ConvolutionSemigroup

The Convolution Semigroup and Integral Transforms

In the context of integral transforms, a convolution semigroup is a family of measures $\lbrace \mu_t \rbrace_{t \geq 0}$ on a locally compact abelian group $G$ (often $\mathbb{R}^n$ or $\mathbb{C}$) such that the following properties hold:

1. Identity Element: $\mu_0 = \delta_0$, the Dirac delta measure at the identity element $0$ of $G$.

2. Convolution Property:

$$ \mu_{s+t} = \mu_s * \mu_t \quad \forall s,t \geq 0 $$

where $*$ denotes the convolution of measures.

3. Continuity: The mapping $t \mapsto \mu_t(f)$ is continuous for every continuous function $f$ with compact support on $G$.

When the measures $\mu_t$ are absolutely continuous with respect to the Lebesgue measure, they can be represented by their densities $p_t(x)$. In this case, the convolution property can be expressed as:

$$ p_{s+t}(y) = \int_{-\infty}^{\infty} p_s(x) p_t(y - x) dx $$

This is a continuous-time analogue of the Chapman-Kolmogorov equation in the theory of Markov processes and provides a way to understand the evolution of the density $p_t(x)$ over time.

In the context of integral transforms like the Fourier or Laplace transform, convolution semigroups are particularly useful because they often lead to simple expressions when transformed. For example, if $\mu_t$ is a convolution semigroup and $\hat{\mu_t}$ is its Fourier transform, then the Fourier transform of $\mu_{s+t}$ can be easily expressed in terms of $\hat{\mu_s}$ and $\hat{\mu_t}$ due to the convolution property.

The Fourier transform of a measure $\mu$ is defined as:

$$ \hat{\mu}(\xi) = \int_{G} e^{-i \langle x, \xi \rangle} d\mu(x) $$

where $\langle \cdot, \cdot \rangle$ is the inner product on $G$.

The convolution property then implies that:

$$ \hat{\mu_{s+t}}(\xi) = \hat{\mu_s}(\xi) \hat{\mu_t}(\xi) $$

which is a simple multiplicative relationship.

Convolution semigroups are a key concept in the theory of stochastic processes, harmonic analysis, and functional analysis, among other areas. They provide a way to understand the evolution of distributions under the action of a linear operator, often a differential operator.

Convolution Semigroups and Transience on Locally Compact Commutative Groups

In the setting of locally compact abelian (LCA) groups, a convolution semigroup is a family of Radon measures $(\mu_t)_{t \geq 0}$ on the LCA group $G$ that satisfies the following properties:

1. Identity: $\mu_0$ is the Dirac measure at the identity element $e$ of $G$.

$$ \mu_0(A) = \begin{cases} 1, & \text{if}\ e \in A \\ 0, & \text{otherwise} \end{cases} $$

2. Semigroup Property: For all $t, s \geq 0$

$$ \mu_{t+s} = \mu_t * \mu_s $$

where $*$ denotes the convolution of measures.

$$ (\mu_t * \mu_s)(A) = \int_G \int_G \chi_A(x * y) d\mu_t(x) d\mu_s(y) $$

3. Continuity: The mapping $t \mapsto \mu_t(A)$ is continuous for every Borel set $A$ in $G$.

Transience

The term "transience" is not directly related to the concept of a convolution semigroup. In the context of stochastic processes, particularly Markov chains, a state is said to be "transient" if, starting from that state, the process is expected to return to it only a finite number of times. In contrast, a state is "recurrent" if the process is expected to return to it an infinite number of times.

Given this definition, a convolution semigroup $(\mu_t)_{t \geq 0}$ on an LCA group $G$ is said to be "transient" if:

$$ \int_G \left( \int_0^\infty \mu_t(x) dt \right) dx < \infty $$

The concept of transience or recurrence is more about the long-term behavior of a stochastic process, whereas a convolution semigroup is a mathematical construct that helps to describe the evolution of a distribution over time. Convolution semigroups can be used to model both transient and recurrent behavior, depending on the specifics of the semigroup and the associated generator (often a differential operator).