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 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 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 study Brownian motion in R^d conditioned so that the time average of a continuous confining potential remains below a fixed level. On every fixed initial time interval, we prove that the conditioned process converges in total variation to the ground-state diffusion associated with a suitable Schr\"odinger operator. We also obtain sharp asymptotics for the probability of the conditioning event, including bounded perturbations of the constraint. The proof is based on a local limit theorem for the corresponding Feynman-Kac measures. Our results extend the previously known one-dimensional quadratic case to arbitrary finite dimension and a broad class of confining potentials, therefore resolving a conjecture of Aurzada, Lifshits and Schickentanz. The presented approach also works when Brownian motion is replaced by suitable reversible Markov processes, including multidimensional Ornstein-Uhlenbeck processes, CIR processes and continuous-time Markov chains.
We initiate a geometric theory of $p$-Brownian motion, the nonlinear Markov process associated with the $p$-Laplacian introduced by Barbu-Rehmeier-R\"ockner. More precisely, we analyze its radial processes thoroughly, relative to an arbitrary center and in every dimension. On the one hand, we show explicit Tanaka--Meyer semimartingale formulas; our consequential characterization of nontriviality of the associated local times reveals notable differences to classical Brownian motion. In parallel, we establish sharp exit time estimates as well as scaling estimates for the self-similarly rescaled radial process. We also prove isometry of the corresponding marginal laws in each Wasserstein distance. Our contributions equally cover the Leibenson process - the nonlinear Markov process associated with the"porous medium equation"with $p$-Laplacian - recently introduced by Barbu-Grube-Rehmeier-R\"ockner.
We construct and characterize the one-parameter family of real-valued processes that allow to recover the law of a Brownian loop-soup of any intensity on the real line conditionally on its occupation time field. These processes generalize the Brownian burglar constructed by Warren and Yor (that corresponds to our process in the limiting case where the intensity vanishes and there is just one Brownian motion). These processes are closely related to the Bass-Burdzy flow and recent work of A\"id\'ekon, Hu and Shi on the stochastic Jacobi flow. Our approach uses a formalism, building on the notion of driver processes, where one considers the evolving object to be the line that we ``stretch", instead of the burglar itself and its remaining occupation time. This leads to a characterization of these (generalized) Brownian burglars by simple natural axioms. It also allows to treat ``negative intensities''and provides fairly direct derivations of several properties of the burglars, and in particular a ``target-independence/locality"property at the special intensity where the local time is the square of a Gaussian Free Field.
Arthur Dremaux· 0 citations
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