arXiv subjects

Beom-Seok Han

Publications and source records attributed to Beom-Seok Han.

11 recordsLinked to original sources

arXiv2026

Instantaneous shrinking of supports for stochastic PDEs

We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation \[ \partial_t u=a(t,x)\,\partial_x^2 u + b(t,x)\,\partial_x u + c(t,x)\,u +\sigma(u)\,\xi(t,x), \qquad (t,x)\in(0,\infty)\times\mathbb R, \] where $\xi$ is space-time white noise, the coefficients $a$, $b$, $c$ may be random, and the noise coefficient $\sigma$ vanishes at the origin and is sublinear there. The model case is $\sigma(u)=u^\gamma$ with $\gamma\in(0,1)$. We show that, under a uniqueness-in-law assumption, if the initial datum has a sufficiently light spatial tail, then every nonnegative solution has compact support at every positive time, even though the initial support is not compact. The initial datum may also be a measure, such as a Dirac mass. When $\gamma\in(0,1/2]$, finite initial mass suffices; this covers the super-Brownian case $\gamma=1/2$. When $\gamma\in(1/2,1)$, we identify a polynomial moment condition on the initial state whose order diverges as $\gamma\uparrow1$, quantifying the trade-off between the strength of the noise near zero and the decay of the initial data required for instantaneous shrinking. As a step of independent interest, we establish weak existence of solutions started from measure-valued initial data for non-Lipschitz $\sigma$ and random operators. Our results provide a stochastic counterpart of the instantaneous shrinking phenomenon of Evans and Knerr for deterministic parabolic equations with strong absorption, in which the role of the absorption term is played entirely by the noise.

math.PR↗

arXiv2026

A Multiplicative Neural Network Architecture: Locality and Regularity of Approximation

We introduce a multiplicative neural network architecture in which multiplicative interactions constitute the fundamental representation, rather than appearing as auxiliary components within an additive model. We establish a universal approximation theorem for this architecture and analyze its approximation properties in terms of locality and regularity in Bessel potential spaces. To complement the theoretical results, we conduct numerical experiments on representative targets exhibiting sharp transition layers or pointwise loss of higher-order regularity. The experiments focus on the spatial structure of approximation errors and on regularity-sensitive quantities, in particular, the convergence of Zygmund-type seminorms. The results show that the proposed multiplicative architecture yields residual error structures that are more tightly aligned with regions of reduced regularity and exhibit more stable convergence in regularity-sensitive metrics. These results demonstrate that adopting a multiplicative representation format has concrete implications for the localization and regularity behavior of neural network approximations, providing a direct connection between architectural design and analytical properties of the approximating functions.

math.FA↗

arXiv2024

On the support of solutions to nonlinear stochastic heat equations

We investigate the strict positivity and the compact support property of solutions to the one-dimensional nonlinear stochastic heat equation: $$\partial_t u(t,x) = \frac{1}{2}\partial^2_x u(t,x) + \sigma(u(t,x))\dot{W}(t,x), \quad (t,x)\in \mathbf{R}_+\times\mathbf{R},$$ with nonnegative and compactly supported initial data $u_0$, where $\dot{W}$ is the space-time white noise and $\sigma:\mathbf{R} \to \mathbf{R} $ is a continuous function with $\sigma(0)=0$. We prove that (i) if $v/ \sigma(v)$ is sufficiently large near $v=0$, then the solution $u(t,\cdot)$ is strictly positive for all $t>0$, and (ii) if $v/\sigma(v)$ is sufficiently small near $v= 0$, then the solution $u(t,\cdot)$ has compact support for all $t>0$. These findings extend previous results concerning the strict positivity and the compact support property, which were analyzed only for the case $\sigma(u)\approx u^\gamma$ for $\gamma>0$. Additionally, we establish the uniqueness of a solution and the weak comparison principle in case (i).

math.PR↗

arXiv2024

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 H\"older type regularity of solutions.

math.PR↗

arXiv2023

$L_p$-regularity theory for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

We study the existence, uniqueness, and regularity of the solution to the stochastic reaction-diffusion equation (SRDE) with colored noise $\dot{F}$: $$ \partial_t u = a^{ij}u_{x^ix^j} + b^i u_{x^i} + cu - \bar{b} u^{1+β} + ξu^{1+γ}\dot F,\quad (t,x)\in \mathbb{R}_+\times\mathbb{R}^d; \quad u(0,\cdot) = u_0, $$ where $a^{ij},b^i,c, \bar{b}$ and $ξ$ are $C^2$ or $L_\infty$ bounded random coefficients. Here $β>0$ denotes the degree of the strong dissipativity and $γ>0$ represents the degree of stochastic force. Under the reinforced Dalang's condition on $\dot{F}$, we show the well-posedness of the SRDE provided $γ< \frac{κ(β+1)}{d+2}$ where $κ>0$ is the constant related to $\dot F$. Our result assures that strong dissipativity prevents the solution from blowing up. Moreover, we provide the maximal Hölder regularity of the solution in time and space.

math.PR↗

arXiv2023

The compact support property for solutions to the stochastic partial differential equations with colored noise

We study the compact support property for solutions of the following stochastic partial differential equations: $$\partial_t u = a^{ij}u_{x^ix^j}(t,x)+b^{i}u_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),\quad (t,x)\in (0,\infty)\times{\bf{R}}^d,$$ where $\dot{F}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $h(t, x, u)$ satisfies $K^{-1}|u|^λ\leq h(t, x, u)\leq K(1+|u|)$ for $λ\in(0,1)$ and $K\geq 1$. We show that if the initial data $u_0\geq 0$ has a compact support, then, under the reinforced Dalang's condition on $\dot{F}$ (which guarantees the existence and the Hölder continuity of a weak solution), all nonnegative weak solutions $u(t, \cdot)$ have the compact support for all $t>0$ with probability 1. Our results extend the works by Mueller-Perkins [Probab. Theory Relat. Fields, 93(3):325--358, 1992] and Krylov [Probab. Theory Relat. Fields, 108(4):543--557, 1997], in which they show the compact support property only for the one-dimensional SPDEs driven by space-time white noise on $(0, \infty)\times \bf{R}$.

math.PR↗

arXiv2023

$L_p$-solvability and Hölder regularity for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise

We present the $L_p$-solvability for stochastic time fractional Burgers' equations driven by multiplicative space-time white noise: $$ \partial_t^αu = a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + \bar b^i u u_{x^i} + \partial_t^β\int_0^t σ(u)dW_t,\,t>0;\,\,u(0,\cdot) = u_0, $$ where $α\in(0,1)$, $β< 3α/4+1/2$, and $d< 4 - 2(2β-1)_+/α$. The operators $\partial_t^α$ and $\partial_t^β$ are the Caputo fractional derivatives of order $α$ and $β$, respectively. The process $W_t$ is an $L_2(\mathbb{R}^d)$-valued cylindrical Wiener process, and the coefficients $a^{ij}, b^i, c$ and $σ(u)$ are random. In addition to the existence and uniqueness of a solution, we also suggest the Hölder regularity of the solution. For example, for any constant $T<\infty$, small $\varepsilon>0$, and almost sure $ω\inΩ$, we have $$ \sup_{x\in\mathbb{R}^d}|u(ω,\cdot,x)|_{C^{[ \fracα{2}( ( 2-(2β-1)_+/α-d/2 )\wedge1 )+\frac{(2β-1)_{-}}{2} ]\wedge 1-\varepsilon}([0,T])}<\infty \quad\text{and}\quad \sup_{t\leq T}|u(ω,t,\cdot)|_{C^{( 2-(2β-1)_+/α-d/2 )\wedge1 - \varepsilon}(\mathbb{R}^d)} < \infty. $$ The Hölder regularity of the solution in time changes behavior at $β= 1/2$. Furthermore, if $β\geq1/2$, then the Hölder regularity of the solution in time is $α/2$ times the one in space.

math.PR↗

arXiv2022

Lp-regularity theory for semilinear stochastic partial differential equations with multiplicative white noise

We establish the $L_p$-regularity theory for a semilinear stochastic partial differential equation with multiplicative white noise: $$ du = (a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + \bar b^{i}|u|^λu_{x^i})dt + σ^k(u)dw_t^k,\quad (t,x)\in(0,\infty)\times\bR^d; \quad u(0,\cdot) = u_0, $$ where $λ>0$, the set $\{ w_t^k,k=1,2,\dots \}$ is a set of one-dimensional independent Wiener processes, and the function $u_0 = u_0(ω,x)$ is a nonnegative random initial data. The coefficients $a^{ij},b^i,c$ depend on $(ω,t,x)$, and $\bar b^i$ depends on $(ω,t,x^1,\dots,x^{i-1},x^{i+1},\dots,x^d)$. The coefficients $a^{ij},b^i,c,\bar{b}^i$ are uniformly bounded and twice continuous differentiable. The leading coefficient $a$ satisfies ellipticity condition. Depending on the diffusion coefficient $σ^k(u)$, we consider two different cases; (i) $λ\in(0,\infty)$ and $σ^k(u)$ has Lipschitz continuity and linear growth in $u$, (ii) $λ,λ_0\in(0,1/d)$ and $σ^k(u) = μ^k |u|^{1+λ_0}$ ($σ^k(u)$ is super-linear). Each case has different regularity results. For example, in the case of $(i)$, for $\varepsilon>0$ $$u \in C^{1/2 - \varepsilon,1 - \varepsilon}_{t,x}([0,T]\times\bR^d)\quad \forall T<\infty, $$ almost surely. On the other hand, in the case of $(ii)$, if $λ,λ_0\in(0,1/d)$, for $\varepsilon>0$ $$ u \in C^{\frac{1-(λd) \vee (λ_0 d)}{2} - \varepsilon,1-(λd) \vee (λ_0 d) - \varepsilon}_{t,x}([0,T]\times\bR^d)\quad \forall T<\infty $$ almost surely. It should be noted that $λ$ can be any positive number and the solution regularity is independent of nonlinear terms in case $(i)$. In case $(ii)$, however, $λ,λ_0$ should satisfy $λ,λ_0\in(0,1/d)$ and the regularities of the solution are affected by $λ,λ_0$ and $d$.

math.PR↗

arXiv2022

A regularity theory for stochastic generalized Burgers' equation driven by a multiplicative space-time white noise

We introduce the uniqueness, existence, $L_p$-regularity, and maximal Hölder regularity of the solution to semilinear stochastic partial differential equation driven by a multiplicative space-time white noise: $$ u_t = au_{xx} + bu_{x} + cu + \bar b|u|^λu_{x} + σ(u)\dot W,\quad (t,x)\in(0,\infty)\times\mathbb{R}; \quad u(0,\cdot) = u_0, $$ where $λ> 0$. The function $σ(u)$ is either bounded Lipschitz or super-linear in $u$. The noise $\dot W$ is a space-time white noise. The coefficients $a,b,c$ depend on $(ω,t,x)$, and $\bar b$ depends on $(ω,t)$. The coefficients $a,b,c,\bar{b}$ are uniformly bounded, and $a$ satisfies ellipticity condition. The random initial data $u_0 = u_0(ω,x)$ is nonnegative. We have the maximal Hölder regularity by employing the Hölder embedding theorem. For example, if $λ\in(0,1]$ and $σ(u)$ has Lipschitz continuity, linear growth, and boundedness in $u$, for $T<\infty$ and $\varepsilon>0$, $$u \in C^{1/4 - \varepsilon,1/2 - \varepsilon}_{t,x}([0,T]\times\mathbb{R})\quad(a.s.). $$ On the other hand, if $λ\in(0,1)$ and $σ(u) = |u|^{1+λ_0}$ with $λ_0\in[0,1/2)$, for $T<\infty$ and $\varepsilon>0$, $$u \in C^{\frac{1/2-(λ-1/2) \vee λ_0}{2} - \varepsilon,1/2-(λ-1/2) \vee λ_0 - \varepsilon}_{t,x}([0,T]\times\mathbb{R})\quad (a.s.).$$ It should be noted that if $σ(u)$ is bounded Lipschitz in $u$, the Hölder regularity of the solution is independent of $λ$. However, if $σ(u)$ is super-linear in $u$, the Hölder regularities of the solution are affected by nonlinearities, $λ$ and $λ_0.$

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↗

arXiv2019

Weighted $L_q(L_p)$-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

We present a weighted $L_{q}(L_{p})$-theory ($p,q\in(1,\infty)$) with Muckenhoupt weights for the equation $$ \partial_{t}^αu(t,x)=Δu(t,x) +f(t,x), \quad t>0, x\in \mathbb{R}^d. $$ Here, $α\in (0,2)$ and $\partial_{t}^α$ is the Caputo fractional derivative of order $α$. In particular we prove that for any $p,q\in (1,\infty)$, $w_{1}(x)\in A_p$ and $w_{2}(t)\in A_q$, $$ \int^{\infty}_0\left(\int_{\mathbb{R}^d} |u_{xx}|^p \,w_{1} dx \right)^{q/p}\,w_{2}dt \leq N \int^{\infty}_0\left(\int_{\mathbb{R}^d} |f|^p \,w_{1} dx \right)^{q/p}\,w_{2}dt, $$ where $A_p$ is the class of Muckenhoupt $A_p$ weights. Our approach is based on the sharp function estimates of the derivatives of solutions.

math.AP↗