CS23: Stochastic Processes Under Constraints

Organizer: Dominic T. Schickentanz (Paderborn University)

Persistence of Strongly Correlated Stationary Gaussian fields: From Universal Probability Decay to Entropic Repulsion

Ohad Noy Feldheim

We study the persistence event (i.e., the event that the field remains positive on a large ball) for stationary Gaussian fields on \(\mathbb{R}^d\) or \(\mathbb{Z}^d\), whose correlation function is non-summable and asymptotically positively correlated (a spectral singularity at the origin).

Firstly, we give precise log-asymptotics for the persistence probability in terms of capacity, expressed in terms of the order of the singularity at the origin. We then establish that the field conditioned to persist is propelled to height \(C\sqrt{\log T}\) and takes the shape of a deterministic function (the “equilibrium potential”).

This generalises a classical result of Bolthausen, Deuschel and Zeitouni for the Gaussian free field (GFF) on \(\mathbb{Z}^d\), \(d \ge 3\), to a wide class of Gaussian fields with spectral singularity, showing that both the leading order of persistence probability decay and entropic repulsion are, in a sense, universal, depending only on the order of the spectral singularity.

Partially-homogeneous reflected random walk on the quadrant

Andrew Wade

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by finitely-many transition laws near each boundary, together with an interior transition law that applies at sufficient distance from both boundaries. Under mild assumptions, in the (most subtle) setting in which the mean drift in the interior is zero, we classify recurrence and transience and provide power-law bounds on tails of passage times; the classification depends on the interior covariance matrix, the (finitely many) drifts near the boundaries, and stationary distributions derived from two one-dimensional Markov chains associated to each of the two boundaries. As an application, we consider reflected random walks related to multidimensional variants of the Lindley process, for which no previous quantitative results on passage-times appear to be known.

Based on joint work with Conrado da Costa and Mikhail Menshikov (Durham University).

Brownian Motion Subject to Time-Inhomogeneous Additive Penalizations

Dominic T. Schickentanz

Consider a Brownian motion \(B=(B_t)_{t \ge 0}\), started in \(x \in \mathbb{R}\), as well as a positive random variable \(\xi\) independent of \(B\) and a measurable, locally bounded function \(u: \mathbb{R}\times [0,\infty) \to~[0,\infty)\). Let \[\tau:= \inf\left\{T \ge 0: \int_0^T u(B_s,s) \mathrm{d}s \ge \xi\right\}\] be the first time the time-inhomogeneous additive Brownian functional associated with \(u\) reaches the threshold \(\xi\). We will analyze the asymptotic behavior of \(\mathbb{P}_x(\tau >T)\) as \(T \to \infty\) and, in particular, provide sufficient criteria for this probability to decay like a multiple of \(\frac{1}{\sqrt{T}}\). Subsequently, we will discuss the existence and long-term behavior of the associated conditioned process, i.e., of \(B\) conditioned on the rare event \[\{\tau=\infty\} = \left\{\int_0^t u(B_s,s) \mathrm{d}s <\xi \text{ for all } t \ge 0\right\}.\] Our framework, in particular, covers occupation times below a wide range of moving barriers. Further, it covers the case where \(u\) is a modified solution of the FKPP equation. This will be the key to upcoming results concerning branching Brownian motions with critically large maximum, a joint project with Bastien Mallein (Toulouse).