Gauss kernel

The $ n $- dimensional Gauss (or Weierstrass) kernel

$$ G ( x,y,t;D ) = ( 4 \pi Dt ) ^ {- n/2 } { \mathop{\rm exp} } \left ( - { \frac{1}{4Dt } } \left | {x - y } \right | ^ {2} \right ) , $$

with $ D $ a positive constant, $ t > 0 $, $ x,y \in \mathbf R ^ {n} $, is the fundamental solution of the $ n $- dimensional heat equation $ u _ {t} = D \Delta u $. Moreover, this kernel is an approximate identity in that the Gauss–Weierstrass singular integral at the function $ f $,

$$ u ( x,t;D ) = \int\limits _ {\mathbf R ^ {n} } {G ( x,y,t;D ) f ( y ) } {dy } , $$

satisfies $ {\lim\limits } _ {t \rightarrow 0 ^ {+} } u ( x,t;D ) = f ( x ) $ almost everywhere, for example, whenever $ \int _ {\mathbf R ^ {n} } {e ^ {- A | y | ^ {2} } | {f ( y ) } | } {dy } < \infty $ for some $ A > 0 $; see [a4]. Thus $ u ( x,t;D ) $ is a solution of the heat equation for $ 0 < t < {1 / {( 4AD ) } } $, $ x \in \mathbf R ^ {n} $ having the initial "temperature" $ f $.

In the theory of Markov processes (cf. Markov process) the Gauss kernel gives the transition probability density of the Wiener–Lévy process (or of Brownian motion). The semi-group property of the Gauss kernel

$$ G ( x,z,t _ {1} + t _ {2} ;D ) = \int\limits _ {\mathbf R ^ {n} } {G ( x,y,t _ {1} ;D ) G ( y,z,t _ {2} ;D ) } {dy } , $$

$$ t _ {1} ,t _ {2} > 0, \quad x,z \in \mathbf R ^ {n} , $$

is essential here.

References

[a1]	P. Butzer, R. Nessel, "Fourier analysis and approximation", I, Birkhäuser (1971)
[a2]	R. Courant, D. Hilbert, "Methods of mathematical physics", II, Wiley (1962)
[a3]	W. Feller, "An introduction to probability theory and its applications", 2, Springer (1976) (Edition: Second)
[a4]	E.C. Titchmarsh, "Introduction to the theory of Fourier integrals", Clarendon Press (1937)
[a5]	K. Weierstrass, "Ueber die analytische Darstellbarkeit sogenannter willkurlichen Functionen reeler Argumente" Berliner Sitzungsberichte (1985) pp. 633–639; 789–805