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.
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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
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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.
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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