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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
In mathematics , logarithmic Sobolev inequalities are a class of inequalities involving the norm of a function f , its logarithm, and its gradient
∇
f
{\displaystyle \nabla f}
. These inequalities were discovered and named by Leonard Gross , who established them in dimension-independent form,[1] [2] in the context of constructive quantum field theory . Similar results were discovered by other mathematicians before and many variations on such inequalities are known.
Gross[3] proved the inequality:
∫
R
n
|
f
(
x
)
|
2
log
|
f
(
x
)
|
d
ν
(
x
)
≤
∫
R
n
|
∇
f
(
x
)
|
2
d
ν
(
x
)
+
‖
f
‖
2
2
log
‖
f
‖
2
,
{\displaystyle \int _{\mathbb {R} ^{n}}{\big |}f(x ){\big |}^{2}\log {\big |}f(x ){\big |}\,d\nu (x )\leq \int _{\mathbb {R} ^{n}}{\big |}\nabla f(x ){\big |}^{2}\,d\nu (x )+\|f\|_{2}^{2}\log \|f\|_{2},}
where
‖
f
‖
2
{\displaystyle \|f\|_{2}}
is the
L
2
(
ν
)
{\displaystyle L^{2}(\nu )}
-norm of
f
{\displaystyle f}
, with
ν
{\displaystyle \nu }
being standard Gaussian measure on
R
n
.
{\displaystyle \mathbb {R} ^{n}.}
Unlike classical Sobolev inequalities , Gross's log-Sobolev inequality does not have any dimension-dependent constant, which makes it applicable in the infinite-dimensional limit.
In particular, a probability measure
μ
{\displaystyle \mu }
on
R
n
{\displaystyle \mathbb {R} ^{n}}
is said to satisfy the log-Sobolev inequality with constant
C
>
0
{\displaystyle C>0}
if for any smooth function f
Ent
μ
(
f
2
)
≤
C
∫
R
n
|
∇
f
(
x
)
|
2
d
μ
(
x
)
,
{\displaystyle \operatorname {Ent} _{\mu }(f^{2})\leq C\int _{\mathbb {R} ^{n}}{\big |}\nabla f(x ){\big |}^{2}\,d\mu (x ),}
where
Ent
μ
(
f
2
)
=
∫
R
n
f
2
log
f
2
∫
R
n
f
2
d
μ
(
x
)
d
μ
(
x
)
{\displaystyle \operatorname {Ent} _{\mu }(f^{2})=\int _{\mathbb {R} ^{n}}f^{2}\log {\frac {f^{2}}{\int _{\mathbb {R} ^{n}}f^{2}\,d\mu (x )}}\,d\mu (x )}
is the entropy functional.
Notes
[ edit ]
^ Gross 1975a
References
[ edit ]
Gross, Leonard (1975a), "Logarithmic Sobolev inequalities", American Journal of Mathematics , 97 (4 ): 1061–1083, doi :10.2307/2373688 , JSTOR 2373688
Gross, Leonard (1975b), "Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form" , Duke Mathematical Journal , 42 (3 ): 383–396, doi :10.1215/S0012-7094-75-04237-4
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Logarithmic_Sobolev_inequalities&oldid=1235817953 "
C a t e g o r i e s :
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● S h o r t d e s c r i p t i o n m a t c h e s W i k i d a t a
● T h i s p a g e w a s l a s t e d i t e d o n 2 1 J u l y 2 0 2 4 , a t 1 1 : 1 6 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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