Potential-gradient cartoons

Choose your favorite type of potential-driven differential equation

• Stochastic ODE
\[d x_t = -\nabla V(x_t) dt + A\cos(t) + \sqrt{\sigma}dW_t,\]
• ODE 2nd order (friction and inertia)
\[ x'_t = -\nabla V(x_t) dt + A\cos(t) + \gamma x''(t ) - \zeta x'(t),\]
• Stochastic DDE
\[dx_t = -\nabla V(x_t)dt + \beta \nabla U(x_{t-r}) dt + \sqrt{\sigma}dW_t,\]
• Stochastic DDE (replacement potential)
\[dx_t = -\nabla V(x_t)dt + \beta \nabla U(x_{t-r}) dt + \sqrt{\sigma}dW_t.\]