We study a variant of the voter model on a coevolving network in which interactions of two individuals with differing opinions only lead to an agreement on one of these opinions with a fixed probability q. Otherwise, with probability 1−q, both individuals become offended in the sense that they never interact again, i.e. the corresponding edge is removed from the underlying network. Eventually, these dynamics reach an absorbing state at which there is only one opinion present in each connected component of the network. If globally both opinions are present at absorption we speak of “segregation”, otherwise of “consensus”. We rigorously show that segregation and a weaker form of consensus both occur with positive probability for every q∈(0,1) and that the segregation probability tends to 1 as q→0. Furthermore, we establish that, if q→1 fast enough, with high probability the population reaches consensus while the underlying network is still densely connected. We provide results from simulations to assess the obtained bounds and to discuss further properties of the limiting state.

The transition matrix of a Markov chain (Xk,k≥0) on a finite or infinite rooted tree is said to be almost upper-directed if, given Xk, the node Xk+1 is either a descendant of Xk or the parent of Xk. It is said to be almost lower-directed if given Xk, Xk+1 is either an ancestor of Xk or a child of Xk. These models include nearest neighbor Markov chains on trees. Under an irreducibility assumption, we show that every almost upper-directed transition matrix on infinite (locally finite) trees has some invariant measures. An invariant measure π is expressed thanks to a determinantal formula. We give general explicit criteria for recurrence and positive recurrence. An efficient algorithm (the leaf addition algorithm) of independent interest allows π to be computed on many trees, without resorting to linear algebra considerations. Flajolet, in a series of papers (starting from [12]), provided some relations between continuous fractions, generating functions of weighted Mötzkin paths, and used them in connection with the analysis of birth and death processes. These fruitful representations made it possible to establish many formulae for continuous fractions. Analogous considerations appear here: this type of study can be extended to weighted paths on trees, whose generating functions can also be expressed, this time in terms of multicontinuous fractions.

Let W={W(t):t∈R+N} be an (N,d)-Brownian sheet and let E⊂(0,∞)N and F⊂Rd be compact sets. We prove a necessary and sufficient condition for W(E) to intersect F with positive probability and determine the essential supremum of the Hausdorff dimension of the intersection set W(E)∩F in terms of the thermal capacity of E×F. This extends the previous results of Khoshnevisan and Xiao (2015) for the Brownian motion and Khoshnevisan and Shi (1999) for the Brownian sheet in the special case when E⊂(0,∞)N is an interval.

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We provide a complete theory of reflected backward stochastic differential equations driven by G-Brownian motion (reflected G-BSDEs), suggested in Li et al. (2018) [15]. We put forward a new definition of reflected G-BSDEs by introducing a non-Skorohod-type minimality condition. Unlike Li et al. (2018) [15], the alternative formulation of this paper is nice in the sense that it is a natural extension of the definitions of standard G-BSDEs as well as reflected BSDEs. We then prove the existence, uniqueness and some properties of solution under Lipschitz-type assumptions on generator. We also provide some applications including probabilistic interpretation of obstacle problems for fully nonlinear PDEs and super-hedging strategy for American options in nonlinear markets with volatility uncretainty. The key argument is the dynamic programming approach, analyzed in the accompanying paper (O et al. (2023) [22]), and we develop it to the G-expectation framework. Our approach is completely different from the penalization method in Li et al. (2018) [15].

We consider a family of Pfaffian Schur processes whose first coordinate marginal relates to the half–space geometric last passage percolation. We show that the line ensembles corresponding to the Pfaffian Schur processes with geometric weights converge uniformly over compact sets to the Airy line ensemble. By detailed asymptotic analysis of the kernels, we can verify the conditions for the finite dimensional weak convergence introduced in [Dim24a]. By utilizing the tightness criteria of the line ensembles established in [Dim24b], we can further improve the finite dimensional convergence to the uniform convergence over compact sets. Moreover, using the same methodology we also show that sequences of spiked Pfaffian Schur processes converge uniformly over compact sets to the Airy wanderer line ensembles constructed in [Dim24a].