arXiv subjects

Johanna Weinberger

Publications and source records attributed to Johanna Weinberger.

3 recordsLinked to original sources

arXiv2026

Compact Support Property of Super-Brownian Motion with Irregular Drift

We study the one-dimensional stochastic partial differential equation \[ d_t X_t(x)=\frac{1}{2}ΔX_t(x) +b_1\unicode{x1D7D9}_{\{X_t(x)>0\}} +\sqrt{X_t(x)}\dot W(t,x), \] 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.

math.PR↗

arXiv2026

Weak Existence and Uniqueness for Super-Brownian Motion with Irregular Drift

We establish weak existence and uniqueness for random field solutions of the one-dimensional SPDE \[ d_tX_t = \frac{1}{2}ΔX_t +h(X_t)+ \sqrt{X_t}\dot{W}, \quad t\geq 0,\] where $\dot{W}$ is space-time white noise and $h$ is a bounded drift with $h(0)\geq 0$. The proof relies on an extension of the duality relation of the super-Brownian motion, which allows us to treat a broad class of admissible drifts, including functions that are non-Lipschitz or discontinuous at zero. In particular, well-posedness is derived for certain drifts that are Hölder continuous at zero with exponent $α\in(0,1)$. We also allow discontinuous drifts of the form $h(x) = b_0\unicode{x1D7D9}_{x = 0} + b_1\unicode{x1D7D9}_{x>0},$ where $b_0 \geq 0$, $b_1 \in \mathbb{R}$. Additionally, if $h(0)=0$ and the initial condition is continuous and compactly supported, we show that the Lebesgue measure of the non-zero set of $X$ is finite. The proofs are based on duality. We use a log-Laplace equation, which is perturbed by jump noise as the equation for the dual process, and the jumps of the dual process are allowed to take infinite values. We believe that the results for the dual process are also of independent interest. Under suitable assumptions on $h$ we also prove survival of $X$ with positive probability, using rescaling and comparison to the KPP-equation with branching noise.

math.PR↗

arXiv2024

Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes

We consider the one-dimensional stochastic differential equation \begin{equation*} X_t = x_0 + L_t + \int_0^t μ(X_s)ds, \quad t \geq 0, \end{equation*} where $μ$ is a finite measure of Kato class $K_η$ with $η\in (0,α-1]$ and $(L_t)_{t \geq 0}$ is a symmetric $α$-stable process with $α\in (1,2)$. We derive weak and strong well posedness for this equation when $η\leqα-1$ and $η< α-1$, respectively, and show that the condition $η\leq α-1$ is sharp for weak existence. We furthermore reformulate the equation in terms of the local time of the solution $(X_{t})_{t \geq 0}$ and prove its well posedness. To this end, we also derive a Tanaka-type formula for a symmetric, $α$-stable processes with $α\in (1,2)$ that is perturbed by an adapted, right-continuous process of finite variation.

math.PR↗