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## T̃Z~i[

FQOOVNTPTi΁jPTFOO
ꏊFLwwaaVOP
uҁFOqv (LH)
ځFStochastic control with fixed marginal distributions
AuXgNgF
@Œ肳ꂽӕz̉ł̊mœḰA܂łقƂǌ ȂA ̖ǴAmʎq̌ɌB {uł́Â悤Ȗւ̈̏Ⳃ^B ܂ApƂāAmߒɑ΂ӕzlB

FQOOVNTQXi΁jPTFOO
ꏊFLwwaaVOP
ځFOn Some Diffusion Processes with Large Drifts
AuXgNgF
@We consider the following diffusion processes on $d$-dim torus,

$$dX^{(c)}(t)=c b(X^{(c)}(t))dt + dB(t),$$

$B(t)$ is the Brownian motion, $b(\cdot)$ is a divergence free ($div(b)=0$) smooth vector field with period 1, $c$ is a large parameter. This diffusion process has Lebsegue measure (on torus) as invariance measure due to the condition $div(b)=0$. This is a particular example of more general class of diffusion processes,

$$dX^{(c)}(t)=(-\nabla U(X^{(c)}(t))+ c b(X^{(c)}(t))dt + dB(t),$$

with $U, b$ periodic and satisfying

$$div(b\exp(-2U))=0,$$

such that they have $\mu$ as the invariance measure,

$$d\mu=\frac 1Z \exp(-2U(x)) dx.$$

Such diffusion processes appear in MCMC(Markoc Chain Monte Carlo) that one chooses particular $b$ to simulate the underlying distribution $\mu$. A main concern is how well the distribution of $X^{(c)}(t)$ approximate $\mu$ and how to choose a better $b$. We are able to say some quantitative behaviors of such processes by taking $c$ large.

## 6̃Z~i[

FQOOVNUPXi΁jPTFOO
ꏊFLwwaaVOP
uҁFsK (R厩R)
ځFHamilton-Jacobỉ̒ԋɂ
AuXgNgF
@Hamilton-Jacobiɑ΂CauchyɂāC ԌoߌɉԂɎ邽߂́C n~gjAƏɊւl@D

## ṼZ~i[

FQOOVNVPOi΁jPTFOO
ꏊFLwwaaVOP
uҁF Y (aw)
ځFLOt̐̔핢
AuXgNgF
@LOt̏̐Ot̑SĂ̒_K ŏԂł핢ԂɂāCցE䗼ɂčŋߓꂽ ʂЉƂƂɁC̉pƂĂ̔핢Ԃ̊Ғlxz Ot̊􉽍\̉͂c_D

## PP̃Z~i[

FQOOVNPPQOi΁jPTFOO
ꏊFLwwaaVOP
uҁFց@玁 (B吔)
ځFLOt̃_EH[N̔핢
AuXgNgF
@LOt̃_EH[NSĂ̒_ Kŏ̎Ԃ핢ԂƂԁD̔핢Ԃ̊mz rEX]瓾邱ƂЉC̃Ot ΂핢ԂƂ̊􉽍\Ƃ̊֌Wɂčl@D

FQOOVNPPQVi΁jPTFOO
ꏊFLwwaaVOP
uҁFL@GT (É)
ځFPKGS[h^Bellmanɂcritical value̕\ɂ
AuXgNgF
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## P̃Z~i[

FQOOWNPQXi΁jPTFOO
ꏊFLwwaaVOP
uҁF@V (B吔)
ځFA survey on Schramm-Loewner-Evolution
AuXgNgF
@Schramm-Loewner-Evolution(SLE)Stochastic Loewner Equation Ƃ ΂֐_ŌÂ猤Ă@Loewner uE^ pă_̂ŁALoop-erased randomwalk p[R[V Ȃǂ̗Uf̃XP[ɌLqAfƂ O.Schramm ēꂽ̂łBułSLE̒`Ɗ{IȂ̐ ďЉ\łB