We give asymptotic analysis for probability of absorbtion P(τ0 ≤ T ) on the interval [0, T ], where τ0 = inf{t : Xt = 0} and Xt is a nonnegative diffusion process relative to Brownian motion Bt, dXt = μXtdt+ σX γ t dBt. X0 = K > 0 Diffusion parameter σx , γ ∈ [ 1 2 , 1) is not Lipschitz continuous and assures P(τ0 > T ) > 0. Our main result: lim K→∞ 1 K2(1−γ) logP(τ0 ≤ T ) = − 1 2EM T , where M...

Absftact-The idea of thermodynamic depth put forth by Lloyd and Pageh requires the computation of the entropy af Markov tqjectorim Toward this end we consider an irreducible finite state Markov chain with transition matrix P and pssoeiated entropy rate H ( X ) = E,,, p,P,, log P,,, where p is the stationary distribution given by the solution of p = p P . A wectory T,, of the Markov chain is a p...

Absftact-The idea of thermodynamic depth put forth by Lloyd and Pageh requires the computation of the entropy af Markov tqjectorim Toward this end we consider an irreducible finite state Markov chain with transition matrix P and pssoeiated entropy rate H ( X ) = E,,, p,P,, log P,,, where p is the stationary distribution given by the solution of p = p P . A wectory T,, of the Markov chain is a p...

This paper is a continuation of our previous (Liu and Pagès, 2020). In this paper, we establish the monotone convex order (see further (1.1)) between two R-valued McKean–Vlasov processes X=(Xt)t∈[0,T] Y=(Yt)t∈[0,T] defined on filtered probability space (Ω,F,(Ft)t≥0,P) by dXt=b(t,Xt,μt)dt+σ(t,Xt,μt)dBt,X0∈Lp(P)withp≥2,dYt=β(t,Yt,νt)dt+θ(t,Yt,νt)dBt,Y0∈Lp(P),where∀t∈[0,T],μt=P∘Xt−1,νt=P∘Yt−1.If m...

We present serial and parallel algorithms for solving a system of equations that arises from the discretization of the Hamilton-Jacobi equation associated to a trajectory optimization problem of the following type. A vehicle starts at a prespecified point zo and follows a unit speed trajectory ~ ( t ) inside a region in P. until an unspecified time T that the region is exited. A trajectory mini...

We study the presence of obstacles in computing BCP (s, t) (Best Coverage Path between two points s and t) in a 2D field under surveillance by sensors. Consider a set of m line segment obstacles and n point sensors on the plane. For any path between s to t, p is the least protected point along the path such that the Euclidean distance between p and its closest sensor is maximum. This distance (...

In this paper we fix 7 types of undirected graphs: paths, paths with prescribed endvertices, circuits, forests, spanning trees, (not necessarily spanning) trees and cuts. Given an undirected graph G = (V, E) and two “object types” A and B chosen from the alternatives above, we consider the following questions. Packing problem: can we find an object of type A and one of type B in the edge set E ...

We are given a graph G = (V, E) and k " commodities ". Each commodity i has a source s i and a destination t i. The goal is to find an (s i , t i) path in the graph G for every commodity i, while minimizing the maximum congestion along any edge. The latter can be written formally as Minimize congestion C = max e |{i : P i e}|, where P i is the (s i , t i) path of commodity i and e is an edge. I...