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Litan Yan

Publications and source records attributed to Litan Yan.

13 recordsLinked to original sources
arXiv2026

Gaussian fluctuation for spatial average of the space--time fractional stochastic heat equation

We study spatial averages of the mild solution to a one-dimensional space--time fractional stochastic heat equation driven by space--time white noise. For fixed \(t>0\), we prove a quantitative central limit theorem for the normalized spatial average over \([-R,R]\): as \(R\to\infty\), its law converges to the standard normal law at rate \(R^{-1/2}\) in total variation distance. The proof relies on the Malliavin--Stein method, combined with precise estimates for the space--time fractional heat kernel and for the Malliavin derivative of the mild solution. We further establish a functional central limit theorem.

math.PR↗
arXiv2024

Temporal quartic variation for non-linear stochastic heat equations with piecewise constant coefficients

We consider a stochastic partial differential equation with piecewise constant coefficients driven by a multiplicative space-time white noise. The existence and uniqueness of the mild solution in Walsh sense is established. We mainly study the limit behavior of the temporal quartic variation of the mild solution. As an application, we deduce a consistent estimator based on corresponding results.

math.PR↗
arXiv2020

Delay-dependent Asymptotic Stability of Highly Nonlinear Stochastic Differential Delay Equations Driven by $G$-Brownian Motion

Based on the classical probability, the stability criteria for stochastic differential delay equations (SDDEs) where their coefficients are either linear or nonlinear but bounded by linear functions have been investigated intensively. Moreover, the dependent stability of the highly nonlinear hybrid stochastic differential equations is recently studied. In this paper, by using the nonlinear expectation theory, we explore the dependent stability of a class of highly nonlinear hybrid stochastic differential delay equations driven by $G$-Brownian motion ($G$-SDDEs). Firstly, we give preliminaries of sublinear expectation. Then, the delay-dependent criteria of the stability and boundedness of solutions to $G$-SDDEs is provided. Finally, an illustrative example is analyzed by the $\varphi$-max-mean algorithm.

math.OC↗
arXiv2016

The quadratic covariation for a weighted fractional Brownian motion

Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-1 0$, we consider the generalized quadratic covariation $\bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}$ defined by $$ \bigl[f(B^{a,b}),B^{a,b}\bigr]^{(a,b)}_t=\lim_{\varepsilon\downarrow 0}\frac{1+a+b}{\varepsilon^{1+b}}\int_\varepsilon^{t+\varepsilon} \left\{f(B^{a,b}_{s+\varepsilon}) -f(B^{a,b}_s)\right\}(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)s^{b}ds, $$ provided the limit exists uniformly in probability. We construct a Banach space ${\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2(\Omega)$ and the generalized Bouleau-Yor identity $$ [f(B^{a,b}),B^{a,b}]^{(a,b)}_t=-\frac1{(1+b){\mathbb B}(a+1,b+1)} \int_{\mathbb R}f(x){\mathscr L}^{a,b}(dx,t) $$ holds for all $f\in {\mathscr H}$, where ${\mathscr L}^{a,b}(x,t)=\int_0^t\delta(B^{a,b}_s-x)ds^{1+a+b}$ is the weighted local time of $B^{a,b}$ and ${\mathbb B}(\cdot,\cdot)$ is the Beta function.

math.PR↗
arXiv2016

Quadratic covariations for the solution to a stochastic heat equation

Let $u(t,x)$ be the solution to a stochastic heat equation $$ \frac{\partial}{\partial t}u=\frac12\frac{\partial^2}{\partial x^2}u+\frac{\partial^2}{\partial t\partial x}X(t,x),\quad t\geq 0, x\in {\mathbb R} $$ with initial condition $u(0,x)\equiv 0$, where $X$ is a time-space white noise. This paper is an attempt to study stochastic analysis questions of the solution $u(t,x)$. In fact, the solution is a Gaussian process such that the process $t\mapsto u(t,\cdot)$ is a bi-fractional Brownian motion seemed a fractional Brownian motion with Hurst index $H=\frac14$ for every real number $x$. However, the properties of the process $x\mapsto u(\cdot,x)$ are unknown. In this paper we consider the quadratic covariations of the two processes $x\mapsto u(\cdot,x),t\mapsto u(t,\cdot)$. We show that $x\mapsto u(\cdot,x)$ admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with "It\^o's integral", but it is not a semimartingale. Moreover, some generalized It\^o's formulas and Bouleau-Yor identities are introduced.

math.PR↗
arXiv2016

An integral functional driven by fractional Brownian motion

Let $B^H$ be a fractional Brownian motion with Hurst index $0 \varepsilon\}}\frac1{B^H_s-a}ds^{2H}\equiv \frac1{\pi}{\mathscr H}{\mathscr L}^H(\cdot,t)(a) $$ in $L^2(\Omega)$ with $ a\in {\mathbb R}, t\geq 0$ and ${\mathscr H}$ denoting the Hilbert transform. We show that $$ {\mathcal C}^H_t(a)=2\left((B^H_t-a)\log|B^H_t-a|-B^H_t+a\log|a| -\int_0^t\log|B^H_s-a|\delta B^H_s\right) $$ for all $a\in {\mathbb R}, t\geq 0$ which is the fractional version of Yamada's formula, where the integral is the Skorohod integral. Moreover, we introduce the following {\it occupation type formula}: $$ \int_{\mathbb R}{\mathcal C}^H_t(a)g(a)da=2H\pi\int_0^t({\mathscr H}g)(B^H_s)s^{2H-1}ds $$ for all continuous functions $g$ with compact support.

math.PR↗
arXiv2014

Derivative for the intersection local time of fractional Brownian Motions

Let $B^{H_1}$ and $\tilde{B}^{H_2}$ be two independent fractional Brownian motions on ${\mathbb R}$ with respective indices $H_i\in (0,1)$ and $H_1\leq H_2$. In this paper, we consider their intersection local time $\ell_t(a)$. We show that $\ell_t(a)$ is differentiable in the spatial variable if $\frac1{H_1}+\frac1{H_2}>3$, and we introduce the so-called {\it hybrid quadratic covariation} $[f(B^{H_1}-\tilde{B}^{H_2}),B^{H_1}]^{(HC)}$. When $H_1<\frac12$, we construct a Banach space ${\mathscr H}$ of measurable functions such that the quadratic covariation exists in $L^2(\Omega)$ for all $f\in {\mathscr H}$, and the Bouleau-Yor type identity $$ [f(B^{H_1}-\tilde{B}^{H_2}),B^{H_1}]^{(HC)}_t=-\int_{\mathbb R}f(a)\ell_t(da) $$ holds. When $H_1\geq \frac12$, we show that the quadratic covariation exists also in $L^2(\Omega)$ and the above Bouleau-Yor type identity holds also for all H\"older functions $f$ of order $\nu>\frac{2H_1-1}{H_1}$.

math.PR↗
arXiv2012

Integral with respect to the $G$-Brownian local time

Let ${\mathscr L}$ be the local time of $G$-Brownian motion $B$. In this paper, we prove the existence of the quadratic covariation $ _{t}$ and the integral $\int_{\mathbb R}f(x){\mathscr L}(dx,t)$. Moreover, a sublinear version of the Bouleau-Yor identity $$ \int_{\mathbb R}f(x){\mathscr L}(dx,t)=- _{t} $$ is showed to hold under some suitable conditions. These allow us to write the It\^o's formula for $C^1$-functions.

math.PR↗
arXiv2012

The Bouleau-Yor identity for a bi-fractional Brownian motion

Let $B$ be a bi-fractional Brownian motion with indices $H\in (0,1),K\in (0,1]$, $2HK=1$ and let ${\mathscr L}(x,t)$ be its local time process. We construct a Banach space ${\mathscr H}$ of measurable functions such that the quadratic covariation $[f(B),B]$ and the integral $\int_{\mathbb R}f(x){\mathscr L}(dx,t)$ exist provided $f\in {\mathscr H}$. Moreover, the Bouleau-Yor identity $$ [f(B),B]_t=-2^{1-K}\int_{\mathbb R}f(x){\mathscr L}(dx,t),\qquad t\geq 0, $$ holds for all $f\in {\mathscr H}$.

math.PR↗
arXiv2011

The generalized quadratic covariation for fractional Brownian motion with Hurst index less than 1/2

Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1/2$. In this paper we study the {\it generalized quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t=\lim_{\epsilon\downarrow 0}\frac{2H}{\epsilon^{2H}}\int_0^t\{f(B^{H}_{s+\epsilon})-f(B^{H}_s)\}(B^{H}_{s+\epsilon}- B^{H}_s)s^{2H-1}ds, $$ where the limit is uniform in probability and $x\mapsto f(x)$ is a deterministic function. We construct a Banach space ${\mathscr H}$ of measurable functions such that the generalized quadratic covariation exists in $L^2$ and the Bouleau-Yor identity takes the form $$ [f(B^H),B^H]_t^{(W)}=-\int_{\mathbb {R}}f(x){\mathscr L}^{H}(dx,t) $$ provided $f\in {\mathscr H}$, where ${\mathscr L}^{H}(x,t)$ is the weighted local time of $B^H$. This allows us to write the fractional It\^{o} formula for absolutely continuous functions with derivative belonging to ${\mathscr H}$. These are also extended to the time-dependent case.

math.PR↗
arXiv2008

Integration with respect to fractional local times with Hurst index $H$ greater than 1/2

Let ${\mathscr L}^H(x,t)=2H\int_0^tδ(B^H_s-x)s^{2H-1}ds$ be the weighted local time of fractional Brownian motion $B^H$ with Hurst index $1/2<H<1$. In this paper, we use Young integration to study the integral of determinate functions $\int_{\mathbb R}f(x){\mathscr L}^H(dx,t)$. As an application, we investigate the {\it weighted quadratic covariation} $[f(B^H),B^H]^{(W)}$ defined by $$ [f(B^H),B^H]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\{f(B^H_{t_{k+1}})-f(B^H_{t_{k}})\}(B^H_{t_{k+1}}-B^H_{t_{k}}), $$ where the limit is uniform in probability and $t_k=kt/n$. We show that it exists and $$ [f(B^H),B^H]^{(W)}_t=-\int_{\mathbb R}f(x){\mathscr L}^H(dx,t), $$ provided $f$ is of bounded $p$-variation with $1\leq p<\frac{2H}{1-H}$. Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.

math.PR↗
arXiv2007

On the linear fractional self-attracting diffusion

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L^2 if $\frac12<H<\frac3{4}$.

math.PR↗