arXiv subjects

Evangelos A. Nikitopoulos

Publications and source records attributed to Evangelos A. Nikitopoulos.

11 recordsLinked to original sources

arXiv2026

Noncommutative $L^p$-integrators and stochastic differential equations in $L^p$

We propose a new framework for noncommutative stochastic integration in $L^p$, inspired by Bichteler's notion of $L^p$-integrators, that massively generalizes our previous theory of $L^2$-valued stochastic integration against $L^2$-decomposable processes. Central to our framework is a noncommutative analog of a predictable integrand we call a predictable linear process. We develop basic properties of stochastic integrals against "noncommutative $L^p$-integrators" and establish a useful criterion for a noncommutative stochastic process to be a noncommutative $L^p$-integrator. This criterion applies, in particular, to a rich class of processes we call measured decomposable processes, which includes free Brownian motion and, more generally, the $q$-Brownian motions. We also show that, when specialized appropriately, our framework recovers the classical notion of an $L^p$-integrator up to the fact that the noncommutative theory sees only the modification class of a classical stochastic process. As an application, we use our theory to study noncommutative stochastic differential equations (SDEs) in $L^p$. Under natural local-Lipschitz and boundedness assumptions, we establish the existence and uniqueness of maximal solutions and show that a maximal solution with finite lifetime must blow up in $L^p$ norm. As corollaries, we obtain new existence and uniqueness results for free SDEs and SDEs driven by $q$-Brownian motion. Notably, in the free case, our results apply to coefficients arising through the continuous functional calculus from merely locally Lipschitz scalar functions, as opposed to locally operator-Lipschitz functions.

math.PR↗

arXiv2026

Lexicographic functional calculus and its application to functional calculus calculus

Let $A$ be a unital $C^*$-algebra and $I$ be a symmetrically normed ideal of $A$. I introduce and study lexicographic functional calculus (LFC), a kind of multivariate functional calculus for tuples $(a_1,\ldots,a_m)$ of noncommuting self-adjoint elements of $A$ ''acting in lexicographic order,'' i.e., from left to right, with an element $b_i \in I$ ''inserted'' between the action of $a_i$ and $a_{i+1}$ for each $i=1,\ldots,m-1$. The reason for its introduction is an application to ''functional calculus calculus,'' the differential calculus of maps induced by single-variate (continuous) functional calculus. Specifically, I prove that if $f\colon\mathbb{R}\to\mathbb{C}$ is sufficiently regular and $a\in A_{\mathrm{sa}}:=\{c\in A:c^*=c\}$, then $f_{a,I}(b):=f(a+b)-f(a)\in I$ for all $b\in I_{\mathrm{sa}}:=I\cap A_{\mathrm{sa}}$, the map $f_{a,I}\colon I_{\mathrm{sa}}\to I$ is Fréchet $C^k$, and the $k^{\text{th}}$ Fréchet derivative of $f_{a,I}$ may be written in terms of LFC applied to the $k^{\text{th}}$ divided difference of $f$, a function of $k+1$ variables. This result recovers or vastly generalizes nearly all comparable results in the literature. For example, it simultaneously recovers the following three highly related results on the regularity of the function $f_A\colon A_{\mathrm{sa}}\to A$ defined by $a\mapsto f(a)$: (1) If $A$ is commutative and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; (2) if $A$ is finite dimensional and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; and (3) if $f\colon\mathbb{R}\to\mathbb{C}$ is ''slightly better than $C^k$,'' e.g., belongs to the homogeneous Besov space $\dot{B}_1^{k,\infty}(\mathbb{R})$, then $f_A$ is Fréchet $C^k$ no matter the choice of $A$. Before LFC, there was no unifying framework for results (1)--(3); in particular, there was no single result of which they were all corollaries.

math.FA↗

arXiv2026

Multidimensional stochastic liquidity in Kyle's model of informed trading

We develop a variational formulation of Kyle's model of informed trading that accommodates stochastic liquidity and multiple traded assets. The main equilibrium result is stated first: under a martingale dual condition, a matrix-valued martingale depth process generates a linear-Gaussian equilibrium with stochastic matrix-valued price impact. We derive this martingale from a primal-dual problem, inspired by causal optimal transport, that characterizes the endogenous speed at which the insider injects private information into prices; in general, this problem admits only local martingale optimizers, and the martingale dual condition is the hypothesis that the optimizer is a true martingale. We interpret informed trading as the optimal liquidation of private information and verify the construction in the scalar and common-eigenbasis cases. The fully general matrix-valued case reduces to a coupled matrix FBSDE, which we isolate as the remaining obstruction. Along the way, we establish an independently interesting Doob-Meyer decomposition for general (not necessarily symmetric) matrix-valued submartingales.

q-fin.TR↗

arXiv2026

Asymptotic expansion for transport maps between laws of multimatrix models

We study the large-$N$ behavior of random matrix tuples $Y^N = (Y_1^N,\dots,Y_d^N)$ with joint density proportional to $e^{-N^2 V}$ for some convex function $V$ in non-commuting variables satisfying certain bounds on its second derivative. We give an asymptotic expansion in powers of $1/N^2$ of the trace of noncommutative smooth functions of $Y^N$. We also give an asymptotic expansion for a family of maps $T^N$ that transport the law of a tuple of independent GUE random matrices to the law of $Y^N$ and, as a consequence, show strong convergence for the multimatrix models $Y^N$. Our proof is based on an asymptotic expansion for the heat semigroup associated to the measure, which is expressed in terms of smooth functions of a matrix Brownian motion $(S^{N}_t)_{t \geq 0}$. We introduce spaces of noncommutative smooth functions that unify and generalize the cases of polynomials and single-variable smooth functions and allow the systematic application of asymptotic expansion techniques to multimatrix models with convex interaction.

math.PR↗

arXiv2025

Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systems

We develop an Euler-type method to predict the evolution of a time-dependent probability measure without explicitly learning an operator that governs its evolution. We use linearized optimal transport theory to prove that the measure-valued analog of Euler's method is first-order accurate when the measure evolves ``smoothly.'' In applications of interest, however, the measure is an empirical distribution of a system of stochastic particles whose behavior is only accessible through an agent-based micro-scale simulation. In such cases, this empirical measure does not evolve smoothly because the individual particles move chaotically on short time scales. However, we can still perform our Euler-type method, and when the particles' collective distribution approximates a measure that \emph{does} evolve smoothly, we observe that the algorithm still accurately predicts this collective behavior over relatively large Euler steps, thus reducing the number of micro-scale steps required to step forward in time. In this way, our algorithm provides a ``macro-scale timestepper'' that requires less micro-scale data to still maintain accuracy, which we demonstrate with three illustrative examples: a biological agent-based model, a model of a PDE, and a model of Langevin dynamics.

math.NA↗

arXiv2025

A martingale approach to noncommutative stochastic calculus

We present a new approach to noncommutative stochastic calculus that is, like the classical theory, based primarily on the martingale property. Using this approach, we introduce a general theory of stochastic integration and quadratic (co)variation for a certain class of noncommutative processes, analogous to semimartingales, that includes both the $q$-Brownian motions and classical matrix-valued Brownian motions. As applications, we obtain Burkholder--Davis--Gundy inequalities (with $p \geq 2$) for continuous-time noncommutative martingales and a noncommutative Itô's formula for "adapted $C^2$ maps," including trace $\ast$-polynomial maps and operator functions associated to the noncommutative $C^2$ scalar functions $\mathbb{R} \to \mathbb{C}$ introduced by Nikitopoulos, as well as the more general multivariate tracial noncommutative $C^2$ functions introduced by Jekel, Li, and Shlyakhtenko.

math.OA↗

arXiv2025

Matrix Random Walks and the Lima Bean Law

A matrix random walk is a stochastic process of the form $B_k = (I+A_1)\cdots(I+A_k)$ where $A_j$ are independent ``step'' matrices in $\mathrm{M}_N(\mathbb{C})$. With the right entry-covariance, a rescaled matrix random walk converges to Brownian motion $B(t)$ on a matrix Lie group. In this paper, we study the eigenvalues of such rescaled matrix random walks, as $N\to\infty$ and $k\to\infty$. The standard Brownian motion $W(t)$ on $\mathrm{M}_N(\mathbb{C})$ has independent Gaussian entries at each $t$. It is bi-invariant: mutiplying on the left or right by a unitary does not change the distribution. We prove that the empirical eigenvalue distribution of any matrix random walk $B_k$ with bi-invariant steps $A_j$ and initial distribution converges (for fixed $k$ as $N\to\infty$) to a probability measure on $\mathbb{C}$: the Brown measure of the free probability $\ast$-distribution limit $b_k$ of the random walk. If the steps $A_j$ are identically distributed with normalized Hilbert--Schmidt norm $\|A_j\|_2 = t$, the limit law of eigenvalues is supported on a compact ``lima bean'' shaped region. We explicitly compute the limit measure and region, and characterize their phase transitions as $t$ evolves. We prove that the Brown measure of $b_k$ converges as $k\to\infty$, to the Brown measure of the free multiplicative Brownian motion, assuming only that the steps are bi-invariant and normalized in Hilbert--Schmidt norm. Thus the Brownian motion is the universal limit of rescaled matrix random walks, under very general assumptions on the distribution of steps.

math.PR↗

arXiv2023

Noncommutative $C^k$ functions and Fréchet derivatives of operator functions

Fix a unital $C^*$-algebra $\mathscr{A}$, and write $\mathscr{A}_{sa}$ for the set of self-adjoint elements of $\mathscr{A}$. Also, if $f:\mathbb{R}\to\mathbb{C}$ is a continuous function, then write $f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$ for the operator function $a\mapsto f(a)$ defined via functional calculus. In this paper, we introduce and study a space $NC^k(\mathbb{R})$ of $C^k$ functions $f:\mathbb{R}\to\mathbb{C}$ such that, no matter the choice of $\mathscr{A}$, the operator function $f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$ is $k$-times continuously Fréchet differentiable. In other words, if $f\in NC^k(\mathbb{R})$, then $f$ "lifts" to a $C^k$ map $f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$, for any (possibly noncommutative) unital $C^*$-algebra $\mathscr{A}$. For this reason, we call $NC^k(\mathbb{R})$ the space of noncommutative $C^k$ functions. Our proof that $f_\mathscr{A}\in C^k(\mathscr{A}_{sa};\mathscr{A})$, which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estimates for "multiple operator integrals" (MOIs), is more elementary than the standard approach; nevertheless, $NC^k(\mathbb{R})$ contains all functions for which comparable results are known. Specifically, we prove that $NC^k(\mathbb{R})$ contains the homogeneous Besov space $\dot{B}_1^{k,\infty}(\mathbb{R})$ and the Hölder space $C_{loc}^{k,\varepsilon}(\mathbb{R})$. We highlight, however, that the results in this paper are the first of their type to be proven for arbitrary unital $C^*$-algebras, and that the extension to such a general setting makes use of the author's recent resolution of certain "separability issues" with the definition of MOIs. Finally, we prove by exhibiting specific examples that $W_k(\mathbb{R})_{loc}\subsetneq NC^k(\mathbb{R})\subsetneq C^k(\mathbb{R})$, where $W_k(\mathbb{R})_{loc}$ is the "localized" $k^{th}$ Wiener space.

math.FA↗

arXiv2023

Higher derivatives of operator functions in ideals of von Neumann algebras

Let $\mathscr{M}$ be a von Neumann algebra and $a$ be a self-adjoint operator affiliated with $\mathscr{M}$. We define the notion of an "integral symmetrically normed ideal" of $\mathscr{M}$ and introduce a space $OC^{[k]}(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of functions $\mathbb{R} \to \mathbb{C}$ such that the following result holds: for any integral symmetrically normed ideal $\mathscr{I}$ of $\mathscr{M}$ and any $f \in OC^{[k]}(\mathbb{R})$, the operator function $\mathscr{I}_{\mathrm{sa}} \ni b \mapsto f(a+b)-f(a) \in \mathscr{I}$ is $k$-times continuously Fréchet differentiable, and the formula for its derivatives may be written in terms of multiple operator integrals. Moreover, we prove that if $f \in \dot{B}_1^{1,\infty}(\mathbb{R}) \cap \dot{B}_1^{k,\infty}(\mathbb{R})$ and $f'$ is bounded, then $f \in OC^{[k]}(\mathbb{R})$. Finally, we prove that all of the following ideals are integral symmetrically normed: $\mathscr{M}$ itself, separable symmetrically normed ideals, Schatten $p$-ideals, the ideal of compact operators, and -- when $\mathscr{M}$ is semifinite -- ideals induced by fully symmetric spaces of measurable operators.

math.OA↗

arXiv2023

Itô's formula for noncommutative $C^2$ functions of free Itô processes

In a recent paper, the author introduced a rich class $NC^k(\mathbb{R})$ of "noncommutative $C^k$" functions $\mathbb{R} \to \mathbb{C}$ whose operator functional calculus is $k$-times differentiable and has derivatives expressible in terms of multiple operator integrals (MOIs). In the present paper, we explore a connection between free stochastic calculus and the theory of MOIs by proving an Itô formula for noncommutative $C^2$ functions of self-adjoint free Itô processes. To do this, we first extend P. Biane and R. Speicher's theory of free stochastic calculus -- including their free Itô formula for polynomials -- to allow free Itô processes driven by multidimensional semicircular Brownian motions. Then, in the self-adjoint case, we reinterpret the objects appearing in the free Itô formula for polynomials in terms of MOIs. This allows us to enlarge the class of functions for which one can formulate and prove a free Itô formula from the space originally considered by Biane and Speicher (Fourier transforms of complex measures with two finite moments) to the strictly larger space $NC^2(\mathbb{R})$. Along the way, we also obtain a useful "traced" Itô formula for arbitrary $C^2$ scalar functions of self-adjoint free Itô processes. Finally, as motivation, we study an Itô formula for $C^2$ scalar functions of $N \times N$ Hermitian matrix Itô processes.

math.OA↗

arXiv2023

Multiple operator integrals in non-separable von Neumann algebras

A multiple operator integral (MOI) is an indispensable tool in several branches of noncommutative analysis. However, there are substantial technical issues with the existing literature on the "separation of variables" approach to defining MOIs, especially when the underlying Hilbert spaces are not separable. In this paper, we provide a detailed development of this approach in a very general setting that resolves existing technical issues. Along the way, we characterize several kinds of "weak" operator valued integrals in terms of easily checkable conditions and prove a useful Minkowski-type integral inequality for maps with values in a semifinite von Neumann algebra.

math.OA↗