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 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 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.
We study a penalisation problem for two-dimensional Brownian motion. Starting from the Wiener measure, we consider a family of probability measures obtained by weighting paths by a nonnegative functional $F_t$ depending on $t \geq 0$, $F_t$ being measurable with respect to the $\sigma$-algebra generated by the path up to time $t$. Under suitable assumptions on the penalisation process, we establish the weak convergence of these measures when $t \rightarrow \infty$. The limiting law is identified explicitly in terms of a $\sigma-$finite measure $\mathbf{W}^{(2)}$, which admits a path decomposition involving the last hitting time of a circle. This decomposition plays a central role in the analysis and yields a martingale representation of the limiting measure. where ordering and local time techniques are no longer available. The proofs rely on Laplace transform methods and Tauberian theorems, which replace excursion-theoretic tools and allow a precise identification of the limiting measure and its structural properties.
Joseph Najnudel, Thammadol Tansrivorarat· 0 citations
We examine the modulus of continuity, in the spatial variable, of the local time process of Feller Brownian motion (FBM) on the half-line $[0,\infty)$. Briefly, a FBM is a strong Markov process on $[0,\infty)$ that moves like standard Brownian motion on $[0,\infty)$ up until it first encounters the state $0$. The process returns to $(0,\infty)$, either continuously (like reflecting Brownian motion) or by jumping to a (random) positive state chosen according to a specified measure. The present work is a continuation and application of our earlier work with Michael Marcus on the moduli of continuity for the local times of a Markov process built by piecing together (``rebirthing") the paths of another Markov process with finite lifetime. We first establish a general result on the resolvent and local times for a rebirthed process with a special holding state (the state $0$ for FBM). We show how our earlier approach using the Eisenbaum Isomorphism Theorem on an assemblage of excursions works out in this context. This knowledge is then used as an approximation device to obtain our main result on the exact uniform moduli of continuity for the local time of FBM on a spatial interval of the form $(0,1]$. Extensions are made, under certain conditions, to the more delicate situation of the spatial interval $[0,1]$. We also consider briefly the case of more general diffusions on $[0,\infty)$.
We establish the sharp logarithmic order $(\log N)^{-1/2}$ for the expected $p$-Wasserstein distance, induced by the supremum norm, between the empirical law of $N$ independent copies of a continuous It\^o process and their common path law. We only assume that the initial condition and the drift and diffusion integrands are controlled by a time-uniform random upper bound with a finite $\rho$-moment for some $\rho>p\geq1$. Under this assumption, we use an adaptive random time interval partition argument, which leads to a $(\log n)^{-1/2}$ functional quantization rate. A general transfer principle then converts the quantization estimate into a mean estimate and nonasymptotic deviation bounds for equal-weight empirical laws. Applications include empirical path-law estimates for path-dependent SDEs and a path-space propagation-of-chaos estimate for path-dependent McKean--Vlasov interacting particle systems.
We study the discrepancy of balancing $n$ independent sample paths of fractional Brownian motion with Hurst exponent $H\in(0,1)$ on $[0,1]$, an infinite-dimensional analogue of balancing Gaussian vectors. We establish a phase transition at $H=1/2$: with high probability, the discrepancy is $\Omega(n^{1/2-H})$ and $\mathcal O(n^{1/2-H}(\log n)^{c(H)})$, where $c(H)=H+1/2$ if $H\geq 1/2$ and $c(H)=1/2$ otherwise. At the critical exponent $H=1/2$, we show that the discrepancy is $\Theta(1)$ with constant probability as $n\to\infty$. In this regime, we further characterize the geometry of the solution space by computing the expected number of local minima, establishing an overlap gap property near the existence threshold, and proving its absence at every diverging optimality threshold. We also give randomized polynomial-time algorithms that compute signings with discrepancy $\mathcal O(n^{1/2-H}\sqrt{\log n})$ for $H<1/2$, $\mathcal O((\log n)^{3/2})$ for $H=1/2$, and $\mathcal O(\sqrt{\log n})$ for $H>1/2$, with high probability. Our analysis combines a truncated balancing argument based on a wavelet representation of fractional Brownian motion with probabilistic methods.