Infinite-dimensional diffusion processes as gibbs measures on $$C[0,1]^{Z^d }$$

1987 ◽  
Vol 76 (3) ◽  
pp. 325-340 ◽  
Author(s):  
J. D. Deuschel
Keyword(s):  
Diffusion Processes ◽  
Gibbs Measures ◽  
Infinite Dimensional ◽  
Dimensional Diffusion

FINITE DIMENSIONAL APPROXIMATIONS OF DIFFUSION PROCESSES ON BANACH SPACES

2001 ◽  
Vol 04 (04) ◽  
pp. 521-531
Author(s):  
T. S. ZHANG
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Quadratic Forms ◽  
Diffusion Processes ◽  
Existence Proof ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Diffusion ◽  
Weak Limits

In this paper, we prove that the laws of diffusion processes on a Banach space X associated with quadratic forms (not necessarily symmetric) are the weak limits of laws of finite dimensional diffusions. A new existence proof for the infinite dimensional diffusion processes is obtained as a by-product.

scholarly journals A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

2007 ◽  
Vol 44 (04) ◽  
pp. 938-949 ◽  
Author(s):  
Shui Feng ◽  
Feng-Yu Wang
Keyword(s):  
Population Genetics ◽  
Sobolev Inequality ◽  
Diffusion Processes ◽  
Exponential Convergence ◽  
Sobolev Inequalities ◽  
Abstract Space ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Reasonable Alternative

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := { x ∈ [0, 1] N : ∑ i≥1 x i = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

DIRICHLET FORMS IN TERMS OF WHITE NOISE ANALYSIS II: CLOSABILITY AND DIFFUSION PROCESSES

1989 ◽  
Vol 01 (02n03) ◽  
pp. 313-323 ◽  
Author(s):  
S. ALBEVERIO ◽  
T. HIDA ◽  
J. POTTHOFF ◽  
M. RÖCKNER ◽  
L. STREIT
Keyword(s):  
White Noise ◽  
Diffusion Processes ◽  
Noise Analysis ◽  
Dirichlet Forms ◽  
White Noise Analysis ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
White Noise Functionals ◽  
And Diffusion

It is shown that infinite dimensional Dirichlet forms as previously constructed in terms of (generalized) white noise functionals fit into the general framework of classical Dirichlet forms on topological vector spaces. This entails that all results obtained there are applicable. Admissible functionals give rise to infinite dimensional diffusion processes.

scholarly journals A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

2007 ◽  
Vol 44 (4) ◽  
pp. 938-949 ◽  
Author(s):  
Shui Feng ◽  
Feng-Yu Wang
Keyword(s):  
Population Genetics ◽  
Sobolev Inequality ◽  
Diffusion Processes ◽  
Exponential Convergence ◽  
Sobolev Inequalities ◽  
Abstract Space ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
Reasonable Alternative

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := {x ∈ [0, 1]N: ∑i≥1xi = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

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