scholarly journals Jump processes as generalized gradient flows

2022 ◽  
Vol 61 (1) ◽  
Author(s):  
Mark A. Peletier ◽  
Riccarda Rossi ◽  
Giuseppe Savaré ◽  
Oliver Tse
Keyword(s):  
State Space ◽  
Uniqueness Result ◽  
Jump Processes ◽  
Gradient Flows ◽  
Kolmogorov Equations ◽  
Generalized Gradient ◽  
Markov Jump ◽  
Polish Spaces ◽  
Markov Jump Processes ◽  
Definition Of

AbstractWe have created a functional framework for a class of non-metric gradient systems. The state space is a space of nonnegative measures, and the class of systems includes the Forward Kolmogorov equations for the laws of Markov jump processes on Polish spaces. This framework comprises a definition of a notion of solutions, a method to prove existence, and an archetype uniqueness result. We do this by using only the structure that is provided directly by the dissipation functional, which need not be homogeneous, and we do not appeal to any metric structure.

Markov jump processes with a singularity

2000 ◽  
Vol 32 (03) ◽  
pp. 779-799 ◽  
Author(s):  
Ole E. Barndorff-Nielsen ◽  
Fred Espen Benth ◽  
Jens Ledet Jensen
Keyword(s):  
State Space ◽  
Laser Cooling ◽  
Quantum Physics ◽  
Jump Processes ◽  
Main Question ◽  
Markov Jump ◽  
Markov Jump Processes ◽  
Equation Problem ◽  
Continuous State Space ◽  
Continuous State

Certain types of Markov jump processes x(t) with continuous state space and one or more absorbing states are studied. Cases where the transition rate in state x is of the form λ(x) = |x|δ in a neighbourhood of the origin in ℝ d are considered, in particular. This type of problem arises from quantum physics in the study of laser cooling of atoms, and the present paper connects to recent work in the physics literature. The main question addressed is that of the asymptotic behaviour of x(t) near the origin for large t. The study involves solution of a renewal equation problem in continuous state space.

IFRA Properties of Some Markov Jump Processes with General State Space

1987 ◽  
Vol 12 (3) ◽  
pp. 562-568 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
State Space ◽  
Jump Processes ◽  
General State ◽  
Markov Jump ◽  
Markov Jump Processes ◽  
General State Space

Markov jump processes with a singularity

2000 ◽  
Vol 32 (3) ◽  
pp. 779-799 ◽  
Author(s):  
Ole E. Barndorff-Nielsen ◽  
Fred Espen Benth ◽  
Jens Ledet Jensen
Keyword(s):  
State Space ◽  
Laser Cooling ◽  
Quantum Physics ◽  
Jump Processes ◽  
Main Question ◽  
Markov Jump ◽  
Markov Jump Processes ◽  
Equation Problem ◽  
Continuous State Space ◽  
Continuous State

Certain types of Markov jump processes x(t) with continuous state space and one or more absorbing states are studied. Cases where the transition rate in state x is of the form λ(x) = |x|δ in a neighbourhood of the origin in ℝd are considered, in particular. This type of problem arises from quantum physics in the study of laser cooling of atoms, and the present paper connects to recent work in the physics literature. The main question addressed is that of the asymptotic behaviour of x(t) near the origin for large t. The study involves solution of a renewal equation problem in continuous state space.

scholarly journals Analysis of Markov Jump Processes under Terminal Constraints

2021 ◽  
pp. 210-229
Author(s):  
Michael Backenköhler ◽  
Luca Bortolussi ◽  
Gerrit Großmann ◽  
Verena Wolf
Keyword(s):  
State Space ◽  
Rare Event ◽  
Jump Processes ◽  
Markov Jump ◽  
Stochastic Filtering ◽  
Markov Jump Processes ◽  
Inference Problems ◽  
Terminal Constraints ◽  
Finite State ◽  
Endpoint Constraints

AbstractMany probabilistic inference problems such as stochastic filtering or the computation of rare event probabilities require model analysis under initial and terminal constraints. We propose a solution to this bridging problem for the widely used class of population-structured Markov jump processes. The method is based on a state-space lumping scheme that aggregates states in a grid structure. The resulting approximate bridging distribution is used to iteratively refine relevant and truncate irrelevant parts of the state-space. This way, the algorithm learns a well-justified finite-state projection yielding guaranteed lower bounds for the system behavior under endpoint constraints. We demonstrate the method’s applicability to a wide range of problems such as Bayesian inference and the analysis of rare events.

scholarly journals Gradient flows in metric random walk spaces

SeMA Journal ◽  
2021 ◽  
Author(s):  
José M. Mazón ◽  
Marcos Solera ◽  
Julián Toledo
Keyword(s):  
Random Walk ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Neumann Boundary ◽  
Jump Processes ◽  
Gradient Flows ◽  
Markov Jump ◽  
Partial Differential ◽  
Markov Jump Processes ◽  
Nonlocal Models

AbstractRecently, motivated by problems in image processing, by the analysis of the peridynamic formulation of the continuous mechanic and by the study of Markov jump processes, there has been an increasing interest in the research of nonlocal partial differential equations. In the last years and with these problems in mind, we have studied some gradient flows in the general framework of a metric random walk space, that is, a Polish metric space (X, d) together with a probability measure assigned to each $$x\in X$$ x ∈ X , which encode the jumps of a Markov process. In this way, we have unified into a broad framework the study of partial differential equations in weighted discrete graphs and in other nonlocal models of interest. Our aim here is to provide a summary of the results that we have obtained for the heat flow and the total variational flow in metric random walk spaces. Moreover, some of our results on other problems related to the diffusion operators involved in such processes are also included, like the ones for evolution problems of p-Laplacian type with nonhomogeneous Neumann boundary conditions.

scholarly journals Fast Reaction Limits via $$\Gamma $$-Convergence of the Flux Rate Functional

2021 ◽  
Author(s):  
Mark A. Peletier ◽  
D. R. Michiel Renger
Keyword(s):  
Evolution Equations ◽  
Detailed Balance ◽  
Approximate Solutions ◽  
Reaction Rates ◽  
Convergence Result ◽  
Rate Function ◽  
Jump Processes ◽  
Kolmogorov Equations ◽  
Markov Jump ◽  
Fast Reaction

AbstractWe study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as $$1/\epsilon $$ 1 / ϵ , and we prove the convergence in the fast-reaction limit $$\epsilon \rightarrow 0$$ ϵ → 0 . We establish a $$\Gamma $$ Γ -convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the $$\Gamma $$ Γ -convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

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