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Jae-Hwan Choi

Publications and source records attributed to Jae-Hwan Choi.

18 recordsLinked to original sources
arXiv2026

Well-Posedness for Cauchy Problems with Singular Time-Measurable Pseudo-Differential Operators in Quasi-decreasing Weighted $\mathrm{L}_2$-Spaces

This study examines Cauchy problems governed by highly singular, time-measurable pseudo-differential operators (singular measurable families of Fourier multipliers). We show that the symbols of these operators can exhibit arbitrary blow-up behavior. In particular, we prove the existence and uniqueness of solutions even when the symbols grow super-exponentially in time and frequency. As a concrete application, we solve evolutionary equations driven by fractional Laplacians of any negative order. Additionally, we establish unique strong solutions under the sole condition that the symbol is locally integrable in frequency, even in the presence of severe blow-up at the initial time.

math.AP↗
arXiv2026

Neural Network Approximation of Solutions to Fractional Parabolic Partial Differential Equations

We establish a dimension-efficient neural network approximation theory for solutions to fractional parabolic equations with lower-order drift and potential terms. By introducing anisotropic spectral Barron spaces, which measure temporal and spatial regularity separately in frequency space, we first develop a dimension-independent maximal regularity theory for these equations, using dimension-independent multiplication estimates and the method of continuity to incorporate the lower-order terms. A key technical novelty is the application of the Vandermonde matrix to the global-in-time extension of the finite-time fractional heat semigroup with sufficient regularity at the initial time, thereby enabling analysis of the forward-in-time evolution via the global space-time Fourier structure of anisotropic Barron norms. We also show that a corresponding uniform-in-time estimate of the spectral Barron regularity generally fails. Finally, we derive $n^{-1/2}$ two-layer approximation bounds in mixed Sobolev norms for non-constant periodic activations and, under additional anisotropic Barron regularity, for non-periodic activations satisfying a polynomial-decay condition.

math.AP↗
arXiv2026

The stochastic Keller--Segel system in critical spaces

We study stochastic, parabolic-parabolic Keller--Segel equations on the $d$-dimensional torus in scaling critical Besov spaces, for $d \geq 3$. Using stochastic maximal regularity estimates, we prove local well-posedness of the equation, i.e., that there exists a unique solution up to a maximal time of existence. We show that the time of existence can be made arbitrarily large with arbitrarily high probability, provided that the initial data is sufficiently small in these critical spaces. Contribution to the Oberwolfach Report for the Seminar 'Stochastic Partial Differential Equations in Critical Spaces' organized by Antonio Agresti and Mark Veraar.

math.PR↗
arXiv2026

Neural Optimal Transport in Hilbert Spaces: Characterizing Spurious Solutions and Gaussian Smoothing

We study Neural Optimal Transport in infinite-dimensional Hilbert spaces. In non-regular settings, Semi-dual Neural OT often generates spurious solutions that fail to accurately capture target distributions. We analytically characterize this spurious solution problem using the framework of regular measures, which generalize Lebesgue absolute continuity in finite dimensions. To resolve ill-posedness, we extend the semi-dual framework via a Gaussian smoothing strategy based on Brownian motion. Our primary theoretical contribution proves that under a regular source measure, the formulation is well-posed and recovers a unique Monge map. Furthermore, we establish a sharp characterization for the regularity of smoothed measures, proving that the success of smoothing depends strictly on the kernel of the covariance operator. Empirical results on synthetic functional data and time-series datasets demonstrate that our approach effectively suppresses spurious solutions and outperforms existing baselines.

cs.LG↗
arXiv2025

A regularity theory for second-order parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces

We develop an optimal regularity theory for parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces. The results extend the classical Schauder estimates to coefficients that are merely measurable in time and to the critical case of integer-order regularity. In addition, nonzero initial data are treated in the optimal trace space via a sharp trace theorem.

math.AP↗
arXiv2025

Characterizations of weighted Besov and Triebel-Lizorkin spaces with variable smoothness

In this paper, we study different types of weighted Besov and Triebel-Lizorkin spaces with variable smoothness. The function spaces can be defined by means of the Littlewood-Paley theory in the field of Fourier analysis, while there are other norms arising in the theory of partial differential equations such as Sobolev-Slobodeckij spaces. It is known that two norms are equivalent when one considers constant regularity function spaces without weights. We show that the equivalence still holds for variable smoothness and weights, which is accomplished by making use of shifted maximal functions, Peetre's maximal functions, and the reverse Hölder inequality. Moreover, we obtain a weighted regularity estimate for time-fractional evolution equations and a generalized Sobolev embedding theorem without weights.

math.CA↗
arXiv2025

A regularity theory for an initial value problem with a time-measurable pseudo-differential operator in a weighted $L_p$-space

In this study, we investigate the existence, uniqueness, and maximal regularity estimates of solutions to homogeneous initial value problems involving time-measurable pseudo-differential operators within the framework of weighted mixed norm Lebesgue spaces. The class of temporal weights in our regularity estimates contains Muckenhoupt's class, and the initial data is in weighted Besov spaces with variable order.

math.AP↗
arXiv2025

An existence and uniqueness result to evolution equations with sign-changing pseudo-differential operators and its applications to logarithmic Laplacian operators and second-order differential operators without ellipticity

We broaden the domain of the Fourier transform to contain all distributions without using the Paley-Wiener theorem and devise a new weak formulation built upon this extension. This formulation is applicable to evolution equations involving pseudo-differential operators, even when the signs of their symbols may vary over time. Notably, our main operator includes the logarithmic Laplacian operator $\log (-Δ)$ and a second-order differential operator whose leading coefficients are not positive semi-definite.

math.AP↗
arXiv2025

Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions

This article investigates the existence, uniqueness, and regularity of solutions to nonlinear stochastic reaction-diffusion-advection equations (SRDAEs) with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions in mixed norm $L_q(L_p)$-spaces. We introduce a new condition (strongly reinforced Dalang's condition) on colored noise, which facilitates a deeper understanding of the complicated relation between nonlinearities and stochastic forces. Additionally, we establish the space-time Hölder type regularity of solutions.

math.PR↗
arXiv2025

A regularity theory for evolution equations with space-time anisotropic non-local operators in mixed-norm Sobolev spaces

In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The primary focus is on space-time non-local equations where the spatial operator is the infinitesimal generator of a vector of independent subordinate Brownian motions, making it the sum of subdimensional non-local operators. A representative example of such an operator is $(Δ_{x})^{β_{1}/2}+(Δ_{y})^{β_{2}/2}$. We establish existence, uniqueness, and precise estimates for solutions in corresponding Sobolev spaces. Due to singularities arising in the Fourier transforms of our operators, traditional methods involving Fourier analysis are not directly applicable. Instead, we employ a probabilistic approach to derive solution estimates. Additionally, we identify the optimal initial data space using generalized real interpolation theory.

math.AP↗
arXiv2024

An existence and uniqueness theory to stochastic partial differential equations involving pseudo-differential operators driven by space-time white noise

In this paper, we aim to develop a new weak formulation that ensures well-posedness for a broad range of stochastic partial differential equations with pseudo-differential operators whose symbols depend only on time and spatial frequencies. The main focus of this paper is to relax the conditions on the symbols of pseudo-differential operators and data while still ensuring that the stochastic partial differential equations remain well-posed in a weak sense. Specifically, we allow symbols to be random and remove all regularity and ellipticity conditions on them. As a result, our main operators include many interesting rough operators that cannot generate any regularity gain or integrability improvement from the equations. In addition, our data do not need to be regular or possess finite stochastic moments.

math.PR↗
arXiv2024

A regularity theory for parabolic equations with anisotropic non-local operators in $L_{q}(L_{p})$ spaces

In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $$ \partial_t u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ Here, $\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: $$ \mathcal{L}^{\vec{a},\vec{0}}(t)u(x)=\sum_{i=1}^{d} \int_{\mathbb{R}}\left( u(x^{1},\dots,x^{i-1},x^{i}+y^{i},x^{i+1},\dots,x^{d}) - u(x) \right) \frac{a_{i}(t,y^{i})}{|y^{i}|^{1+α_{i}}} \mathrm{d}y^{i} . $$ To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators.

math.AP↗
arXiv2023

On the trace theorem to Volterra-type equations with local or non-local derivatives

This paper considers traces at the initial time for solutions of evolution equations with local or non-local derivatives in vector-valued $L_p$ spaces with $A_p$ weight. To achieve this, we begin by introducing a generalized real interpolation method. Within the framework of generalized interpolation theory, we make use of stochastic process theory and two-weight Hardy's inequality to derive our trace and extension theorems. Our results encompass findings applicable to time-fractional equations with broad temporal weight functions.

math.AP↗
arXiv2023

A weighted $L_p$-regularity theory for parabolic partial differential equations with time measurable pseudo-differential operators

We obtain the existence, uniqueness, and regularity estimates of the following Cauchy problem \begin{equation}\label{ab eqn} \begin{cases} \partial_t u(t,x)=ψ(t,-i\nabla)u(t,x)+f(t,x),\quad &(t,x)\in(0,T)\times\mathbb{R}^d,\\ u(0,x)=0,\quad & x\in\mathbb{R}^d \end{cases} \end{equation} in (Muckenhoupt) weighted $L_p$-spaces with time-measurable pseudo-differential operators \begin{equation} \label{ab op} ψ(t,-i\nabla)u(t,x):=\mathcal{F}^{-1}\left[ψ(t,\cdot)\mathcal{F}[u](t,\cdot)\right](x). \end{equation} More precisely, we find sufficient conditions of the symbol $ψ(t,ξ)$ (especially depending on the smoothness of the symbol with respect to $ξ$) to guarantee that equation is well-posed in (Muckenhoupt) weighted $L_p$-spaces. Here the symbol $ψ(t,ξ)$ is merely measurable with respect to $t$, and the sufficient smoothness of $ψ(t,ξ)$ with respect to $ξ$ is characterized by a property of each weight. In particular, we prove the existence of a positive constant $N$ such that for any solution $u$ to the equation, \begin{equation} \label{ab est} \int_0^T \int_{\mathbb{R}^d} |(-Δ)^{γ/2} u(t,x) |^p (t^2 + |x|^2)^{α/2} \mathrm{d}x\mathrm{d}t \leq N\int_0^T \int_{\mathbb{R}^d} |f(t,x)|^p (t^2 + |x|^2)^{α/2} \mathrm{d}x\mathrm{d}t \end{equation} and \begin{equation} \label{ab est 2} \int_0^T \left(\int_{\mathbb{R}^d} |(-Δ)^{γ/2} u(t,x) |^p |x|^{α_2} \mathrm{d}x \right)^{q/p} t^{α_1}\mathrm{d}t \leq N\int_0^T \left(\int_{\mathbb{R}^d} |f(t,x) |^p |x|^{α_2} \mathrm{d}x \right)^{q/p} t^{α_1}\mathrm{d}t, \end{equation} where $p,q\in(1,\infty)$, $-d-1<α< (d+1)(p-1)$, $-1 < α_1 < q-1$, $-d <α_2< d(p-1)$, and $γ$ is the order of the operator $ψ(t,-i\nabla)$.

math.AP↗
arXiv2023

Sobolev regularity theory for the non-local elliptic and parabolic equations on $C^{1,1}$ open sets

We study the zero exterior problem for the elliptic equation $$ Δ^{α/2}u-λu=f, \quad x\in D\,; \quad u|_{D^c}=0 $$ as well as for the parabolic equation $$ u_t=Δ^{α/2}u+f, \quad t>0,\, x\in D \,; \quad u(0,\cdot)|_D=u_0, \,u|_{[0,T]\times D^c}=0. $$ Here, $α\in (0,2)$, $λ\geq 0$ and $D$ is a $C^{1,1}$ open set. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and Hölder estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and Hölder regularities of solutions and their arbitrary derivatives using a system of weights consisting of appropriate powers of the distance to the boundary. The range of admissible powers of the distance to the boundary is sharp.

math.AP↗
arXiv2023

A maximal $L_p$-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes

Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,Λ_t)_{t\geq0}$. Suppose that for any Schwartz function $φ$ on $\mathbb{R}^d$ whose Fourier transform is in $C_c^{\infty}(B_{c_s} \setminus B_{c_s^{-1}} )$, there exist positive constants $N_0$, $N_1$, and $N_2$ such that \begin{equation*} \int_{\mathbb{R}^d}|\mathbb{E}[φ(x+r^{-1}Z_t)]|dx\leq N_0 e^{- \frac{N_1 t}{s(r)}},\quad \forall (r,t)\in(0,1)\times[0,T], \end{equation*} and $$ \|ψ^μ(r^{-1}D)φ\|_{L_1(\mathbb{R}^d)}\leq \frac{N_2}{s(r)},\quad \forall r\in(0,1). $$ where $s$ is a scaling function (Definition 2.4), $c_s$ is a positive constant related to $s$, $μ$ is a symmetric Lévy measure on $\mathbb{R}^d$, $ψ^μ(r^{-1}D)φ(x)= \mathcal{F}^{-1} \left[ ψ^μ(r^{-1}ξ) \mathcal{F}[φ]\right](x)$ and $$ψ^μ(ξ):=\int_{\mathbb{R}^d}(e^{iy\cdotξ}-1-iy\cdotξ1_{|y|\leq 1})μ(dy).$$ In this paper, we establish the $L_p$-solvability to the initial value problem \begin{equation} \frac{\partial u}{\partial t}(t,x)=\mathcal{A}_Z(t)u(t,x),\quad u(0,\cdot)=u_0,\quad (t,x)\in(0,T)\times\mathbb{R}^d, \end{equation} In other words, there exists a unique solution $u$ to equation satisfying $$ \|u\|_{L_q((0,T);H_p^{μ;γ}(\mathbb{R}^d))}\leq N\|u_0\|_{B_{p,q}^{s;γ-\frac{2}{q}}(\mathbb{R}^d)}, $$ where $N$ is independent of $u$ and $u_0$, and the spaces $B_{p,q}^{s;γ-\frac{2}{q}}(\mathbb{R}^d)$ and $H_p^{μ;γ}(\mathbb{R}^d)$ are scaled Besov spaces (see Definition 2.8) and generalized Bessel potential spaces (see Definition 2.3), respectively.

math.PR↗
arXiv2021

A regularity theory for stochastic partial differential equations with a super-linear diffusion coefficient and a spatially homogeneous colored noise

Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+ξ|u|^{1+λ}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d, $$ where $λ\geq 0$ and the coefficients depend on $(ω,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $λ$, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of $F$ and the spatial dimension $d$.

math.PR↗