Symmetries of Kolmogorov Backward Equation

This paper revisits the state vector of an autonomous underwater vehicle (AUV) dynamics coupled with the underwater Markovian stochasticity in the ‘non-linear filtering’ context. The underwater stochasticity is attributed to atmospheric turbulence, planetary interactions, sea surface conditions and astronomical phenomena. In this paper, we adopt the Itô process, a homogeneous Markov process, to describe the AUV state vector evolution equation. This paper accounts for the process noise as well as observation noise correction terms by considering the underwater filtering model. The non-linear filtering of the paper is achieved using the Kolmogorov backward equation and the evolution of the conditional characteristic function. The non-linear filtering equation is the cornerstone formalism of stochastic optimal control systems. Most notably, this paper introduces the non-linear filtering theory into an underwater vehicle stochastic system by constructing a lemma and a theorem for the underwater vehicle stochastic differential equation that were not available in the literature.

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A Numerical Realization of the Wiener–Hopf Method for the Kolmogorov Backward Equation

Modern Methods in Operator Theory and Harmonic Analysis - Springer Proceedings in Mathematics & Statistics ◽

10.1007/978-3-030-26748-3_23 ◽

2019 ◽

pp. 399-426

Author(s):

Oleg Kudryavtsev ◽

Vasily Rodochenko

Keyword(s):

Numerical Realization ◽

Backward Equation ◽

Hopf Method

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Kolmogorov Backward Equation

10.1007/springerreference_1793 ◽

2011 ◽

Keyword(s):

Backward Equation

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The infinitesimal generator of the stochastic Burgers equation

Probability Theory and Related Fields ◽

10.1007/s00440-020-00996-5 ◽

2020 ◽

Vol 178 (3-4) ◽

pp. 1067-1124

Author(s):

Massimiliano Gubinelli ◽

Nicolas Perkowski

Keyword(s):

Burgers Equation ◽

Martingale Problem ◽

Uniqueness Result ◽

Stochastic Pdes ◽

Martingale Approach ◽

Infinite Dimensional ◽

Cylinder Test ◽

Stochastic Burgers Equation ◽

Trivial Intersection ◽

Backward Equation

Abstract We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for “energy solutions” of the stochastic Burgers equation of Gubinelli and Perkowski (J Am Math Soc 31(2):427–471, 2018) to a wider class of equations. As applications of our approach we prove that the stochastic Burgers equation on the torus is exponentially $$L^2$$ L 2 -ergodic, and that the stochastic Burgers equation on the real line is ergodic.

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Computation of stochastic wellhead protection zones by combining the first-order second-moment method and Kolmogorov backward equation analysis

Journal of Hydrology ◽

10.1016/s0022-1694(00)00281-x ◽

2000 ◽

Vol 237 (3-4) ◽

pp. 127-146 ◽

Cited By ~ 29

Author(s):

H. Kunstmann ◽

W. Kinzelbach

Keyword(s):

Moment Method ◽

Second Moment ◽

Protection Zones ◽

First Order ◽

Wellhead Protection ◽

Equation Analysis ◽

First Order Second Moment ◽

Backward Equation ◽

Second Moment Method

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Moments of the first-passage time of a Wiener process with drift between two elastic barriers

Journal of Applied Probability ◽

10.2307/3215214 ◽

1995 ◽

Vol 32 (4) ◽

pp. 1007-1013 ◽

Cited By ~ 17

Author(s):

Marco Dominé

Keyword(s):

Wiener Process ◽

First Passage Time ◽

Passage Time ◽

First Passage ◽

One Dimensional ◽

Special Cases ◽

Backward Equation ◽

The One ◽

First Passage Problem ◽

Reflecting Barriers

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

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A stochastic theory of particle transport

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1977.0189 ◽

1977 ◽

Vol 358 (1692) ◽

pp. 105-120 ◽

Cited By ~ 15

Keyword(s):

Particle Physics ◽

Neutron Transport ◽

Stochastic Theory ◽

Single Chain ◽

Amorphous Medium ◽

History Of ◽

Backward Equation ◽

Mean And Variance ◽

The Green Function ◽

Regeneration Point

A probability balance equation is formulated for the number of particles present in a cascade resulting from multiple births at each collision. Janossy’s regeneration point method is used and it leads to an integro differential equation for the generating function from which statistical information can readily be extracted. The technique is applied to the interpretation of radiation damage cascades in a homogeneous, amorphous medium in which two particles are ‘born’ per collision. The history of a single chain is followed and equations for the mean and variance are obtained as well as for individual probabilities. It is further shown how the backward and forward forms of the Boltzmann equation are related via the Green function of the system. Additional study shows that the variance also obeys a forward type of equation although its solution is not obtained as conveniently as that of the corresponding backward equation. Several analogies are made with other branches of particle physics; in particular, cosmic rays and neutron transport.

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