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# 数学代写|随机分析代写Stochastic Calculus代考|IMSE760 Preview

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## 数学代写|随机分析代写Stochastic Calculus代考|Preview

In Section 2 we introduce the Langevin-Smoluchowski measure $P$ on path space, under which the canonical process $(X(t))_{t \geqslant 0}$ has dynamics (3) with initial distribution $P(0)$. Then, in Section 3, this process is studied under time-reversal. That is, we fix a terminal time $T \in(0, \infty)$ and consider the time-reversed process $\bar{X}(s)=X(T-s)$, $0 \leqslant s \leqslant T$. Standard time-reversal theory shows that $\bar{X}$ is again a diffusion, and gives an explicit description of its dynamics.

Section 4 develops our main result, Theorem 1. An equivalent change of probability measure $\mathbb{P} \gamma \sim \mathbb{P}$ adds to the drift of $\bar{X}$ a measurable, adapted process $\gamma(T-s)$, $0 \leqslant s \leqslant T$. In broad brushes, this allows us to define, in terms of relative entropies, the quantities
$$H^\gamma:=\left.H\left(\mathbb{P}^\gamma \mid \mathbb{Q}\right)\right|{\sigma(\bar{X}(T))}, \quad D^\gamma:=\left.H\left(\mathbb{P}^\gamma \mid \mathbb{P}\right)\right|{\sigma(\bar{X})}-\left.H\left(\mathbb{P}^\gamma \mid \mathbb{P}\right)\right|_{\sigma(\bar{X}(T))}$$

Here, $\mathbb{Q}$ is the probability measure on path space, inherited from the LangevinSmoluchowski dynamics (3) with initial distribution given by the invariant Gibbs probability measure Q. Theorem 1 establishes then the variational characterization
$$\inf \gamma\left(H^\gamma+D^\gamma\right)=H(P(T) \mid \mathrm{Q}),$$ where $P(T)$ denotes the distribution of the random variable $X(T)$ under $\mathbb{P}$. The process $\gamma$ that realizes the infimum in $\underline{(2)}$ gives rise to a probability measure $\mathbb{P}{\gamma}$, under which the time-reversed diffusion $\bar{X}$ is of Langevin-Smoluchowski type in its own right, but now with initial distribution $P(T)$. In other words, with the constraint of minimizing the sum of the entropic quantities $H^\gamma$ and $D^\gamma$ of (1), LangevinSmoluchowski measure on path space is invariant under time-reversal.

## 数学代写|随机分析代写Stochastic Calculus代考|The setting

Let us consider a Langevin-Smoluchowski diffusion process $(X(t)){t \geqslant 0}$ of the form $$\mathrm{d} X(t)=-\nabla \Psi(X(t)) \mathrm{d} t+\mathrm{d} W(t)$$ with values in $\mathbb{R}^n$. Here $(W(t)){t \geqslant 0}$ is standard $n$-dimensional Brownian motion, and the “potential” $\Psi: \mathbb{R}^n \rightarrow[0, \infty)$ is a $C^{\infty}$-function growing, along with its derivatives of all orders, at most exponentially as $|x| \rightarrow \infty$; we stress that no convexity assumptions are imposed on this potential. We posit also an “initial condition” $X(0)=\Xi$, a random variable independent of the driving Brownian motion and with given distribution $P(0)$. For concreteness, we shall assume that this initial distribution has a continuous probability density function $p_0(\cdot)$.

Under these conditions, the Langevin-Smoluchowski equation (3) admits a pathwise unique, strong solution, up until an “explosion time” e; such explosion never happens, i.e., $\mathbb{P}(\mathrm{e}=\infty)=1$, if in addition the second-moment condition (12) and the coercivity condition (11) below hold. The condition (11) propagates the finiteness of the second moment to the entire collection of time-marginal distributions $P(t)=\operatorname{Law}(X(t)), t \geqslant 0$, which are then determined uniquely. In fact, adapting the arguments in [42] to the present situation, we check that each time-marginal distribution $P(t)$ has probability density $p(t, \cdot)$ such that the resulting function $(t, x) \mapsto p(t, x)$ is continuous and strictly positive on $(0, \infty) \times \mathbb{R}^n$; differentiable with respect to the temporal variable $t$ for each $x \in \mathbb{R}^n$; smooth in the spatial variable $x$ for each $t>0$; and such that the logarithmic derivative $(t, x) \mapsto \nabla \log p(t, x)$ is continuous on $(0, \infty) \times \mathbb{R}^n$. These arguments also lead to the Fokker-Planck [20, 21, 41, 43], or forward Kolmogorov [28], equation
$$\partial p(t, x)=\frac{1}{2} \Delta p(t, x)+\operatorname{div}(\nabla \Psi(x) p(t, x)), \quad(t, x) \in(0, \infty) \times \mathbb{R}^n$$
with initial condition $p(0, x)=p_0(x)$, for $x \in \mathbb{R}^n$.
Here and throughout this paper, $\partial$ denotes differentiation with respect to the temporal argument; whereas $\nabla, \Delta$ and div stand, respectively, for gradient, Laplacian and divergence with respect to the spatial argument.

## 数学代写|随机分析代写Stochastic Calculus代考|Preview

$$\inf \gamma\left(H^\gamma+D^\gamma\right)=H(P(T) \mid \mathrm{Q}),$$

## 数学代写|随机分析代写Stochastic Calculus代考|The setting

$$\mathrm{d} X(t)=-\nabla \Psi(X(t)) \mathrm{d} t+\mathrm{d} W(t)$$

$$\partial p(t, x)=\frac{1}{2} \Delta p(t, x)+\operatorname{div}(\nabla \Psi(x) p(t, x)), \quad(t, x) \in(0, \infty) \times \mathbb{R}^n$$

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