Explicit contraction rates for a class of degenerate and infinite-dimensional diffusions

Explicit contraction rates for a class of degenerate and infinite-dimensional diffusions Given a separable and real Hilbert space $${\mathbb {H}}$$ H and a trace-class, symmetric and non-negative operator $$\mathscr {G}\,{:}\,{\mathbb {H}}\rightarrow {\mathbb {H}}$$ G : H → H , we examine the equation $$\begin{aligned} dX_t = -X_t dt + b(X_t) dt + \sqrt{2} dW_t, \quad X_0=x\in {\mathbb {H}}, \end{aligned}$$ d X t = - X t d t + b ( X t ) d t + 2 d W t , X 0 = x ∈ H , where $$(W_t)$$ ( W t ) is a $$\mathscr {G}$$ G -Wiener process on $${\mathbb {H}}$$ H and $$b\,{:}\,{\mathbb {H}}\rightarrow {\mathbb {H}}$$ b : H → H is Lipschitz. We assume there is a splitting of $${\mathbb {H}}$$ H into a finite-dimensional space $${\mathbb {H}}^l$$ H l and its orthogonal complement $${\mathbb {H}}^h$$ H h such that $$\mathscr {G}$$ G is strictly positive definite on $${\mathbb {H}}^l$$ H l and the non-linearity b admits a contraction property on $${\mathbb {H}}^h$$ H h . Assuming a geometric drift condition, we derive a Kantorovich ( $$L^1$$ L 1 Wasserstein) contraction with an explicit contraction rate for the corresponding Markov kernels. Our bounds on the rate are based on the eigenvalues of $$\mathscr {G}$$ G on the space $${\mathbb {H}}^l$$ H l , a Lipschitz bound on b and a geometric drift condition. The results are derived using coupling methods. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Stochastical Partial Differential Equations Springer Journals

Explicit contraction rates for a class of degenerate and infinite-dimensional diffusions

Loading next page...
 
/lp/springer_journal/explicit-contraction-rates-for-a-class-of-degenerate-and-infinite-F5n0U0FsI6
Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media New York
Subject
Mathematics; Probability Theory and Stochastic Processes; Partial Differential Equations; Statistical Theory and Methods; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Numerical Analysis
ISSN
2194-0401
eISSN
2194-041X
D.O.I.
10.1007/s40072-017-0091-8
Publisher site
See Article on Publisher Site

Abstract

Given a separable and real Hilbert space $${\mathbb {H}}$$ H and a trace-class, symmetric and non-negative operator $$\mathscr {G}\,{:}\,{\mathbb {H}}\rightarrow {\mathbb {H}}$$ G : H → H , we examine the equation $$\begin{aligned} dX_t = -X_t dt + b(X_t) dt + \sqrt{2} dW_t, \quad X_0=x\in {\mathbb {H}}, \end{aligned}$$ d X t = - X t d t + b ( X t ) d t + 2 d W t , X 0 = x ∈ H , where $$(W_t)$$ ( W t ) is a $$\mathscr {G}$$ G -Wiener process on $${\mathbb {H}}$$ H and $$b\,{:}\,{\mathbb {H}}\rightarrow {\mathbb {H}}$$ b : H → H is Lipschitz. We assume there is a splitting of $${\mathbb {H}}$$ H into a finite-dimensional space $${\mathbb {H}}^l$$ H l and its orthogonal complement $${\mathbb {H}}^h$$ H h such that $$\mathscr {G}$$ G is strictly positive definite on $${\mathbb {H}}^l$$ H l and the non-linearity b admits a contraction property on $${\mathbb {H}}^h$$ H h . Assuming a geometric drift condition, we derive a Kantorovich ( $$L^1$$ L 1 Wasserstein) contraction with an explicit contraction rate for the corresponding Markov kernels. Our bounds on the rate are based on the eigenvalues of $$\mathscr {G}$$ G on the space $${\mathbb {H}}^l$$ H l , a Lipschitz bound on b and a geometric drift condition. The results are derived using coupling methods.

Journal

Stochastical Partial Differential EquationsSpringer Journals

Published: Jan 30, 2017

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off