We determine the composition factors of the polynomial representation of DAHA, conjectured by M. Kasatani in [Kasa, Conjecture 6.4.]. He constructed an increasing sequence of subrepresentations in the polynomial representation of DAHA using the “multi-wheel condition'', and conjectured that it is a composition series. On the other hand, DAHA has two degenerate versions called the “degenerate DAHA'' and the “rational DAHA''. The category $\mathcal{O}$ of modules over these three algebras and the category of modules over the $v$-Schur algebra are closely related. By using this relationship, we reduce the determination of composition factors of polynomial representations of DAHA to the determination of the composition factors of the Weyl module $W_v^{(n)}$ for the $v$-Schur algebra. By using the LLT-Ariki type theorem of $v$-Schur algebra proved by Varagnolo-Vasserot, we determine the composition factors of $W_v^{(n)}$ by calculating the upper global basis and crystal basis of Fock space of $U_q(\widehat{\mathfrak{sl}}_\ell)$ when $v$ is a primitive $\ell$-th root of unity.

This result gives a different way from the determination of decomposition number of $W_v^{(n)}$ by H. Miyachi or B. Ackermann via the modular representation theory of the general linear groups.

We extend to abstract Wiener spaces the variational representation \[ \mathbb{E}[e^{F}] = \exp \left( \sup_{v\in\mathcal{H}^a}\mathbf{E} \left[ F(\cdot +v) - \frac{1}{2}\|v\|^2_{\mathbb{H}} \right] \right), \] proved by Boué and Dupuis [1] on the classical Wiener space. Here $F$ is any bounded measurable function on the abstract Wiener space $(\mathbb{W},\mathbb{H},\mu)$, and $\mathcal{H}^a$ denotes the space of $\mathcal{F}_t$-adapted $\mathbb{H}$-valued random fields in the sense of Üstünel and Zakai [11]. In particular, we simplify the proof of the lower bound given in [1, 3] by using the Clark-Ocone formula. As an application, a uniformly Laplace principle is established.

We obtain the tail estimation of the quadratic variation of a local martingale with no assumption with respect to positive jumps. Moreover, applying it, we also discuss a tail property of the first-passage times of stochastic integrals.

Suppose $M$ is a complete n-dimensional manifold, $n\ge 2$, with a metric $\overline{g}_{ij}(x,t)$ that evolves by the Ricci flow $\partial_t \overline{g}_{ij}=-2\overline{R}_{ij}$ in $M\times (0,T)$. For any $0<p<1$, $(p_0,t_0)\in M\times (0,T)$, $q\in M$, we define the $\mathcal{L}_p$-length between $p_0$ and $q$, $\mathcal{L}_p$-geodesic, the generalized reduced distance $l_p$ and the generalized reduced volume $\widetilde{V}_p(\tau)$, $\tau=t_0-t$, corresponding to the $\mathcal{L}_p$-geodesic at the point $p_0$ at time $t_0$. Under the condition $\overline{R}_{ij} \ge -c_1\overline{g}_{ij}$ on $M\times (0,t_0)$ for some constant $c_1>0$, we will prove the existence of a $\mathcal{L}_p$-geodesic which minimize the $\mathcal{L}_p(q,\overline{\tau})$-length between $p_0$ and $q$ for any $\overline{\tau}>0$. This result for the case $p=1/2$ was mentioned and used many times by G. Perelman but no proof of it was given in Perelman's papers on Ricci flow. Let $g(\tau)=\overline{g}(t_0-\tau)$ and let $\widetilde{V}_p^{\overline{\tau}}(\tau)$ be the rescaled generalized reduced volume. Suppose $M$ also has nonnegative curvature operator with respect to the metric $\overline{g}(t)$ for any $t\in (0,T)$ and when $1/2<p<1$, $M$ has uniformly bounded scalar curvature on $(0,T)$. Let $0<c<1$ and let $\tau_0=\min ((2(1-p))^{-1/(2p-1)},t_0)$. For any $1/2\le p<1$ we prove that there exists a constant $A_0\ge 0$ with $A_0=0$ for $p=1/2$ such that $e^{-A_0\tau}\widetilde{V}_p(\tau)$ is a monotone decreasing function in $(0,\overline{\tau}_1)$ where $\overline{\tau}_1=(1-c)\tau_0$ if $1/2<p<1$ and $\overline{\tau}_1=t_0$ if $p=1/2$. When $(M,\overline{g})$ is an ancient $\kappa$-solution of the Ricci flow, we will prove a monotonicity property of the rescaled generalized volume $\widetilde{V}_p^{\overline{\tau}}(\tau)$ with respect to $\overline{\tau}$ for any $1/2\le p<1$. When $p=1/2$, the $\mathcal{L}_p$-length, $\mathcal{L}_p$-geodesic, the $l_p$ function and $\widetilde{V}_p(\tau)$ are equal to the $\mathcal{L}$-length, $\mathcal{L}$-geodesic, the reduced distance $l$ and the reduced volume $\widetilde{V}(\tau)$ introduced by Perelman in his papers on Ricci flow. We will also prove a result on the reduced distance $l$ and the reduced volume $\widetilde{V}$ which was used by Perelman without proof in [18].

A result of R. Durrett, D. Iglehart and D. Miller states that Brownian meander is Brownian motion conditioned to stay positive for a unit of time, in the sense that it is the weak limit, as $x$ goes to $0$, of Brownian motion started at $x>0$ and conditioned to stay positive for a unit of time. We extend this limit theorem to the case of multidimensional Brownian motion conditioned to stay in a smooth convex cone.

In this paper, we consider the strong unique continuation for normal elliptic systems whose coefficients are Gevrey class. By using Lerner's lemma, we prove the Carleman estimate with some weight function.

The joint law of the first hitting time of a half-line, the first hitting place, and some local time for the two-dimensional Brownian motion is studied. The method is based on the continuous-time version of the fluctuation identity (the Wiener-Hopf decomposition) in two dimension.

In this paper, we give a characterization of the direct product of balls by its holomorphic automorphism group. Using a result on the standardization of certain compact group actions on complex manifolds, we show that, for a connected Stein manifold $M$ of dimension $n$, if its holomorphic automorphism group contains a topological subgroup that is isomorphic to the holomorphic automorphism group of the direct product $\mathbf B$ of balls in $\mathbf C^n$, then $M$ itself is biholomorphically equivalent to $\mathbf B$.

Let $P(\omega)$ be a uniqueness polynomial of degree $q$ without multiple zeros whose derivative has mutually distinct $k$ zeros $d_l$ with multiplicities $q_l$ for $l=1, 2, \ldots, k$ respectively, and let $S:=\{a_1, a_2, \cdots, a_q\}$ be the zero set of $P(\omega)$. Under the assumption that $P(d_{l_s})\neq P(d_{l_t})$ $(1\leq l_s &lt; l_t\leq k)$, we give some sufficient conditions for the set $S$ to be a unique range set with some weak value-sharing hypothesis, namely, to satisfy the condition that $\sum_{j=1}^q\nu_{f,m_0)}^{a_j}\equiv\sum_{j=1}^q\nu_{g,m_0)}^{a_j}$ ($m_0\in\mathbb{Z}^+\cup\{\infty\}$) implies $f\equiv g$ for any two nonconstant meromorphic or entire functions $f$ and $g$ on $\mathbb{C}$, which improve a result of H. Fujimoto. Also, we discuss some other related topics.