P. BIANE, R. SPEICHER ; Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Prob. Th. Rel. Fields, 112 (1998), 373-409.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....case and WN (t) SN (t) in the symmetric case. Then, WN (1) is a complex (respectively, real) Wigner matrix. Let t denote the spectral measure of XN (t) DN WN (t) note that 1 = then t can be studied by use of It o s calculus as we now explain. It was proved in [3, 6] that : satis es an It o s formula (in the special case where D = 0 , assumption which is in fact clearly irrelevant) Then, if we denote, for any f; g 2 C (IR [0; 1] any s t 2 [0; 1] and any : 2 C( 0; 1] P(IR) f) f(x; t)d t (x) f(x; s)d s (x) u f(x; ....

....=2 Na Choosing = c = and = N it follows (applying the above once for hm and once for hm ) that XN (h m )j aN ) 2 c Since is arbitrary, 5.14) follows, thus completing the proof of the lemma. 6 Free Probability: Properties of V ( and I( We have following [3] and [4] that if we let s be the di erential operator on C (IR) with values in C given by s f(x) x y s (y) then (1.1) reads s f s (x) s x f s (x) 6.1) Let M(IR) denote the space of nite, complex, Borel measures on IR. Consider the following vector subspaces of C ....

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P. BIANE, R. SPEICHER ; Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Prob. Th. Rel. Fields, 112 (1998), 373-409.

....case and WN (t) SN (t) in the symmetric case. Then, WN (1) is a complex (respectively, real) Wigner matrix. Let t denote the spectral measure of XN (t) DN WN (t) note that 1 = then t can be studied by use of It o s calculus as we now explain. It was proved in [3, 6] that : satis es an It o s formula (in the special case where D = 0 , assumption which is in fact clearly irrelevant) Then, if we denote, for any f; g 2 C b (IR [0; 1] any s t 2 [0; 1] and any : 2 C( 0; 1] P(IR) f) f(x; t)d t (x) f(x; s)d s (x) u f(x; ....

....=2 Na Choosing = c = and = N it follows (applying the above once for hm and once for hm ) that XN (h m )j aN ) 2 c Since is arbitrary, 5.14) follows, thus completing the proof of the lemma. 6 Free Probability: Properties of V ( and I( We have following [3] and [4] that if we let s be the di erential operator on C (IR) with values in C given by s f(x) x y s (y) then (1.1) reads s f s (x) s x f s (x) 6.1) Let M(IR) denote the space of nite, complex, Borel measures on IR. Consider the following vector subspaces of C ....

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P. BIANE, R. SPEICHER ; Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Prob. Th. Rel. Fields, 112 (1998), 373-409.

....tries to develop a free theory which goes parallel to classical probability theory. Astonishingly, this analogy is very far reaching and there exist a lot of (non trivial) free counterparts of classical results. In the following I want to illuminate this general statement by a recent joint work [BSp1, BSp2] with Philippe Biane on free di usion. 3.1. Classical di usion. Let me rst explain what I mean with the corresponding classical notion. If V : R R is a suciently nice function (called potential in the following) one can consider the classical di usion in this potential. On one side there is a ....

....p 1. Whereas such kind of estimates are also true for other kind of stochastic calculi, a very speci c feature of the free calculus is that one can also derive L 1 estimates, i.e. one can estimate the integrals in operator norm. FREE PROBABILITY THEORY AND FREE DIFFUSION 9 Theorem 3.3.1. [BSp1]) Let (A t ) t 0 and (B t ) t 0 be adapted processes. Then we have k Z A t dS t B t k 2 p 2 Z kA t k 2 kB t k 2 dt 1=2 : 21) Having established the existence of the free stochastic integrals in nice topologies one can continue to investigate the corresponding stochastic ....

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P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Probab. Theory Relat. Fields 112 (1998), 373-409.

....also split S t into its two summands and develop a free stochastic calculus for l t and l t , in analogy to the HudsonParthasarathy calculus for a t and a t . This was done by Kummerer and Speicher [4] The free stochastic calculus with respect to S t , which is due to Biane and Speicher [2], however, has some advantages and we will here restrict to that theory. In our presentation we will put the emphasis on two main points: ffl appropriate norms: on a linear level all stochastic calculi have formally the same structure, the main point lies in establishing the integrals with ....

P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, Preprint (1997).

....t, one has X t 2 A t for all t 0, and for all s; t with s t, the element S t Gamma S s is free with A s , and has semi circular distribution of mean zero and variance t Gamma s. Once this brownian motion exists, one can define stochastic integrals of biprocesses with respect to S, as in [BS]. The main results about stochastic integrals that we shall use are the free Burkholder Gundy inequality (Theorem 3.2.1 of [BS] and the free Ito s formula (Theorem 4.1.2 of [BS] or the functional form, see section 4.3) 2.3 Operator Lipschitz function. Let f : R C be a locally bounded ....

....distribution of mean zero and variance t Gamma s. Once this brownian motion exists, one can define stochastic integrals of biprocesses with respect to S, as in [BS] The main results about stochastic integrals that we shall use are the free Burkholder Gundy inequality (Theorem 3.2. 1 of [BS]) and the free Ito s formula (Theorem 4.1.2 of [BS] or the functional form, see section 4.3) 2.3 Operator Lipschitz function. Let f : R C be a locally bounded measurable function, it is called an operator Lipschitz function if there exists a constant K 0 such that (2.3.1) kf(X) Gamma f(Y ....

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P. Biane and R. Speicher, Stochastic calculus with respect to free brownian motion and analysis on Wigner space, Prob. Th. Rel. Fields 112 (1998), 373--409.

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