Convolution algebra for extended Feller convolution

Abstract

We apply the recently introduced framework of admissible homomorphisms in the form of a convolution algebra of $\mathbb{C}^2$-valued admissible homomorphisms to handle two-dimensional uni-directional homogeneous stochastic kernels. The algebra product is a non-commutative extension of the Feller convolution needed for an adequate operator representation of such kernels: a pair of homogeneous transition functions uni-directionally intertwined by the extended Chapman–Kolmogorov equation is a convolution empathy; the associated Fokker–Planck equations are re-written as an implicit Cauchy equation expressed in terms of admissible homomorphisms. The conditions of solvability of such implicit evolution equations follow from the consideration of generators of a convolution empathy.

Appendix: Dynamic boundary condition for heat equation

A dynamic boundary condition for a partial differential equation is a boundary condition in which the time derivative of the unknown appears. For example, such a boundary condition for the heat equation arises if the boundary of the body is considered as a thin conducting film. With its own thermal property, the boundary thermally interacts with the body in a one-way manner like an absorbing-barrier: the body is an external source for the boundary but the boundary is not an external source for the body (see Sect. A.1).

In this appendix, we give a physical motivation of how the implicit evolution equation (1) originates from such a two-space approach to the heat equation. Full details are in [6]. In particular, we give a heuristic formulation of how non-perfect thermal contact between the body and its boundary results in a pair of coupled heat equations that can be formulated as an implicit evolution equation of the form (1). For a mathematically exact treatment please see [4].

1.1 Dynamic boundary condition

For the body material $\varOmega$, let $u(x,t), \overrightarrow{\varphi }(x,t)$ and f(x, t) denote the temperature field, the ambient flux and the source at the internal point x of the body $\varOmega$ at time t, respectively. For the boundary $\varGamma := \partial \varOmega$, the corresponding thermal properties at $X\in \varGamma$ at time t are denoted by $U(X,t), \overrightarrow{\varPhi }(X,t)$ and F(X, t).

If $\varOmega$ is three dimensional, then the boundary $\varGamma$ is a two-dimensional manifold. Then the traditional choice for the ambient flux $\overrightarrow{\varphi }(x,t) = - k\overrightarrow{\nabla }u(x,t)$ is a three-vector, whereas $\overrightarrow{\varPhi }(X,t) = - K\overrightarrow{\nabla }_{_{^{S}}}U(X,t)$ is a two-vector which lives in the tangent plane of $\varGamma$ at point X; here $\overrightarrow{\nabla }_{_{^{S}}}$ denotes the surface gradient.

Applying the law of conservation of energy to the body $\varGamma$,

$$\begin{aligned} Q\tfrac{d}{dt}\int _{\mathscr{B}} U(X,t)dX = -\int _{\partial \mathscr{B}} \overrightarrow{\varPhi }(X,t) \cdot \widehat{N}(X) ds + \int _{\mathscr{B}} F(X,t) + \overrightarrow{\varphi }(X,t)\cdot \widehat{n}(X)dX \ \ \ \ \end{aligned}$$

(85)

where $\mathscr{B}$ denotes the ‘volume’ patch of the boundary $\varGamma$ taken as a body and $Q=1$ without loss of generality. Since $\mathscr{B}$ is a surface patch, the integral on the left hand side of Eq. (85) is a surface integral which we denote by $\int _{\mathscr{B}} dX$.

Unlike F(X, t), which is a per-volume per-time quantity, $\overrightarrow{\varPhi }(X,t)$ is a per-surface per-time quantity. Thus the first integral on the right hand side of (85) is a surface integral of (the surface) $\varGamma$ which we denote by $\int _{\partial \mathscr{B}} ds$. Assuming sufficient smoothness on $\mathscr{B}$, with the traditional choices of flux, we have

$$\begin{aligned} -\int _{\partial \mathscr{B}} \overrightarrow{\varPhi }(X,t) \cdot \widehat{N}(X) ds = \int _{\mathscr{B}} K\varDelta _{_{^{S}}}U(X,t)dX, \end{aligned}$$

(86)

where $\widehat{N}(X)$ is the unit exterior of the body $\varGamma = \partial \varOmega$; $\widehat{n}(X)$ denotes the ambient unit exterior of the original body $\varOmega$; $\varDelta _{_{^{S}}}$ is the surface Laplacian or the Laplace–Beltrami operator. Thus,

$$\begin{aligned} \tfrac{\partial }{\partial t}U(X,t) = K\varDelta _{_{^{S}}}U(X,t) + F(X,t) + k D_{\widehat{n}} u(X,t) \end{aligned}$$

(87)

where $D_{\widehat{n}} u(X,t)$ is the directional derivative of u(X, t) in the direction of the ambient unit exterior $\widehat{n}(X)$.

Similarly applying the law of conservation of energy to the body $\varOmega$,

$$\begin{aligned} q\tfrac{d}{dt}\int _{\mathscr{G}} U(x,t)dx = \int _{\mathscr{G}} k\varDelta u(x,t)dx + \int _{\mathscr{G}} f(x,t)dx \end{aligned}$$

(88)

where $\mathscr{G}$ denotes the volume patch of the body $\varOmega$ and $q=1$ without loss of generality. The last integral in Eq. (88), in contradistinction to the corresponding last integral of Eq. (85), lacks an analogous term to the term $\overrightarrow{\varphi }(X,t)\cdot \widehat{n}(X)$, which captures the body $\varOmega$ acting as a source internal to $\varGamma$. The surface flux $\overrightarrow{\varPhi }(X,t)$ lies in the tangent plane at X of the boundary surface $\varGamma$ and therefore has no internal impact on the body $\varOmega$. Therefore $\varGamma$ is an absorbing barrier in the sense that $\varGamma$ is not a source for $\varOmega$, whereas $\varOmega$ is a source of $\varGamma$. The dynamic boundary condition (87) can then be viewed as one-way or one-term coupled.

Consequently, the law of conservation of energy for the body $\varOmega$ remains intact as

$$\begin{aligned} \tfrac{\partial }{\partial t}u(x,t) = k\varDelta u(x,t) + f(x,t), \end{aligned}$$

(89)

so that the pair $\langle {u, U}\rangle$ is intertwined by a pair of heat equations:

$$\begin{aligned} \left\{ \begin{aligned}&\tfrac{\partial }{\partial t}u(x,t) = k\varDelta u(x,t) + f(x,t); x\in \varOmega \\&\tfrac{\partial }{\partial t}U(X,t) = K\varDelta _{_{^{S}}}U(X,t) + F(X,t) + k D_{\widehat{n}} u(X,t); X \in \varGamma . \end{aligned} \right. \end{aligned}$$

(90)

Now each point $X\in \varGamma$ will be taken as a ‘point of contact’ between the two bodies. We introduce a pair of trace operators $\langle {\gamma _{0}, \gamma _{1}}\rangle$ to measure the quality of thermal contact between the two bodies

$$\begin{aligned} \left\{ \begin{aligned}&\gamma _{0} u(X) := \lim _{x\rightarrow X}u(x); x\in \varOmega \\&\gamma _{1} u(X) := D_{\widehat{n}} u(X); X \in \varGamma . \end{aligned} \right. \end{aligned}$$

(91)

The pair $\langle {\gamma _{0}, \gamma _{1}}\rangle$ expresses perfect thermal contact and non-perfect thermal contact, respectively, as

$$\begin{aligned} U(X)= & {} \gamma _{0} u(X), \end{aligned}$$

(92)

$$\begin{aligned} U(X)= & {} \gamma u(X), \end{aligned}$$

(93)

where $\gamma = \gamma _{0} - C(X)k\gamma _{1}$ and C(X) is a positive function. Equation (93) is implicit in the contact constitutive equation $k\gamma _{1}u(X) = \frac{1}{C(X)} (\gamma _{0} u(X) - U(X))$, where C(X) measures the quality of thermal contact.

1.2 Trace-induced implicit evolution equation

Consider the first temperature field u(x, t) of the pair $\langle {u, U}\rangle$ satisfying the intertwined pair of equations (90). We take u(x, t) as a time evolution $u(t):= u(\cdot , t)$ in an appropriate space, X, of functions in $x\in \varOmega$. Likewise, the second dependent temperature field U(x, t) (see equation (93)) is a time evolution $U(t):= U(\varvec{{\cdot }}, t)$ in a distinct space of functions in $X\in \varGamma$.

Although we considered the body and boundary as separate distinct bodies in the dynamic boundary formulation, the coupled system of heat equations (90) shows that the body and boundary are thermally inseparable. Therefore we re-write the system of Eq. (90) in vector format,

$$\begin{aligned} \tfrac{\partial }{\partial t} \begin{pmatrix}u(x,t) \\ U(X,t)\end{pmatrix} = \begin{pmatrix}k\varDelta u(x,t)\\ K\varDelta _{_{^{S}}}U(X,t)+ k D_{\widehat{n}} u(X,t)\end{pmatrix} + \begin{pmatrix}f(x,t) \\ F(X,t) \end{pmatrix} . \end{aligned}$$

(94)

Thus the solution to the coupled heat equations (90) is the time evolution vector $\begin{pmatrix}u(\cdot ,t) \\ U(\varvec{{\cdot }},t)\end{pmatrix} \in {{Y}}$ evolving from a pair of initial states $y_{0} := \begin{pmatrix}u_{0} \\ U_{0}\end{pmatrix} \in {{Y}}$.

The pair of trace operators $\langle {\gamma _{0}, \gamma _{1}}\rangle$ enables joint consideration of u and U: in the case of perfect thermal contact (see Eq. (92)), the trace operator $\gamma _{0}$ is a transition map from the first body $\varOmega$ into the second body $\varGamma$: $\gamma _{0}$ transitions $u(\cdot , t)\in W^{1}_{2}(\varOmega )$ into $U(\varvec{{\cdot }}, t)\in L_{2}(\varGamma )$; thus, the trace operator $\gamma _{0}$ forces the choice of two distinct spaces $W^{1}_{2}(\varOmega )$ and $L_{2}(\varGamma )$ corresponding to the two distinct systems $\varOmega$ and $\varGamma$. We use the same choice of spaces for the case for non-perfect thermal contact (93) since $\gamma$ is a function of $\gamma _{0}, \gamma _{1}$ and C(X). The space $W^{1}_{2}(\varOmega )$ is the golden mean for the classical heat equation which was preserved in the system (90) by virtue of the absorbing-barrier boundary assumption. So a possible choice for the Banach space X is the associated space of definition, $L_{2}(\varOmega )$. Thus the solution space ${ {Y}} := L_{2}(\varOmega ) \times L_{2}(\varGamma )$.

The vector format of Eq. (94) takes the form

$$\begin{aligned} \tfrac{d}{dt}Bu(\cdot , t) = Au(\cdot , t) \ \ \text {or more concisely,} \ \ \ \tfrac{d}{dt}Bu(t) = Au(t), \end{aligned}$$

(95)

if $f(x,t) = 0 = F(X,t)$. Here $B: { {X}} \rightarrow { {Y}}$ denotes the intertwined trace operator

$$\begin{aligned} 1 \otimes \gamma : u(\cdot , t) \in { {X}} \mapsto \begin{pmatrix}u(\cdot ,t) \\ U(\varvec{{\cdot }},t)\end{pmatrix} \in { {Y}} \end{aligned}$$

and $A: { {X}} \rightarrow { {Y}}$ the intertwined Laplacian operator, where B only adds the information $U(\varvec{{\cdot }},t) = \gamma u(\cdot , t)$:

$$\begin{aligned} u(\cdot , t) \in { {X}} \mapsto \begin{pmatrix}k\varDelta u(\cdot ,t)\\ K\varDelta _{_{^{S}}}U(\varvec{{\cdot }},t)+ k D_{\widehat{n}} u(\varvec{{\cdot }},t)\end{pmatrix} . \end{aligned}$$

Remark 5

The example (see [16, §8]) shows that the intertwined trace operator B is non-closeable even in an elementary case of perfect thermal contact, $\gamma = \gamma _{0}$, in a one dimensional body. Therefore we assume B to be non-closeable and so we study Eq. (95) as it stands.