Related papers: Martingale measure associated with the critical $2…
We consider a continuous-time random walk in the quarter plane for which the transition intensities are constant on each of the four faces $(0,\infty)^2$, $F_1=\{0\}\times(0,\infty)$, $F_2=(0,\infty)\times\{0\}$ and $\{(0,0)\}$. We show…
Using grand canonical Monte Carlo simulations, we investigate the percolation behavior of a square well fluid with an ultra-short range of attraction in three dimension (3D) and in confined geometry. The latter is defined through two…
In this paper, we first establish a new fractional magnetohydrodynamic (MHD) coupled flow and heat transfer model for a generalized second-grade fluid. This coupled model consists of a fractional momentum equation and a heat conduction…
Let $\mu_t$ denote the critical derivative Gibbs measure of branching Brownian motion at time $t$. It has been proved by Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470--502) and Maillard and Zeitouni (Ann. Inst. Henri Poincar\'e…
We establish that the family of polymer measures $M^{\theta}_{[s,t]}$ associated with the Stochastic Heat Flow (SHF), indexed by $\theta\in\mathbb{R}$, has a conditional Gaussian Multiplicative Chaos (GMC) structure. Namely, taking the…
By using stochastic calculus for two-parameter processes and chaos expansion into multiple Wiener-It\^o integrals, we define a 2D-stochastic current over the Brownian sheet. This concept comes from geometric measure theory. We also study…
We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under…
Statistically self-similar measures on $[0,1]$ are limit of multiplicative cascades of random weights distributed on the $b$-adic subintervals of $[0,1]$. These weights are i.i.d, positive, and of expectation $1/b$. We extend these cascades…
Recently, a new approach in the fine analysis of stochastic processes sample paths has been developed to predict the evolution of the local regularity under (pseudo-)differential operators. In this paper, we study the sample paths of…
The current resurgence in the phase diagram study beyond the critical point has questioned the conventional belief of supercritical fluid as a single phase with varying properties. On the same line, a novel two-phase approach has been…
We are concerned with a stochastic mean curvature flow of graphs over a periodic domain of any space dimension. We establish existence of martingale solutions which are strong in the PDE sense and study their large-time behavior. Our…
A recently developed model of random walks on a $D$-dimensional hyperspherical lattice, where $D$ is {\sl not} restricted to integer values, is extended to include the possibility of creating and annihilating random walkers. Steady-state…
In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in…
We consider a branching Brownian motion in $\mathbb{R}^d$. We prove that there exists a random subset $\Theta$ of $\mathbb{S}^{d-1}$ such that the limit of the derivative martingale exists simultaneously for all directions $\theta \in…
We introduce a transform on the class of stochastic exponentials for d-dimensional Brownian motions. Each stochastic exponential generates another stochastic exponential under the transform. The new exponential process is often merely a…
In this paper, we consider a class of stochastic optimal control problems with risk constraints that are expressed as bounded probabilities of failure for particular initial states. We present here a martingale approach that diffuses a risk…
We study the stochastic heat flow with constant initial data and analyze its spatial average on the scale of $\varepsilon\ll1$. We prove that the logarithm of the averaged process satisfies a pointwise central limit theorem: After being…
The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both…
In this paper, we study a disordered pinning model induced by a random walk whose increments have a finite $(2+\kappa)$-th moment for some $\kappa>0$. It is known that this model is marginally relevant, and moreover, it undergoes a phase…
A family of exponential martingales of a stochastic Laplacian growth problem is proposed. Stochastic Laplacian growth describes a regularized interface dynamics in a two-fluid system, where the viscous fluid is incompressible at a large…