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 that its unique weak solution has the compact support property. The drift term lies outside the usual regularity assumptions for the Dawson--Girsanov theorem. Instead, the proof is based on a layer decomposition in which the solution is constructed as the monotone limit of sums of super-Brownian motions with random immigration rates determined recursively by the positivity sets of the preceding layers. By comparison, the compact support property extends to a broader class of bounded nonnegative drifts vanishing at the origin. Finally, support-radius estimates yield a comparison of the total mass of $X$ with a squared Bessel process, which allows us to show that $X$ has a positive extinction probability.
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 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...
We consider a fractional Brownian motion $B$ with Hurst index $0<H<1/2$, and its maximiser $\tau$ on $[0,1]$. We show that the rescaled process $a^H(B_{\tau+\,\cdot\,/a}-B_\tau)$ converges in $C_{\mathrm{loc}}(\mathbb R)$ to a limiting tangent law that is $H$-self-similar, supported on nonpositive paths pinned at zero,...
We prove absolute continuity of the running maximum $X^{\ast}_T=\sup_{0\leq s\leq T}X_s$ of one-dimensional time-inhomogeneous L\'evy--It\^o diffusions driven by a Brownian motion and an independent non-truncated pure-jump L\'evy process. Using Bismut's directional Malliavin calculus on the Wiener--Poisson space togeth...
Consider a one-dimensional branching Brownian motion. Let $W_\infty(\beta)$ denote the limit of the additive martingale in the subcritical regime $\lvert \beta\rvert<\beta_c$ and $Z_\infty$ be the limit of the derivative martingale at criticality. Madaule (Stochastic Process. Appl. 126 (2016), no. 2, 470--502) establis...
We consider the stochastic heat equation $\partial_t u(t,x) = \Delta u(t,x) + \dot{W}_\alpha(t,x)$ on a bounded Lipschitz domain with zero Dirichlet boundary condition and zero initial condition, where $\dot{W}_\alpha$ is a Gaussian noise that is white in time and whose spatial covariance is the kernel of $(-\Delta)^{-...
Jing Hu, Cheuk-Yin Lee· 0 citations
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