18 problems

Let $L_{t}^{s,x}$ be the local time of $\varphi(s)$ at level $x$. Let $(\mathcal{P}_{y})_{y\geq0}$ be the semigroup and let $\mathcal{P}^{+}_{x}$ and $\mathcal{P}^{-}_{-x}$ be the…

Mixed Gaussian limit conjecture. In the general non-zero energy case, this process, standardized by the local time $L_t(\lambda)$, should exhibit an asymptotic mixed Gaussian limit…

Let $L_{t}^{s,x}$ denote the local time of $\varphi(s)$ at level $x$ up to time $t$, and let $\tau^{-,j}_{\ell}(s,x)$ and $\theta^{-,j}_{\ell}(s,x)$ denote the successive downcross…

Let $X$ be $d$-dimensional super-Brownian motion, let $L^x$ be its total occupation local time, and define the random measures $\mathcal{L}^{\lambda}$ on $\mathbb{R}^d$ by … Let…

Let $X=(X_t)_{t\geq 0}$ be a continuous semimartingale, let $I=]a,b[\subset\mathbb{R}_+$ with $b<+\infty$, and let $\gamma:I\to\mathbb{R}$ be a function of class $C^1$. The measure…

Let the inverse local time process of a Markov process have a L'evy measure whose tail is regularly varying. The regularly varying inverse-local-time conjecture. Analogous transien…

The dyadic local-time conjecture. Almost surely, $\widetilde{L}^{\pi_n}_t(x)$ converges uniformly for $(t,x)\in[0,T]\times\mathbb{R}$ to

Let $S(X_t)=\overline{\{x:X(t,x)>0\}}$ be the closed support of $X_t$, let $\partial S(X_t)$ denote its boundary, and let $L_t$ be the local-time measure associated with the bounda…

Let $V$ be a spectrally negative Lévy environment with bounded variation, let $X$ be the associated diffusion, and write $\mathcal{L}^*_X(t)$ for its maximal local time up to time…

Let $X$ be a trajectory for which local time $L_t^x$ is considered, and suppose local time exists as a function of $(t,x)$. A trajectory has locally infinite variation if its varia…

Let $v>\gamma^{\circ}$ and let $\mathbbm{Q}^{U_v}=\lim_{a\to\infty}\mathbbm{W}(\mathord\cdot\mid\mathcal{E}_a^{U_v})$ be any measure supplied by the critical-speed framework. Let…

Let $\mathbbm{Q}^v$ be a limiting conditioned Wiener measure arising from the general framework, and write $L_x(\infty)=\lim_{T\to\infty}L_x(T)$ for the local time at level $x$. Mo…

Let $\mathcal{E}$ be the event that Brownian motion has local time bounded everywhere by $1$, and let $\kappa$ denote the exponent measuring the strength of entropic repulsion thro…

Let $\mathcal{D}=\{w:[1,\infty)\mapsto[1,\infty): w\text{ is increasing to }\infty\}$, let $\mathbb{Q}$ be the recurrent limiting law described above, and define … Here $\tau_h$ is…

Let $\Delta$ be the process considered above, and for $x\in\mathbb{R}$ let $\{t:\Delta_t=x\}$ denote its level set. Khoshnevisan–Lewis conjecture. The Hausdorff dimension of this l…

Let $B=(B(t),t\geq 0)$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$, and let $f\in H_0^{\frac{1}{H}-d}$, where … For $f\in H_0^\beta$, define … The esta…

Let $L_t^x$ denote Brownian local time at position $x$ and time $t$, and define the increment … For $q\geq 1$, set … Let $\alpha_{4,t}$ and $\eta$ be the quantities appearing in th…

Let $f$ satisfy … and let $\mathbb{P}$ be any weak limit of the conditioned Wiener measures $\mathbb{W}_t$. Let $L_t$ denote the local time at the origin at time $t$. Local-time se…