Let $B$ be a standard Brownian motion, $x\ge 0,\, \nu<1$, and $b:[0,\infty)\to\mathbb R$ is a continuous function locally of finite variation starting from 0. We define the perturbed Brownian motion reflected at the boundary $b$ by establishing strong existence and pathwise uniqueness of a solution to the equation \[ W_t=(1-\nu)x+B_t+\nu M_t(W)+\frac12 L_t^0(W-b), \qquad W_t\ge b(t), \] where $M_t(W):=\sup_{0\le s\le t} W_s$ and the process $L^0(W-b)$ is the semimartingale local time at 0 of the process $W-b$. We give a positive result under condition (PB) on the boundary $b$ : for every $T>0$, the upward increment $\sup_{0\le s<t\le T,\,t-s\le h}(b(t)-b(s))^+ = o(\sqrt{h})$ as $h\downarrow 0$. The proof splits into two regimes: the case $\nu<1/2$ is a consequence of the Skorokhod problem in an orthant proved by [Williams 1995], while the case $\nu\ge 1/2$ combines a deterministic comparison estimate and a logarithmic upper bound on the number of completed round-trips, following the strategy of [Chaumont and Doney 1999]. For $\alpha\in(0,1/2)$, we also construct an increasing $\alpha$-H\"older boundary for which no continuous adapted solution starting from zero exists for any $\nu<1$.
We study the long-time behaviour of Brownian motion with drift under a rare-event conditioning of the quadratic additive functional $Z_T=\int_0^T W_s^2ds$, where $(W_t)_{t\geq 0}$ is a Brownian motion with drift $\mu$. More precisely, we consider the conditional law of $W$ given the event $Z_T\le \theta T$, as $T\to\in...
Frank Aurzada, Yuvraj Dutta, Max Wiegand· 0 citations
We study the one-dimensional stochastic partial differential equation \[ d_t X_t(x)=\frac{1}{2}\Delta X_t(x) +b_1\unicode{x1D7D9}_{\{X_t(x)>0\}} +\sqrt{X_t(x)}\dot W(t,x), \] where $b_1>0$, $\dot W$ is space-time white noise, and the initial condition is a nonnegative, compactly supported continuous function. We prove...
We consider beta-Dyson Brownian motion, with $\beta>= 1$, started from a deterministic configuration with uniformly bounded support. Let $\mu_t$ be the semicircular free-convolution flow issued from the initial empirical measure, and set $S_t = supp(\mu_t)$. For every fixed $T, \epsilon>0$, with probability at least $1...
Let $U$ and $H$ be two separable Hilbert spaces and $T>0$. We consider a stochastic differential equation which evolves in the Hilbert space $H$ of the form \begin{align} \label{SDEa} dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal g...
We study instantaneous shrinking of supports for nonnegative solutions of the stochastic partial differential equation \[ \partial_t u=a(t,x)\,\partial_x^2 u + b(t,x)\,\partial_x u + c(t,x)\,u +\sigma(u)\,\xi(t,x), \qquad (t,x)\in(0,\infty)\times\mathbb R, \] where $\xi$ is space-time white noise, the coefficients $a$,...
We prove quantitative pathwise stability estimates for one-dimensional stochastic differential equations driven by a common Brownian motion and a common symmetric $\alpha$-stable process, where $\alpha\in(1,2)$. The comparison is made under a synchronous coupling and is measured by $\sup_{0\le t\le T}\mathbb E|X_t-\wt...
Takuya Nakagawa, Ryoichi Suzuki· 0 citations
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