Makoto Nakashima
We consider one-dimensional stochastic heat equation with nonlinear drift, $\displaystyle \partial_t u=\frac{1}{2}Δu+b(u)u+σ(u)\dot{W}(t,x)$, where $b:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function and $σ:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function with suitable property. We will construct nonnegative solutions to such SPDEs.
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Large Deviations and Phase Transition for Random Walks in Random Nonnegative Potentials
我们建立了大偏差原则和相变的结果淬火和退火设置的最近邻随机游动恒定漂移的随机非负电位$\mathbb Z^d$。我们补充分析的\cite{Zer}，其中的形状定理的李雅普诺夫函数和大偏差原则，在没有漂移实现淬火设置。
Let $A_kA_{k-1}\cdots A_1$ be product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\forall i,j\in\{1,2\},$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim c\varrho(A_k)\varrho(A_{k-1})\cdots \varrho(A_1)$ as $k\rightarrow\infty,$ where $\varrho(A_n)$ is the spectral radius of the matrix $A_n$ and $c\in(0,\infty)$ is some constant, so that the elements of $A_kA_{k-1}\cdots A_1$ can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some deli…
We study the solutions of the stochastic heat equation with multiplicative space-time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is Hölder continuous of index $γ>3/4$, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are Hölder continuous.
By a probabilistic method we provide an explicit fundamental solution of the Cauchy problem associated to the heat equation on the half-line with constant drift and Dirichlet boundary condition at zero.
We define power variation estimators for the drift parameter of the stochastic heat equation with the fractional Laplacian and an additive Gaussian noise which is white in time and white or correlated in space. We prove that these estimators are consistent and asymptotically normal and we derive their rate of convergence under the Wasserstein metric.
We prove the existence and uniqueness of global solutions to the semilinear stochastic heat equation on an unbounded spatial domain with forcing terms that grow superlinearly and satisfy an Osgood condition $\int 1/|f(u)|du = +\infty$ along with additional restrictions. For example, consider the forcing $f(u) = u \log(e^e + |u|)\log(\log(e^e+|u|))$. A new dynamic weighting procedure is introduced to control the solutions, which are unbounded in space.
Gaussian fluctuations for the nonlinear stochastic heat equation with drift
本文研究了具有漂移项的非线性随机热方程空间平均的定量中心极限定理，通过马尔可夫算子和热核乘积估计方法，得出了相关结论。
Consider a random walker on the nonnegative lattice, moving in continuous time, whose positive transition intensity is proportional to the time the walker spends at the origin. In this way, the walker is a jump process with a stochastic and adapted jump intensity. We show that, upon Brownian scaling, the sequence of such processes converges to Brownian motion with inert drift (BMID). BMID was introduced by Frank Knight in 2001 and generalized by White in 2007. This confirms a conjecture of Burdzy and White in 2008 in the one-dimensional setting.
Pathwise uniqueness for stochastic heat and damped equations with Hölder continuous drift
本文在不要求结构条件成立的前提下，证明了无穷维随机微分方程的逐路径唯一性，并覆盖最高到三维的随机热方程、一维随机阻尼波动方程及最高到三维的随机Euler-Bernoulli阻尼梁方程。
In this paper, we study the stochastic heat equation in the spatial domain $\mathbb{R}^d$ subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies {\it Dalang's condition}. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. As a first application of these moments bounds, we find the necessary and sufficient conditions for the solution to have phase transit…
In this work, we study ergodicity of continuous time Markov processes on state space $\mathbb{R}_{\geq 0} := [0,\infty)$ obtained as unique strong solutions to stochastic equations with jumps. Our first main result establishes exponential ergodicity in the Wasserstein distance, provided the stochastic equation satisfies a comparison principle and the drift is dissipative. In particular, it is applicable to continuous-state branching processes with immigration (shorted as CBI processes), possibly with nonlinear branching mechanisms or in Lévy random environments. Our second main result establis…
We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space $\mathcal{B}^β_{q,\infty}({\mathbb R})$, $D\subset{\mathbb R}$ and $\dot W$ is a space-time white noise on ${\mathbb R}_+\times D$. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever $β-1/q\ge-1$, $β>-1$ and $q\in[1,\infty]$. This class includes equations with $b$ being measures, in particular, $b=δ_0$ which corresponds to the …
We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochas…
We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular weight.
Quantitative heat kernel estimates for diffusions with distributional drift
本文研究了具有分布漂移的扩散过程，通过证明热核上下界，建立了关于时间t和漂移项b范数的显式依赖关系。
Heat kernel estimates for kinetic SDEs with drifts being unbounded and in Kato's class
本文证明了具有Hölder扩散和Kato类无界奇异漂移项的动理学随机微分方程弱解的存在唯一性，建立了密度的精确双边估计，并将结果应用于二阶奇异相互作用粒子系统。
This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\partial u /\partial t - \frac{1}{2}Δu = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the noise $\dot{W}$ is a centered and spatially homogeneous Gaussian noise that is white in time. Using the moment formulas obtained in [9, 10], we identify a set of conditions on the initial data, the correlation measure and the weight function $ρ$, which will together guarantee the existence of an invariant measure in the weighted space $L^2_ρ(\mathbb{R}^d)$. In pa…
We construct diffusions with values in the nonnegative orthant, normal reflection along each of the axes, and two pairs of local drift/variance characteristics assigned according to rank; one of the variances is allowed to vanish, but not both. The construction involves solving a system of coupled Skorokhod reflection equations, then ``unfolding'' the Skorokhod reflection of a suitable semimartingale in the manner of Prokaj (Statist. Probab. Lett. 79 (2009) 534-536). Questions of pathwise uniqueness and strength are also addressed, for systems of stochastic differential equations with reflecti…
Heat equation with singular drift which has bounded solution discontinuous at an interior point
本文构造了一个带奇异漂移的一维热方程，其光滑初值柯西问题的解有界但在内部公共点不连续。
We consider a stochastic flow $ϕ_t(x,ω)$ in $\mathbb{R}^n$ with initial point $ϕ_0(x,ω)=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{ F(\|ϕ_t(x)\|)}{\|ϕ_t(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^*,c^*>0$ not depending on $n$, such that if $F>C^*n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F \leq c^*n^{3/4}$, and if the initial set has nonemp…
We consider a stochastic heat equation driven by a space-time white noise and with a singular drift, where a local-time in space appears. The process we study has an explicit invariant measure of Gibbs type, with a non-convex potential. We obtain existence of a Markov solution, which is associated with an explicit Dirichlet form. Moreover we study approximations of the stationary solution by means of a regularization of the singular drift or by a finite-dimensional projection.
Heat kernel estimates for stable-driven SDEs with distributional drift
本文研究了带有Besov分布漂移的α-稳定过程驱动的SDE，通过光滑化和Duhamel展开结合Besov空间性质，证明了鞅解密度在特定条件下具有双侧热核界和梯度估计。
Blowup for the multiplicative stochastic heat equation with superlinear drift
研究了具有超线性漂移的乘法随机热方程在不同区域的爆破条件，证明了有限时间爆破的判定条件及即时爆炸的机制。
Compact Support Property of Super-Brownian Motion with Irregular Drift
该研究针对带不规则漂移的一维超布朗运动对应的随机偏微分方程，通过层分解方法证明其弱解的紧支集性质，推广了适用漂移类，并证明解具有正灭绝概率。
We prove upper and lower bounds of the heat kernel for the operator $Δ-\nabla (\frac{1}{|x|^α})\cdot \nabla $ in $\mathbb{R}^{n}\setminus\{0} $ where $α>0$. We obtain these bounds from an isoperimetric inequality for a measure $\mathrm{e}^{-\frac{1}{|x|^α}}dx$ on $\mathbb{R}^{n}\setminus \{0\}$. The latter amounts to a certain functional isoperimetric inequality for the radial part of this measure.
We study the heat kernel of the supercritical fractional diffusion equation with the drift in the critical Hölder space. We show that such a drift can have point irregularities strong enough to make the heat kernel vanish at a point for all $t>0$.
In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=σ(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $σ$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $α\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-α},\infty)$ and $β\in[α,1]$, $q\in(\frac{d}β,\infty)$. Assume $\|({\mathbb I}-Δ)^{-α/2}b\|_p+\|(-Δ)^{β/2}σ\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of th…
Spatial fluctuation for stochastic heat equation with Hölder coefficients
本文研究带Hölder系数的随机热方程的空间涨落，建立了非负解的结合性、遍历性与中心极限定理，还在γ=1/2时得到矩生成函数极限与解的空间增长下界。
A new proof of a pathwise uniqueness result of Krylov and Röckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift, pathwise continuous dependence on initial conditions may be obtained, by means of this new proof. The proof is formulated in such a way to show that the only major tool is a good regularity theory for the heat equation forced by a function with the same regularity of the drift.
Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow
本文研究了有限马尔可夫链上非负测度的Wasserstein几何，提出广义热方程作为离散熵的梯度流，并证明其收敛性。
We prove the existence of density for the solution to the multiplicative semilinear stochastic heat equation on an unbounded spatial domain, with drift term satisfying a half-Lipschitz type condition. The methodology is based on a careful analysis of differentiability for a map defined on weighted functional spaces.
Nonnegative solutions to nonlocal parabolic equations
本文研究带可测系数的一般非局部抛物方程的非负整体解，证明 Widder 型定理，并用纯变分方法给出基本解双边界与新颖的 Harnack 型估计。
Strong rate of convergence for the Euler--Maruyama scheme of SDEs with unbounded Hölder continuous drift coefficient
本文研究了具有无界Hölder连续漂移系数的SDEs中Euler-Maruyama方案的强收敛速率，采用Itô-Tanaka技巧和热核估计进行分析。
Exact Temporal Variation for Fractional Stochastic Heat Equation Driven by Space-Time White Noise
本文研究分数随机热方程解的时间过程的精确分数变分，应用该结果估计漂移参数。
Analytically weak and mild solutions to stochastic heat equation with irregular drift
本研究调整随机热方程弱解与温和解的定义，将“正则漂移下两类解等价”的结论推广到广义函数漂移情形，还证明p≥1时L_p(ℝ)漂移下方程存在唯一两类解，拓展了经典结果。
On Heat kernel Estimtes for Brownian SDEs with Distributional Drift
本文研究了带有分布漂移的布朗运动SDE的热核估计和正则性估计，通过考虑非退化和退化噪声（如动能型模型）在Young regime中的情况，得出了Schauder估计，并证明了奇异SDE的弱well-posed性。
Rearranged Stochastic Heat Equation: Ergodicity and Related Gradient Descent on ${\mathcal P}({\mathbb R})$
本文研究实数概率测度空间中的重排随机热方程，通过引入漂移项建立带无穷维公共噪声的 McKean-Vlasov 型方程，证明解的唯一存在性、指数遍历性，并揭示噪声趋于零时梯度下降在局部极小值附近的亚稳态行为。
Let $α\in(0,2)$ and $d\in{\mathbb N}$. Consider the following SDE in ${\mathbb R}^d$:$${\rm d}X_t=b(t,X_t){\rm d} t+a(t,X_{t-}){\rm d} L^{(α)}_t,\ \ X_0=x,$$where $L^{(α)}$ is a $d$-dimensional rotationally invariant $α$-stable process, $b:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ and $a:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ are H{ö}lder continuous functions in space, with respective order $β,γ\in (0,1)$ such that $(β\wedge γ)+α>1$, uniformly in $t$. Here $b$ may be unbounded.When $a$ is bounded and uniformly elliptic, we show that the unique solution $X…
Wright-Fisher stochastic heat equations with irregular drifts
本文研究具有不规则漂移项的 Wright-Fisher 随机热方程，利用分支-聚生布朗运动对偶技术证明了在噪声系数非 Lipschitz 且退化时乘性白噪声的正则化效应及解的弱存在唯一性。
Sectional curvature and matrix displacement convexity
该研究证明黎曼流形截面曲率非负等价于熵张量的矩阵位移凸性，还将此结论应用于强化HWI不等式等结果，且内在维数Wasserstein收缩不等式可刻画非负截面曲率
Well-posedness for Wright--Fisher SPDEs with measurable drift
本文证明一类具有可测漂移的 Wright--Fisher 随机热方程解的弱存在性与唯一性，通过随机对偶方法推广了噪声正则化结果。
Multilevel Picard scheme for solving high-dimensional drift control problems with state constraints
针对高维带状态约束的漂移控制问题，提出多层Picard近似方案，证明其克服维数灾难，数值实验验证维度至少20可行。
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