Free Stochastic Calculus and Free Brownian Motion¶

Job: develop the free Brownian motion (the semicircular process), the Biane–Speicher free Itô integral / isometry / formula, the free Ornstein–Uhlenbeck process / free BDG inequalities, and the large-$N$ limit of the random matrix stochastic calculus (the DBM as the free Brownian motion) — with the free Itô formula for $S(t)^2$ / the DBM as the worked example.

Core¶

Free Brownian motion¶

The free Brownian motion $S(t)$ (the semicircular process, the free analog of the Brownian motion / the Wiener process) in a tracial von Neumann algebra $(M,\tau)$: a self-adjoint process with $S(0)=0$, the increments $S(t)-S(s)$ are semicircular of variance $t-s$ (the free analog of the Gaussian increments), and the increments over disjoint intervals are free (the free analog of the independent increments).

Free Brownian motion	Classical Brownian
$S(t)=S(t)^*$, $S(0)=0$	$B(t)=B(t)^*$, $B(0)=0$
Increments $S(t)-S(s)$ semicircular of variance $t-s$	increments Gaussian of variance $t-s$
Increments over disjoint intervals free	increments over disjoint intervals independent
$S(t)$ has the semicircular distribution of variance $t$	$B(t)$ has the Gaussian distribution of variance $t$
$(S(t_1),\dots,S(t_n))$ for disjoint intervals free semicircular family	$(B(t_1),\dots,B(t_n))$ independent Gaussian family

The free stochastic integral (Biane–Speicher)¶

The free Itô integral $\int_0^T f(t)\,dS(t)$ for adapted processes $f(t)$ (the integrand is in the algebra generated by $S(s)$ for $s\le t$, the free analog of the adapted integrand). The construction: the integral on the simple adapted processes (the step functions), the free Itô isometry $$ \tau\Bigl(\Bigl(\int_0^T f(t)\,dS(t)\Bigr)^2\Bigr)=\int_0^T\tau(f(t)^2)\,dt $$ (the $L^2$-isometry, the free analog of the classical Itô isometry), and the extension to the $L^2$-adapted processes.

Free Itô integral	Classical Itô
$\int f\,dS$ (the free Itô, the semicircular integrator)	$\int f\,dB$ (the classical Itô, the Gaussian integrator)
Free Itô isometry $\tau((\int f\,dS)^2)=\int\tau(f^2)\,dt$	$\mathbb{E}[(\int f\,dB)^2]=\int\mathbb{E}[f^2]\,dt$
Free Itô formula: $dP=\partial_t P\,dt+\partial_S P\,dS+\frac12\partial_S^2 P\,dt$	classical Itô: $dP=\partial_t P\,dt+\partial_B P\,dB+\frac12\partial_B^2 P\,dt$
Quadratic variation $\langle dS\,dS\rangle=dt$	$\langle dB\,dB\rangle=dt$ (the same)

The free Itô formula (the noncommutative Itô formula): for the polynomial $P(S(t),t)$, $$ dP=\partial_t P\,dt+\partial_S P\,dS+\frac12\partial_S^2 P\,dt, $$ the Itô formula with the quadratic variation $\langle dS\,dS\rangle=dt$ (the free analog of the classical Itô with the same quadratic variation — the key structural fact, the free and the classical Itô have the same quadratic variation $dt$).

The free Ornstein–Uhlenbeck process and the free BDG¶

Free process / inequality	Description
Free Ornstein–Uhlenbeck	$dX_t=-\theta X_t\,dt+dS_t$, the stationary distribution the semicircle (the free analog of the OU process)
Free Burkholder–Davis–Gundy	the $L^p$-bounds on the free martingale: $\\|X\\|_p\sim\\|\langle X\rangle^{1/2}\\|_p$ (the free BDG, the analog of the classical BDG)
Free SDEs	the SDEs driven by the free Brownian $S_t$; the existence / uniqueness; the free analog of the classical SDE theory

The free stochastic calculus and random matrices¶

The large-$N$ limit of the random matrix stochastic calculus: the Dyson Brownian motion of the Wigner eigenvalues (the random matrices / universality DBM) is the free Brownian motion of the matrix entries in the large-$N$ limit. The free Itô formula as the limit of the matrix Itô formula: the matrix-valued Itô formula (the $N\times N$ matrix SDE) converges to the free Itô formula (the $N\to\infty$ limit, the asymptotic freeness, the free stochastic calculus as the large-$N$ limit of the matrix stochastic calculus).

Free stochastic calculus	Random matrix calculus
Free Brownian $S(t)$ (the semicircular process)	the Wigner matrix $H_t$ with the GOE Brownian entries (the large-$N$ limit $H_t\to S(t)$)
Free Itô formula	the matrix Itô formula (the $N\times N$ SDE, the $N\to\infty$ limit)
DBM (the eigenvalue SDE)	the large-$N$ / the macroscopic limit of the DBM = the free Itô for the matrix $S(t)$
Asymptotic freeness	the free stochastic calculus as the $N\to\infty$ limit of the matrix stochastic calculus

See it¶

Free Itô formula for $S(t)^2$ and the DBM. The free Brownian motion $S(t)$ in $(M,\tau)$, the free Itô formula for $S(t)^2$: $$ d(S^2)=S\,dS+dS\,S+\langle dS\,dS\rangle=S\,dS+dS\,S+dt $$ (the free Itô with the quadratic variation $dt$ — note the noncommutativity: $S\,dS\ne dS\,S$, the free Itô keeps both terms $S\,dS+dS\,S$ plus the quadratic variation $dt$).

The large-$N$ limit: the $N\times N$ matrix Brownian $H_t$ (the Wigner matrix with the GOE Brownian entries, $dH_t=dG_t$ the GOE increment) has $d(H^2)=H\,dH+dH\,H+I\,dt$ (the matrix Itô, the $I\,dt$ the quadratic variation from the GOE normalization), and in the large-$N$ limit (the asymptotic freeness, the $N\to\infty$), this converges to the free Itô formula (the $S\,dS+dS\,S+dt$ with the free Brownian $S$ and the identity $I\to 1$ the trace). The Dyson Brownian motion: the eigenvalues $\lambda_i(t)$ of $H_t$ satisfy the DBM SDE (the Coulomb repulsion $1/(\lambda_i-\lambda_j)$ + the noise $\sqrt{1/N}\,dB_i$), and the free Itô formula is the large-$N$ / the macroscopic limit of the DBM (the $S(t)=H_t$ the free Brownian, the $S(t)^2$ the second moment / the semicircle evolution — the macroscopic / the trace dynamics, not the individual eigenvalues). The key point: the free stochastic calculus is the large-$N$ limit of the random matrix stochastic calculus, with the free Itô formula recovering the macroscopic eigenvalue dynamics (the DBM / the semicircle evolution).

Check¶

Verify the free Itô formula for $S(t)^2$: $d(S^2)=S\,dS+dS\,S+dt$, and contrast with the classical $d(B^2)=2B\,dB+dt$.

The free Itô formula: $dP(S)=\partial_S P\,dS+\frac12\partial_S^2 P\,dt$ for $P(S)=S^2$. $\partial_S(S^2)=S\otimes1+1\otimes S$ (the free difference quotient, the two terms — the noncommutative derivative gives both $S\,dS$ and $dS\,S$), and $\partial_S^2(S^2)=2\otimes1$ (the second derivative, the quadratic variation $\frac12\cdot 2\,dt=dt$). So $d(S^2)=(S\,dS+dS\,S)+dt$, with both $S\,dS$ and $dS\,S$ (the noncommutativity — the order matters, both terms kept). The classical: $d(B^2)=2B\,dB+dt$ (the commutativity $B\,dB=dB\,B$, the two terms combine into $2B\,dB$, the classical Itô). The difference: the free Itô keeps $S\,dS+dS\,S$ (the noncommutative, the order), the classical combines into $2B\,dB$ (the commutative, the order irrelevant). The quadratic variation $dt$ is the same.

Explain why the free Itô quadratic variation $\langle dS\,dS\rangle=dt$ is the same as the classical, and what this says about the free stochastic calculus.

The free Itô quadratic variation $\langle dS\,dS\rangle=\tau(dS\,dS)=\mathrm{Var}(dS)=dt$ (the variance of the semicircular increment $dS=S(t+dt)-S(t)$ is $dt$, the semicircular of variance $dt$, so $\tau((dS)^2)=dt$ — the quadratic variation). The classical $\langle dB\,dB\rangle=\mathbb{E}[(dB)^2]=dt$ (the Gaussian variance). The two are the same (the variance of the increment is $dt$ in both), even though the distribution differs (semicircular vs Gaussian). This says the free stochastic calculus has the same quadratic variation / the same second-order structure as the classical, but the noncommutativity (the $S\,dS\ne dS\,S$) makes the first-order terms different — the free Itô keeps the order, the classical combines. The quadratic variation being the same is the structural reason the free and the classical Itô formulas have the same form (the $dt$ term).

Pitfalls¶

The free Itô formula has the same quadratic variation as the classical ($\langle dS\,dS\rangle=dt$), but the free stochastic calculus is the noncommutative / the large-$N$ limit (not the classical Itô — the $S\,dS\ne dS\,S$ keeps the order, the free is the noncommutative analog).
The free Brownian motion is the semicircular process (not the Gaussian — the free analog, the different distribution: the semicircle, not the Gaussian, the free CLT limit).
The free Itô integral / the free Itô isometry is the $L^2(\tau)$-isometry (the trace, not the expectation), the noncommutative / the tracial analog of the classical $L^2(\mathbb{P})$-isometry.
The free OU / the free BDG are the analogs of the classical OU / BDG, but the stationary distribution is the semicircle (not the Gaussian), and the constants differ.
The large-$N$ limit of the random matrix stochastic calculus is the free stochastic calculus (the asymptotic freeness, the $N\to\infty$), but the finite-$N$ matrix Itô is the classical matrix Itô (the commutative matrix algebra, the finite-$N$ is not free), only the large-$N$ limit is free.
The DBM as the free Brownian motion is the macroscopic / the trace dynamics (the $S(t)^2$ the second moment), not the individual eigenvalue dynamics (the DBM is the $N$ eigenvalues, the free Itô is the macroscopic / the trace — the two are related by the large-$N$ limit, not the same).

Refs¶

Biane–Speicher, "Stochastic calculus with respect to free Brownian motion" (Theory Probab. Appl. 1998)
Biane, "Free Brownian motion, free stochastic calculus and convolution" (Sém. Probab. 1997)
Voiculescu, "The analogues of entropy and of Fisher's information" (the free Brownian / the free stochastic calculus)
Cabanal-Duvillard–Guionnet, "Large deviations and stochastic calculus for large random matrices" (the large-$N$ limit)
Erdős–Yau, A Dynamical Approach to Random Matrix Theory (the DBM as the large-$N$ limit of the free Brownian)

Next¶

FP index · Foundations · Free Entropy & Fisher · Operator-Algebraic Freeness · Subordination & Harmonic Analysis · RMT / universality · Operator algebras

Free Brownian motion	Classical Brownian
\(S(t)=S(t)^*\), \(S(0)=0\)	\(B(t)=B(t)^*\), \(B(0)=0\)
Increments \(S(t)-S(s)\) semicircular of variance \(t-s\)	increments Gaussian of variance \(t-s\)
Increments over disjoint intervals free	increments over disjoint intervals independent
\(S(t)\) has the semicircular distribution of variance \(t\)	\(B(t)\) has the Gaussian distribution of variance \(t\)
\((S(t_1),\dots,S(t_n))\) for disjoint intervals free semicircular family	\((B(t_1),\dots,B(t_n))\) independent Gaussian family

Free Itô integral	Classical Itô
\(\int f\,dS\) (the free Itô, the semicircular integrator)	\(\int f\,dB\) (the classical Itô, the Gaussian integrator)
Free Itô isometry \(\tau((\int f\,dS)^2)=\int\tau(f^2)\,dt\)	\(\mathbb{E}[(\int f\,dB)^2]=\int\mathbb{E}[f^2]\,dt\)
Free Itô formula: \(dP=\partial_t P\,dt+\partial_S P\,dS+\frac12\partial_S^2 P\,dt\)	classical Itô: \(dP=\partial_t P\,dt+\partial_B P\,dB+\frac12\partial_B^2 P\,dt\)
Quadratic variation \(\langle dS\,dS\rangle=dt\)	\(\langle dB\,dB\rangle=dt\) (the same)