The thesis presents two applications of evolution equations for non-commutative variables to the theory of non-commutative probability and von Neumann algebras.

In the first part, non-commutative processes $(X_t)_{t \in [0,T]}$ with boolean, free, monotone, or anti-monotone independent increments, under certain continuity and boundedness assumptions, are classified in terms of certain evolution equations for their $F$-transforms $F_{X_t}(z) = (E[(z - X_t)^{-1}])^{-1}$. This classification is done in the setting of operator-valued non-commutative probability, in which the expectation takes values in a $\mathrm{C}^*$-algebra $\mathcal{B}$ rather than $\mathbb{C}$. Thus, the $F$-transform is a function of an operator variable $z$ from (matrices over) $\mathcal{B}$, and it is understood through the theory of fully matricial or non-commutative functions, an operator-valued analogue of complex analysis. The classification of these processes generalizes previous work on the L {e}vy-Hin\v{c}in formula for processes with independent and stationary increments, and it leads to Bercovici-Pata-type bijections between the processes with independent increments for the four different types of independence. We also describe a canonical model for each process with independent increments using operators on a Fock space. In fact, the interaction between operator models and analytic function theory is a major theme of the first part, and leads to a new ``coupling'' technique to prove estimates for the non-commutative central limit theorem and for Loewner chains.

In the second part, we strengthen the probabilistic, information-theoretic, and transport-theoretic connections between asymptotic random matrix theory and tracial $\mathrm{W}^*$-algebras through the study of functions and differential equations for several non-commuting self-adjoint variables. We consider a random variable $X^{(n)}$ in $M_n(\mathbb{C})_{\text{sa}}^d$ given by a probability distribution

\[

d\mu^{(n)}(x) = \frac{1}{\int e^{-n^2 V^{(n)}}} e^{-n^2 V^{(n)}(x)}\,dx,

\]

where $V^{(n)}: M_n(\C)_{sa}^d \to \R$ is uniformly convex and semi-concave. We assume that $(\nabla V^{(n)})_{n \in \N}$ is asymptotically approximable by trace polynomials, which means that $\nabla V^{(n)}$ behaves asymptotically like some element $f$ from a certain space of ``functions of $d$ self-adjoint variables from a tracial $\mathrm{W}^*$-algebra.''

Then we show first that $X^{(n)}$ almost surely converges in non-commutative law to some $d$-tuple $X$ from a tracial $\mathrm{W}^*$-algebra $(\mathcal{M},\tau)$, meaning that $(1/n) \Tr(p(X^{(n)})) \to \tau(p(X))$ almost surely for every non-commutative polynomial $p$ (which is comparable to earlier known results). The strategy to prove convergence of the expectation $E[(1/n) \Tr(p(X^{(n)}))]$ is to show that the heat semigroup associated to the measure $\mu^{(n)}$ preserves asymptotic approximability by trace polynomials. The same method leads to a new conditional version of this result, which shows that if $k < d$ and if $(f^{(n)})$ is asymptotically approximable by polynomials, then so is the function $g^{(n)}$ given by $g^{(n)}(X_1^{(n)},\dots,X_k^{(n)}) = E[f^{(n)}(X^{(n)}) | X_1^{(n)}, \dots, X_k^{(n)}]$. Understanding the large-$n$ behavior of such conditional expectations is a key step in showing our second main result that the classical entropy of $X^{(n)}$, after renormalization, converges to Voiculescu's non-microstates free entropy $\chi^*(X)$ (and an analogous result for conditional entropy given $X_1^{(n)}$, \dots, $X_k^{(n)}$). In particular, we obtain a new proof of the result from a 2017 preprint of Dabrowski that $\chi^*(X)$ agrees with the microstates free entropy $\chi(X)$ for any $X$ that arises from such random matrix models.

The final main result studies the large-$n$ behavior of certain functions $F^{(n)}$ that transport the measure $\mu^{(n)}$ to the distribution $\sigma_1^{(n)}$ of a standard Gaussian self-adjoint $d$-tuple $Z^{(n)}$. The transport map $F^{(n)}$ is obtained by the same construction as in Otto and Villani's famous proof of the Talagrand inequality based on heat semigroups and transport equations. Using successive conditioning, we can obtain a transport function $F^{(n)}$ that is lower-triangular in the sense that

\[

F^{(n)}(x_1,\dots,x_d) = (F_1^{(n)}(x_1), F_2^{(n)}(x_1,x_2),\dots,F_d^{(n)}(x_1,\dots,x_d)),

\]

where $x = (x_1,\dots,x_d) \in M_n(\C)^d$. We show that $F^{(n)}$ is asymptotically approximable by trace polynomials as $n \to \infty$, and consequently, in the large-$n$ limit, we obtain an isomorphism $\mathrm{W}^*(X_1,\dots,X_d) \to \mathrm{W}^*(Z_1,\dots,Z_d)$ that maps $\mathrm{W}^*(X_1,\dots,X_k)$ to $\mathrm{W}^*(Z_1,\dots,Z_k)$ for every $k = 1, \dots, d$. As an application, we show that this statement holds when $X$ itself is given by $X_j = Z_j + \delta p_j(Z)$ where $Z$ is a free semicircular $d$-tuple, $p_j$ is a self-adjoint non-commutative polynomial, and $\delta$ is sufficiently small, depending on $p_1$, \dots, $p_d$.