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 $a$, $b$, $c$ may be random, and the noise coefficient $\sigma$ vanishes at the origin and is sublinear there. The model case is $\sigma(u)=u^\gamma$ with $\gamma\in(0,1)$. We show that, under a uniqueness-in-law assumption, if the initial datum has a sufficiently light spatial tail, then every nonnegative solution has compact support at every positive time, even though the initial support is not compact. The initial datum may also be a measure, such as a Dirac mass. When $\gamma\in(0,1/2]$, finite initial mass suffices; this covers the super-Brownian case $\gamma=1/2$. When $\gamma\in(1/2,1)$, we identify a polynomial moment condition on the initial state whose order diverges as $\gamma\uparrow1$, quantifying the trade-off between the strength of the noise near zero and the decay of the initial data required for instantaneous shrinking. As a step of independent interest, we establish weak existence of solutions started from measure-valued initial data for non-Lipschitz $\sigma$ and random operators. Our results provide a stochastic counterpart of the instantaneous shrinking phenomenon of Evans and Knerr for deterministic parabolic equations with strong absorption, in which the role of the absorption term is played entirely by the noise.
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 $1 - C \exp(-(log n)^2)$, every particle remains within an epsilon-neighborhood of $S_t$ for all $0<= t<= T$. The result holds from time zero, requires no regularity assumption at the initial spectral edges, and applies to multi-cut supports with macroscopic interior gaps. A key ingredient is a deterministic local resolvent exclusion principle: an $o((n \eta)^(-1))$ comparison of Stieltjes transforms on a complex disc above a real point separated from the reference support excludes eigenvalues from the corresponding real interval. This gives a model-independent mechanism for converting local resolvent estimates into spectral confinement.
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, and rerooting-rescaling invariant: rerooting the limit process at its maximum on any fixed compact interval separated from zero and rescaling again asymptotically reproduces the same law. We also identify the tangent law as the limit of two-sided finite-grid hard-wall laws as the mesh vanishes and both horizons diverge, which can informally be interpreted as conditioning fractional Brownian motion on a nonpositive path. As an application, we consider persistence probabilities for fractional Brownian motion: a tilted variant of $B$ yields a different tangent law with a finite left horizon and an infinite right horizon and we show that \[ \mathbb P(B_t\leq1\text{ for all }0\leq t\leq T) = \big(C+o(1)\big)\,T^{-(1-H)},\quad \text{as }T\to\infty, \] where the leading order coefficient $C\in(0,\infty)$ has an explicit representation in terms of the expected maximum and the tilted tangent law.
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 together with the running-maximum criteria of Song--Xie and Nakagawa--Suzuki, we reduce the problem to constructing an admissible direction $\Theta$. The key is to ensure that the directional derivative $D_\Theta X_t$ is strictly positive for all $t\in(0,T]$. We give explicit directions in two regimes. In the uniformly elliptic case with time-inhomogeneous coefficients, a purely Brownian perturbation yields an explicit positive integral representation for $D_\Theta X_t$, and hence $X^{\ast}_T$ admits a density without truncating the jump component. In a Brownian-degenerate pure-jump model with a bounded deterministic time-dependent jump weight $\kappa(t)$ that may vanish on subintervals, we prove absolute continuity under the minimal nondegeneracy-in-time condition $\int_0^t \kappa(s)^2\,ds>0$ for every $t>0$ and infinite activity of the L\'evy measure. A weighted Poisson positivity lemma is the key new input. Finally, we show that silent initial intervals can create atoms and we derive an explicit atom--density decomposition.
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) established the following convergence \[ \frac{W_\infty(\beta)}{\beta_c-\beta}\xrightarrow[\beta\nearrow \beta_c]{\mathbb{P}} 2Z_\infty. \] The goal of this paper is twofold: firstly, we strengthen this result into an almost sure convergence; secondly, we describe the fluctuations occurring in this convergence by proving \[ \frac{1}{\beta_c-\beta}\left( \frac{W_\infty(\beta)}{\beta_c-\beta} - 2 Z_\infty +2(\beta_c-\beta)\log(\beta_c-\beta) Z_\infty\right) \xrightarrow[\beta\nearrow \beta_c]{(d)} S, \] where, conditionally on $Z_\infty$, $S$ follows a spectrally negative 1-stable distribution with scale and shift parameters proportional to $Z_\infty$. Furthermore, these results are extended to the setting of complex additive martingales and the fluctuations to a multi-dimensional convergence.
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 $(-\Delta)^{-\alpha}$ with $\alpha>0$. We prove that a unique pointwise defined mild solution exists if and only if $\alpha>d/2-1$. In this case, if in addition the domain is $C^2$, we also establish spatial and temporal Holder regularity of the solution. When $d/2-1<\alpha<d/2$, we show that the Holder exponents are optimal and obtain exact local and uniform moduli of continuity, a Chung-type law of the iterated logarithm, and sharp small ball probability estimates for the solution.
Jing Hu, Cheuk-Yin Lee· 0 citations
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