The general isomorphism theorem for transient, non necessarily symmetric, Markov processes conditioned to start and to die at the same point, established by Eisenbaum and Kaspi (2009) has been derived by Fitzsimmons and Rosen (2014) from Palm's disintegration formula for loop soups. By allowing Markov processes to start and to end at different points, the isomorphism theorem of Dynkin (1983) should not a priori be obviously connected to loop measures. Nevertheless we show here that Dynkin's isomorphism theorem can be seen as a consequence of the disintegration formula for loop soups. Finally we show that the used arguments are actually available in the large framework of positive infinitely divisible processes and lead similarly to isomorphism theorems.
This paper is the last in a series devoted to constructing stochastic motions representing the two-dimensional N-body delta-Bose gas for all integers N >= 3 via Feynman-Kac-type formulas; see [2, 3, 4] for the earlier parts of this series. The main result here supplements [2, 3] by establishing an explicitly defined bijective transformation between two classes of Langevin-type SDEs with general coefficients of a particular form. In essence, this transformation provides an SDE-based realization of the principle that the dynamics of the relative motions among N particles and of the center of mass uniquely determine the overall dynamics of the particles.
This work investigates continuous dependence on initial values for 1-dimensional (distribution-dependent) SDEs with H & ouml;lder continuous coefficients. The regularity obtained in this work coincides with the regularity associated with SDEs with Lipschitz continuous coefficients, and hence is sharpest. Our approach depends on an application of Meyer-Tanaka's formula and a subtle estimation of local times for continuous semimartingales.
Let (Xn) be a sequence of random variables with values in a standard Borel space S. We investigate the condition E{f(Xn+1) X1,. . . ,Xn} converges in probability, (*) as n H co, for each bounded Borel function f : S H R. Some consequences of (*) are highlighted and various sufficient conditions for it are obtained. In particular, (*) is characterized in terms of stable convergence. It is also shown that, under (*), there is a random probability measure alpha on S such that E{f(Xn+1) X1,. . . , Xn}-H P f f d alpha for each bounded Borel f. Moreover, since (*) holds whenever (Xn) is conditionally identically distributed, three weak versions of the latter condition are investigated. For each version, our main goal is proving (or disproving) that (*) holds. Several counterexamples are given as well.
The critical beta-splitting tree is a random rooted tree, in which a set of m >= 2 leaves is recursively split into two subsets containing i and m i leaves with probabilities proportional to i-1(m i)-1. We further explore a connection initially unveiled in Iksanov (2025) between the critical beta-splitting tree and an infinite balls-in-boxes scheme. Using this connection, we derive a new joint central limit theorem for components of the height of a leaf chosen uniformly at random in the discrete version of the critical beta-splitting tree. Also, we obtain a joint central limit theorem for the heights in the discrete and continuous versions of the critical beta-splitting tree.
We show that a site percolation is a stronger model than a bond percolation. We use the van den Berg-Kesten (vdBK) inequality to prove that site percolation on a neighborhood of a vertex of degree 4 cannot be simulated even approximately by bond percolation, and develop a decision tree technique to prove the same for a neighborhood of a vertex of degree 3. This technique can be used to obtain inequalities for connectedness probabilities, including a conjectured inequality of Erik Aas.
We study the effect of local perturbations on the recurrence of random walks with long jumps. Such walks serve as discrete models for infinite-horizon Lorentz processes, in which a particle can take arbitrarily long steps in specific directions. Motivated by a question of Sinai in the finite-horizon case and its extension by Sz & aacute;sz to the infinite-horizon setting, we give recurrence and transience criteria for long-jump walks on Z2 and certain classes of graphs, and we prove that local perturbations in a bounded region do not change the recurrence property. Our proofs combine the Markov chain approach with the electrical network method, making the arguments transparent to a broad audience in probability.
Standard gradient-based iteration algorithms for optimization, such as gradient descent and its various proximal-based extensions to nonsmooth problems, are known to converge slowly for ill-conditioned problems, sometimes requiring many tens of thousands of iterations in practice. Since these iterations are computed sequentially, they may present a computational bottleneck in large-scale parallel simulations. In this work, we present a "parallel-in-iteration" framework that allows one to parallelize across these iterations using multiple processors with the objective of reducing the wall-clock time needed to solve the underlying optimization problem. Our methodology is based on re-purposing parallel time integration algorithms for time-dependent differential equations, motivated by the fact that optimization algorithms often have interpretations as discretizations of time-dependent differential equations (such as gradient flow). Specifically in this work, we use the parallel-in-time method of multigrid reduction-in-time (MGRIT), but note that our approach permits in principle the use of any other parallel-in-time method. We numerically demonstrate the efficacy of our approach on two different model problems, including a standard convex quadratic problem and the nonsmooth elastic obstacle problem in one and two spatial dimensions. For our model problems, we observe fast MGRIT convergence analogous to its prototypical performance on partial differential equations of diffusion type. Some theory is presented to connect the convergence of MGRIT to the convergence of the underlying optimization algorithm. Theoretically predicted parallel speedup results are also provided.
We introduce a novel notion of divergence between continuous martingales; the reciprocal specific relative entropy. First, we motivate this definition from multiple perspectives. Thereafter, we solve the reciprocal specific relative entropy minimization problem over the set of win-martingales (used as models for prediction markets [2]). Surprisingly, we show that the optimizer is the renowned neutral Wright-Fisher diffusion. We also justify that this diffusion is in a sense the most salient win-martingale, since it is uniquely selected when we suitably perturb the degenerate martingale optimal transport problem of variance minimization.
We consider a reflected process in the positive orthant driven by an exogenous c & agrave;dl & agrave;g process. For a given input process, we show that solutions to the reflection problem can be continued at a jump time if and only if the proposed jump of the unregulated process is within the dual cone of a linear programming problem associated with the state of the system. As a consequence, for piecewise non-decreasing driving processes there exists a unique minimal strong solution to the given particle system up until the stopping time at which such a non-allowed jump first occurs. We apply this model to study the ruin of interconnected insurance firms, where the stopping time can be interpreted as the failure of a reinsurance agreement between the firms. Our work extends the analysis of the particle system in Baker, Hambly, and Jettkant (2025) to a class of L & eacute;vy processes, and the existence result of Reiman (1984) beyond the case of sub-stochastic reflection matrices.
The Fr & eacute;chet-Shohat Theorem (FST) (1931) is a well-established result concerning determinate moment problems. We endeavor to address an analogous problem within the realm of indeterminate Hamburger and Stieltjes moment problems, focusing exclusively on absolutely continuous random variables. We demonstrate that, under an additional condition stipulating the convergence of the entropy of a sequence of distribution functions to the entropy of the unique maximum entropy distribution, a stronger mode of convergence is achieved, which subsequently implies convergence in distribution. This result is attainable due to the inherent property of indeterminate moment problems possessing a unique density ghmax, distinguishable from other solutions by its maximal entropy. In conclusion, the moment convergence implies weak convergence under determinacy and under indeterminacy, if there is entropy convergence to the ghmax then there is also weak convergence.
The mixed fractional Brownian motion-the sum of independent fractional and standard Brownian motions-is known to be a semimartingale if the Hurst exponent H of its fractional component satisfies H > 3/4. The question posed in the title is motivated by recent findings in quantitative finance. In this note, we show that the drift in its Doob-Meyer decomposition has a derivative that is gamma-H & ouml;lder continuous for any gamma < 2H-3/2.
We extend the Benamou-Brenier formula from classical optimal transport to weak optimal transport and show that the barycentric optimal transport problem studied by Gozlan and Juillet has a dynamic analogue. We also investigate a martingale relaxation of this problem, and relate it to the martingale Benamou-Brenier formula of Backhoff-Veraguas, Beiglb & ouml;ck, Huesmann and K & auml;llblad.
