Brownian Motion
A continuous-time stochastic process with Gaussian increments and continuous sample paths. It is the core driving force of traditional score-based generative models.
Timeline (last 90 days)
On the sphere, thanks to the symmetry of Brownian motion, the ancestral bridge can be expressed exactly without approximation.
The spherical Brownian diffusion paper proposes Brownian motion as the most natural isotropic diffusion process on the sphere, and uses it as the forward process of a continuous diffusion language model.
A Lévy process is a class of stochastic processes with independent stationary increments; Brownian motion is a special case, and the compound Poisson process is the simplest jump-type Lévy process.
At any positive generation time, the distribution produced by α-stable dynamics has a power-law tail with exponent α, whereas Brownian dynamics always remains light-tailed, a qualitative separation that does not depend on the specific target distribution or training details.
The α-stable Lévy process is a generalization of Brownian motion; Brownian motion corresponds to the stable process with α=2, and when α<2 the process allows arbitrarily large jumps and the jumps exhibit heavy-tailed characteristics.
When the step size $\eta \to 0$, the trajectory of the SGD iteration $x_{k+1} = x_k - \eta \nabla f(x_k) + \eta \xi_k$ can be approximated by an SDE: $dX_t = -\nabla f(X_t)\,dt + \sqrt{\eta}\,\Sigma\,dW_t$.
Brownian motion is characterized by independent increments, normally distributed increments, and continuous sample paths, yet the paths are almost everywhere non-differentiable.
Brownian motion is characterized by three main features—independent increments, normally distributed increments, and continuous sample paths—and its paths are almost everywhere non-differentiable.
布朗运动由爱因斯坦在1905年给出了严格的数学描述，奠定了随机过程理论的基础
All Facts (9)
On the sphere, thanks to the symmetry of Brownian motion, the ancestral bridge can be expressed exactly without approximation.
50%UnverifiedThe spherical Brownian diffusion paper proposes Brownian motion as the most natural isotropic diffusion process on the sphere, and uses it as the forward process of a continuous diffusion language model.
50%UnverifiedA Lévy process is a class of stochastic processes with independent stationary increments; Brownian motion is a special case, and the compound Poisson process is the simplest jump-type Lévy process.
50%UnverifiedAt any positive generation time, the distribution produced by α-stable dynamics has a power-law tail with exponent α, whereas Brownian dynamics always remains light-tailed, a qualitative separation that does not depend on the specific target distribution or training details.
50%UnverifiedThe α-stable Lévy process is a generalization of Brownian motion; Brownian motion corresponds to the stable process with α=2, and when α<2 the process allows arbitrarily large jumps and the jumps exhibit heavy-tailed characteristics.
50%UnverifiedWhen the step size $\eta \to 0$, the trajectory of the SGD iteration $x_{k+1} = x_k - \eta \nabla f(x_k) + \eta \xi_k$ can be approximated by an SDE: $dX_t = -\nabla f(X_t)\,dt + \sqrt{\eta}\,\Sigma\,dW_t$.
50%UnverifiedBrownian motion is characterized by independent increments, normally distributed increments, and continuous sample paths, yet the paths are almost everywhere non-differentiable.
50%UnverifiedBrownian motion is characterized by three main features—independent increments, normally distributed increments, and continuous sample paths—and its paths are almost everywhere non-differentiable.
50%Unverified布朗运动由爱因斯坦在1905年给出了严格的数学描述，奠定了随机过程理论的基础
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