Why is the gaussian free field a distribution but Brownian motion is a function?
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?
probability-theory stochastic-processes stochastic-calculus stochastic-pde
asked Nov 18 at 23:12
prdnr
645
add a comment |
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?
probability-theory stochastic-processes stochastic-calculus stochastic-pde
asked Nov 18 at 23:12
prdnr
645
add a comment |
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?
probability-theory stochastic-processes stochastic-calculus stochastic-pde
asked Nov 18 at 23:12
prdnr
645
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?
probability-theory stochastic-processes stochastic-calculus stochastic-pde
probability-theory stochastic-processes stochastic-calculus stochastic-pde
asked Nov 18 at 23:12
prdnr
645
asked Nov 18 at 23:12
prdnr
645
asked Nov 18 at 23:12
prdnr
645
asked Nov 18 at 23:12
prdnr
645
asked Nov 18 at 23:12
prdnr
645
645
add a comment |
add a comment |
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