup XZ B ,
t k t k∞ ≤ 0
→∞
XZ C (Z,S :q [0,t]), t [0,+ ).
t ∈ ∗ q ∈ ∀ ∈ ∞
ii. XZ X0 0 as t .
k t − tk∞ → → ∞
iii. The law of ((XZ) ,XZ) converges to a stationary law µ C X ,...,X ,X ,...,X as
t ∗ t V ∈ h 1 m 1∗ m∗i′
t goes to infinity. µ is the non-commutative law of m variables uniformly bounded by B . If
V 0
V is self-adjoint, µ is the law of m self-adjointvariables.
V
iv. eTqhueatrieosntrwichtiicohnsτtVate=s tµhVa|tCfhoXr1,a...l,lXPmi oCf µXV ,to...C,hXX1,.a.n.,dXamlliisatis1fi,e.s..t,hme Schwinger-Dyson
1 m
∈ h i ∈ { }
τ τ (∂ P)= τ (D VP). (5)
V V i V i
⊗
Moreover, if ν is the law of m variables whose uniform norm is bounded by b and ν|ChX1,...,Xmi
satisfies (5), then ν|ChX1,...,Xmi = τV.
The Schwinger-Dyson equation (5) is exactly the same as the one which characterized the
enumeration of maps (2) from which we deduce that = τ at least for
M(βi,qi)1≤i≤n 12Pmi=1Xi2+Pβiqi
sufficiently small polynomials V. This allows to give some information on thedomain of analyticity
of β = (β ) (P) = (P) (see section 3).
i 1≤i≤n → M(βi,qi)1≤i≤n Mβ
Theorem 1.2. Let V = V = n β q be a polynomial, where β = (β ) are (complex)
β i=1 i i i 1 i n
≤≤
parameters and (q ) are monomials. For c,M positive real numbers and M M with M as
i 1 i n 0 0
in Theorem 1.1, let ≤T(≤c,M) CPn be the interior of the subset of parameters β≥= (β ) such
i 1 i n
⊂ ≤≤
that V is (c,M)-convex.
β
Then , for any P C X ,...,X , β T(c,M) τ (P) is analytic. In particular, β
∈ h 1 mi ∈ → Vβ →
(P) extends analytically to the interior of the set of β ’s where 1 m X2 + β q is (c,M)-
Mβ i 2 i=1 i i i
convex for M M (c) .
0
≥ P P
The laws µ are interesting in their own. In the free probability language we have proved the
V
following.
Theorem 1.3. Let V be a (c,M)-convex potential with M M the constant of Theorem 1.1. If V
0
≥
is self-adjoint, there exists a non-commutative law in C X ,...,X ,X ,...,X with conjuguate
h 1 m 1∗ m∗ i′
variable (D V) . There exists at most one such law satisfying the additional constraint to
i 1 i m
≤≤
be the law of variables bounded by b. For non self-adjoint potential, there a unique law µ
V
∈
C X ,...,X ,X ,...,X which satisfies for all P C X ,...,X ,X ,...,X and all i
h 1 m 1∗ m∗i′ ∈ h 1 m 1∗ m∗i ∈
1,...,m
{ }
7
m m
µ µ ((∂ +∂ )(D +D )P) = µ (D VD P +(D V) D P) (6)
V V i i, i i, V i i i ∗ i,
⊗ ∗ ∗ ∗
i=1 i=1
X X
with (D ,∂ ) the non-commutative deriavtives with respect to X . There exists at most one such
i, i, i∗
∗ ∗
law satisfying the additional constraint to be the law of variables bounded by b.
This statement is deduced from (2.2)(2). Moreover, we let Z be an m-tuple of operators with
law µ . We shall prove that the C -algebra and the von Neumann algebra generated by Z have
V ∗
many properties in common with the one generated by a semi-circular system.
Theorem 1.4. Assume that V is (c,M)-convex with M M the constant of Theorem 1.1. The
0
≥
C -algebra generated by Z is exact, projectionless and its associated von Neumann algebra has the
∗
Haagerup approximation property and admits and embedding into the ultrapower of the hyperfinite
II factor.
1
In particular we have
Corollary 1.5. Assume that V is (c,M)-convex with M M the constant of Theorem 1.1. Let
0
≥
Z be any m-tuple of b-bounded variables with the unique law µ satisfying (2). (a) The algebra
V
C (Z) has no non-trivial projections. (b) The spectrum of any non-commutative *-polynomial P
∗
in the m-tuple Z is connected (in the case that P(Z) is normal, this means that the support of its
spectral measure is connected). (c) If P is any polynomial in Z whose value is self-adjoint, then
the probability measure given by the law of P(Z) has connected support.
2 Existence of free diffusions and convergence to their stationary
measure
We shall show that if M is chosen large enough (depending on c and DV(0).DV(0) ), we can
k k∞
build a bounded solution to some free stochastic differential equation with drift DV provided that
V is (c,M)-convex. This generalizes Langevin dynamics to the context of operators.
Lemma 2.1. Let V be a (c,M)-convex polynomial in X ,...,X . Then there exist finite constants
1 m
M = M (c, DV(0).DV(0) ), B = B (c, DV(0).DV(0) )
0 0 0 0
k k∞ k k∞
b = b(c, DV(0).DV(0) ,M) B
0
k k∞ ≥
so that if M M , and Z is any m-tuple with Z < b, there exists a unique solution X to
0 t
≥ k k
1
dX = dS DV(X )dt, t [0,+ ). (7)
t t t
− 2 ∈ ∞
with the initial data X = Z. Moreover, in this case,
0
X M, t [0,+ ),
t
k k≤ ∀ ∈ ∞
limsup X B ,
t 0
t k k∞ ≤
→∞
X C (Z,S :q [0,+ )), t [0,+ ).
t ∗ q
∈ ∈ ∞ ∀ ∈ ∞
If V is self-adjoint (c,M)-convex, the above results hold under the additional assumption that
X = Z is self-adjoint. In this case, X remains self-adjoint for all t 0.
0 t
≥
8
Proof. We remind the reader (cf. [5]) that if DV is uniformly Lipschitz for the uniform norm,
the existence and uniqueness to (7) is clear by the following Picard argument. For the existence
we consider the sequence Xn, n = 0,1,2,... constructed recursively as follows. Set X0 = Z for all
t t
t, and having defined Xn, let Xn+1 be given by the equation
t t
1
dXn+1 = dS DV(Xn)dt
t t− 2 t
with the initial condition Xn+1 = Z.
0
Subtracting the equations for Xn+1 and Xn from each other, we get for all t 0,
t t ≥
1 t 1 t
Xn+1 Xn DV(Xn) DV(Xn 1) ds DV Xn Xn 1 ds
k t − t k∞ ≤ 2Z0 k s − s− k∞ ≤ 2k kLZ0 k s − s− k∞
where DV denotes the Lipschitz norm of DV
k kL
DV = inf C : DV(Y) DV(Y ) C Y Y .
′ ′
k kL { k − k∞ ≤ k − k∞}
Iterating, we deduce that
DV n tn DV n tn t
kXtn+1−Xtnk∞ ≤ (cid:18)k 2 kL(cid:19) n!suu≤ptkSu+uDV(Z)k∞ ≤ (cid:18)k 2 kL(cid:19) n!(2√t+ 2kDV(Z)k∞)
which proves norm convergence of Xn (note that Xn = Z for all n). Moreover, this limit satisfies
t 0
(7). We also see that Xn C (Z,S :q 0) and therefore the limit X C (Z,S :q 0) as well.
t ∈ ∗ q ≥ t ∈ ∗ q ≥
The proof of uniqueness follows the same lines since any two solutions X ,Y satisfy
t t
1 t
X Y DV X Y ds
t t s s
k − k∞ ≤ 2k kLZ0 k − k∞
which proves that X = Y by Gronwall’s argument.
If V is a self-adjoint polynomial, then D V is also self-adjoint for all i 1,...,m . Therefore,
i
∈ { }
since S is self-adjoint for all t 0, we deduce by induction that Xn is self-adjoint for all n 0
t ≥ t ≥
and all t 0 and so its limit X is also self-adjoint. Hence the solution X of the free stochastic
t t
≥
equation is self-adjoint for all times.
We now return to the case of a (c,M) convex V. By Lemma 3.2 in [5], we can construct a new
function
1
f (X) = DV(X)h( X )
R i
2 k k∞
i
X
so that f is uniformly Lipschitz and f (X) = DV(X) if X R. The Picard argument
R R j
above implies existence and uniqueness of a solution XR to k k∞ ≤
t P
dXR = dS +f (XR)dt
t t R t
for all times t. Clearly, if we show that this solution satisfies (XR) R, it will also be a
jk j tk∞ ≤
solution to the original equation (7) involving DV. Furthermore, if we start with some initial data
P
X with (X ) < R, solutions to (7) always exist for small time (at least up until the time
0 j 0
k k∞
that the operator norm of the solution exceeds R). Thus we may consider a solution up to the time
P
9
that its norm reaches some fixed constant M (with the intent to show that this time is infinite).
By free Itˆo calculus
dX .X = 2X .dS DV(X ).X dt+2dt
t t t t t t
−
= 2X .dS (DV(X ) DV(0)).X dt DV(0).X dt+2dt
t t t t t
− − −
Therefore, for all s 0,
≥
s s
ecsX .X = X .X 2 ectDV(0).X dt ect[(DV(X ) DV(0)).X cX .X ]dt
s s 0 0 t t t t t
− − − −
Z0 Z0
s s
+2 ectdt+2 ect.dS
t
Z0 Z0
s s
X .X ectDV(0).X dt+2c 1ecs+2 ect.dS
0 0 t − t
≤ −
Z0 Z0
where the last inequality holds in the sense of operator and we relied on our hypothesis of (c,M)
convexity. Since also ecsX .X is a non negative operator, we deduce that
s s
A := X .X
s s s
k k∞
s s
1 1
e csA +2c 1 +2 ec(t s)X .dS + DV(0).DV(0) 2 ec(t s)A2dt
≤ − 0 − (cid:13)Z0 − t t(cid:13) k k∞Z0 − t
(cid:13) (cid:13)∞
By Theorem 3.2.1 of [4], we know(cid:13) that the free a(cid:13)nalog of the Burkh¨older-Davis inequality for
(cid:13) (cid:13)
integrals with respect to free Brownian motion holds for the Lp norm even with p = . More
∞
precisely, the following estimate holds:
s s 1 s 1
2 2
ec(t s)X .dS 2√2 e2c(t s)A dt 2√2 ec(t s)A dt .
− t t − t − t
≤ ≤
(cid:13)Z0 (cid:13) (cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19)
(cid:13) (cid:13)∞
Moreover, b(cid:13)y the Cauchy-Sch(cid:13)warz inequality, we obtain the bound
(cid:13) (cid:13)
sec(t−s)At12dt 2 = se12c(t−s)·e12c(t−s)At21dt 2 ≤ 1c sec(t−s)Atdt.
(cid:18)Z0 (cid:19) (cid:18)Z0 (cid:19) Z0
1
Hence, we get the inequality (with C = 4( DV(0).DV(0) 2 +4√2)2 and since A b2m)
c k k∞ 0 ≤
s
A2 4e csm2b4+24c 2+C ec(t s)A dt
s ≤ − − − t
Z0
C s
4e csm2b4+24c 2+ ec(t s)(B 1A2+B)dt
≤ − − 2 − − t
Z0
s
4e csm2b4+24c 2+CB(2c) 1+CB 1 ec(t s)A2dt
≤ − − − − − t
Z0
where B is any positive constant (we just used that for all x, 2x B +B 1x2).
−
≤
We now use Gronwall’s lemma (or simply iterate in the above inequality) to deduce, with
C = 24c 2+CB(c) 1, that
′ − −
C s
A2secs ≤ [4m2b4+C′ecs]+ B eBC(s−u)[4m2b4+C′ecu]du
Z0
10