Greg Markowsky
In this paper we will examine the derivative of intersection local time of Brownian motion and symmetric stable processes in $R^2$. These processes do not exist when defined in the canonical way. The purpose of this paper is to exhibit the correct rate for renormaliztion of these processes.
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On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion
该论文修正了分数布朗运动交局部时导数存在的条件，证明了其指数可积性，并给出了存在条件违反时自交局部时导数的极限定理。
We prove joint Holder continuity and an occupation-time formula for the self-intersection local time of fractional Brownian motion. Motivated by an occupation-time formula, we also introduce a new version of the derivative of self-intersection local time for fractional Brownian motion and prove Holder conditions for this process. This process is related to a different version of the derivative of self-intersection local time studied by the authors in a previous work.
Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion
利用Wiener混沌展开，重新证明了分数布朗运动自相交局部时高阶导数的存在性，并引入Varadhan型重整化以扩展偶次导数的收敛范围。
In this paper we compute the $\frac 43$-variation of the derivative of the self-intersection Brownian local time $γ_t=\int_0^t \int_0^u δ'(B_u-B_s)dsdu\,, t\ge 0$, applying techniques from the theory of fractional martingales.
The existence condition $H<1/d$ for first-order derivative of self-intersection local time for $d\geq3$ dimensional fractional Brownian motion can be obtained in Yu (2021). In this paper, we show a limit theorem under the non-existence critical condition $H=1/d$.
On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes
本文研究 alpha-稳定过程（含布朗运动）相交及自相交局部时间导数的指数可积性，并指出文献中分数布朗运动相关结论的一个反例。
We consider the existence and Hölder continuity conditions for the $k$-th order derivatives of self-intersection local time for $d$-dimensional fractional Brownian motion, where $k=(k_1,k_2,\cdots, k_d)$. Moreover, we show a limit theorem for the critical case with $H=\frac{2}{3}$ and $d=1$, which was conjectured by Jung and Markowsky (2014).
In this article, we obtain sharp conditions for the existence of the high order derivatives ($k$-th order) of intersection local time $ \widehatα^{(k)}(0)$ of two independent d-dimensional fractional Brownian motions $B^{H_1}_t$ and $\widetilde{B}^{H_2}_s$ with Hurst parameters $H_1$ and $H_2$, respectively. We also study their exponential integrability.
We study the object formally defined as γ\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-σ} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-σ} dr ds, where $X_{t}$ is the symmetric stable processes of index $0<β\le 2$ in $R^{d}$. When $β\leσ<\displaystyle\min \Big\{{3\over 2}β, d\Big\}$, this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for $γ$. This is applied to obtain results about stable processes in random potentials.
Let $B_t$ be a one dimensional Brownian motion, and let $α'$ denote the derivative of the intersection local time of $B_t$ as defined in Jay Rosen's work (see references). The object of this paper is to prove the following formula $(1/2)α'_t(x) + (1/2)sgn(x)t = \int_0^t L_s^{B_s - x}dB_s - \int_0^t sgn(B_t - B_u - x) du$ which was given as a formal identity by Rosen without proof.
The derivative of self-intersection local time (DSLT) for Brownian motion was introduced by Rosen and subsequently used by others to study the $L^2$ and $L^3$ moduli of continuity of Brownian local time. A version of the DSLT for fractional Brownian motion (fBm) has also appeared in the literature. In the present work, we further its study by proving a Tanaka-style formula for it. In the course of this endeavor we provide a Fubini theorem for integrals with respect to fBm. The Fubini theorem may be of independent interest, as it generalizes (to Hida distributions) similar results previously se…
We study Poissonian ensembles of Markov loops and the associated renormalized self-intersection local times.
Let $\{B_{t}\}_{t\geq0}$ be a fractional Brownian motion with Hurst parameter $\frac{2}{3}<H<1$. We prove that the approximation of the derivative of self-intersection local time, defined as \begin{align*} α_{\varepsilon} &= \int_{0}^{T}\int_{0}^{t}p'_{\varepsilon}(B_{t}-B_{s})\text{d}s\text{d}t, \end{align*} where $p_\varepsilon(x)$ is the heat kernel, satisfies a central limit theorem when renormalized by $\varepsilon^{\frac{3}{2}-\frac{1}{H}}$. We prove as well that for $q\geq2$, the $q$-th chaotic component of $α_{\varepsilon}$ converges in $L^{2}$ when $\frac{2}{3}<H<\frac{3}{4}$, and sat…
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 $L^2(Ω)$ for all $f\in {\ma…
Asymptotic Properties of the Derivative of Self-Intersection Local Time of Multidimensional Fractional Brownian Motion
针对d维分数布朗运动，证明自相交局部时一阶导数近似在L²意义下存在的充要条件为H<3/[2(1+d)]，并给出不同H范围下该近似归一化后的中心极限定理。
Central limit theorems for the derivatives of self-intersection local time for $d$-dimensional Brownian motion
本文证明了$d$维布朗运动自交局部时高阶导数近似在适当重整化下满足中心极限定理，完整回答了Markowsky的猜想，并推广到Wiener混沌分量。
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