MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

3 added 243 characters in body

Let me assume that you seek for the generalization of Gaussian distribution in order to generalize the Brownian motion.

As far as I know, regarding the heat kernel as the generalization of the Gaussian distribution has long been adopted in many literatures. It comes from the following observation.

In $\mathbb {R}^1$, the following notions coincide:

(1) Gaussian distribution $N(x,t)\sim f(t,x,y)=\frac {1}{\sqrt{2\pi t}}e^{\frac{-(y-x)^2}{2t}}$,

(2) transition function $p(t,x,y)$ of the Brownian motion $B_t$,

(3) (heat kernel) fundamental solution $k_t(x,y)$ of the heat equation $\partial_t k=\Delta_y k$, with initial data $\delta_x$.

Thus, on manifolds, one way to define the Brownian motion is to construct a stochastic Markov process on the manifold whose transition function is exactly the heat kernel (let's identify the heat kernel with the Gaussian distribution in this setting). Since we always have the Laplacian-Beltrami $\Delta$ on a manifold, it is justifiable to talk about the heat equation and thus the heat kernel, and the Brownian motion in this sense is known to exist for a large class of manifolds.

But on metric spaces, we no longer have the Laplacian-Beltrami. So, in order to talk about heat kernel/Gaussian distribution, we need to generalize the notion of Laplacian-Beltrami. The key concept on this line the so-called Dirichlet form. A Dirichlet form on metric measure space $(X,d,\mu)$ a nonnegative-definite closed symmetric bilinear form $(\cdot,\cdot)$ densely defined on $L^2(X,\mu)$. It should further satisfies satisfy a couple of conditions so that it behaves like its prototype $(f,g)=\int (f,g)=\int_{M} {\nabla f\cdot \nabla gd\mu}$ g dx}$on a manifold$M$. Notice that$(f,g)=(-\Delta f,g)_{L^2(M)}$on manifolds, in the general case, one applies the Hille-Yosida theorem to obtain obtains the desired "Laplacian" in by the same wayformula. Therefore, every Dirichlet form corresponds to a "Laplacian" and thus a Gaussian distribution (and thus a Brownian motion). What's more, a reasonable Dirichlet form always exists provided the space is suitably good. In sum, if the space you are considering have both metric and measure structures, then the theory of Dirichlet form may provide you some satisfactory results regarding construction and properties of the Guassian distribution (and thus the Brownian motion). Roughly speaking, if we don't have a presumed measure, we may not be able to construct a reasonable probability space; if we don't have a metric, it would be hard to measure the regularity and decay of the Gaussian distribution. So metric measure structure might be the minimal structure for reasonable construction of Gaussian distribution. Some reference books could be found in the above link. This paper by Sturm may allow you to have a glance at the whole picture. I am not an expert in this field. I apologize in advance for any mistake and naivety. 2 deleted 5 characters in body Let me assume that you seek for the generalization of Gaussian distribution in order to generalize the Brownian motion. As far as I know, regarding the heat kernel as the generalization of the Gaussian distribution has long been adopted in many literatures. It comes from the following observation. In$\mathbb {R}^1$, the following notions coincide: (1) Gaussian distribution$N(x,t)\sim f(t,x,y)=\frac {1}{\sqrt{2\pi t}}e^{\frac{-(y-x)^2}{2t}}$, (2) transition function$p(t,x,y)$of the Brownian motion$B_t$, (3) (heat kernel) fundamental solution$k_t(x,y)$of the heat equation$\partial_t k=\Delta_y k$, with initial data$\delta_x$. (also called heat kernel) Thus, on manifolds, one way to define the Brownian motion is to construct a stochastic process on the manifold whose transition function is exactly the heat kernel (let's consider it as identify the heat kernel with the Gaussian distribution in this setting). Since we always have the Laplacian-Beltrami$\Delta$on a manifold, it is justifiable to talk about heat kernel, and the Brownian motion in this sense is known to exist for a large class of manifolds. But on metric spaces, we no longer have the Laplacian-Beltrami. So, in order to talk about heat kernel (or Gaussian distribution)kernel/Gaussian distribution, we need to generalize the notion of Laplacian-Beltrami. The key notion concept on this line the so-called Dirichlet form. A Dirichlet form on metric measure space$(X,d,\mu)$a nonnegative-definite symmetric bilinear form$(\cdot,\cdot)$densely defined on$L^2(X,\mu)$. It should further satisfies a couple of conditions so that it behaves like its prototype$(f,g)=\int {\nabla f\cdot \nabla gd\mu}$with domain being on a manifold$C_0^{\infty}(M)$. M$. Notice that $(f,g)=(-\Delta f,g)_{L^2(M)}$ on manifolds, in the general case, one applies the Hille-Yosida theorem to obtain the desired "Laplacian" in the same way. Therefore every Dirichlet form corresponds to a "Laplacian" and thus a Gaussian distribution (and finally thus a Brownian motion). What's more, a reasonable Dirichlet form always exists provided the space is suitably good.

In sum, if the space you are considering have both metric and measure structures, then the theory of Dirichlet form may provide you some satisfactory results regarding construction and properties of the Guassian distribution (and thus the Brownian motion). Roughly speaking, if we don't have a presumed measure, we may not be able to construct a reasonable probability space; if we don't have a metric, it would be hard to measure the regularity and decay of the Gaussian distribution. So metric measure structure might be the weakest minimal structure for reasonable construction of Gaussian distribution.

1

Let me assume that you seek for the generalization of Gaussian distribution in order to generalize the Brownian motion.

As far as I know, regarding the heat kernel as the generalization of the Gaussian distribution has long been adopted in many literatures. It comes from the following observation.

In $\mathbb {R}^1$, the following notions coincide:

(1) Gaussian distribution $N(x,t)\sim f(t,x,y)=\frac {1}{\sqrt{2\pi t}}e^{\frac{-(y-x)^2}{2t}}$,

(2) transition function $p(t,x,y)$ of the Brownian motion $B_t$,

(3) fundamental solution $k_t(x,y)$ of the heat equation $\partial_t k=\Delta_y k$, with initial data $\delta_x$. (also called heat kernel)

Thus, on manifolds, one way to define the Brownian motion is to construct a stochastic process on the manifold whose transition function is exactly the heat kernel (let's consider it as the Gaussian distribution in this setting). Since we always have the Laplacian-Beltrami $\Delta$ on a manifold, it is justifiable to talk about heat kernel, and the Brownian motion in this sense is known to exist for a large class of manifolds.

But on metric spaces, we no longer have the Laplacian-Beltrami. So, in order to talk about heat kernel (or Gaussian distribution), we need to generalize the notion of Laplacian-Beltrami. The key notion on this line the so-called Dirichlet form. A Dirichlet form on metric measure space $(X,d,\mu)$ a nonnegative-definite symmetric bilinear form densely defined on $L^2(X,\mu)$. It should further satisfies a couple of conditions so that it behaves like its prototype $(f,g)=\int {\nabla f\cdot \nabla gd\mu}$ with domain being $C_0^{\infty}(M)$. Notice that $(f,g)=(-\Delta f,g)_{L^2(M)}$ on manifolds, in the general case, one applies the Hille-Yosida theorem to obtain the desired "Laplacian" in the same way. Therefore every Dirichlet form corresponds to a "Laplacian" and thus a Gaussian distribution (and finally a Brownian motion). What's more, a reasonable Dirichlet form always exists provided the space is suitably good.

In sum, if the space you are considering have both metric and measure structures, then the theory of Dirichlet form may provide you some satisfactory results regarding construction and properties of the Guassian distribution (and thus the Brownian motion). Roughly speaking, if we don't have a presumed measure, we may not be able to construct a reasonable probability space; if we don't have a metric, it would be hard to measure the regularity and decay of the Gaussian distribution. So metric measure structure might be the weakest structure for reasonable construction of Gaussian distribution.